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Elliptic curves with rational 2-torsion and related ternary Diophantine equations Mulholland, Jamie Thomas 2006

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ELLIPTIC CURVES WITH RATIONAL 2-TORSION A N D RELATED TERNARY DIOPHANTINE EQUATIONS by JAMIE THOMAS MUX-HOLLAND B.Sc. Simon Fraser University, 2000 M.Sc. The University of British Columbia, 2002 A THESIS SUBMITTED IN PARTIAL FULLFTLLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY i n THE FACULTY OF GRADUATE STUDIES Mathematics THE UNIVERSITY OF BRITISH COLUMBIA July 2006 © Jamie Thomas Mulholland, 2006 Abstract Our main result is a classification of elliptic curves with rational 2-torsion and good reduction outside 2, 3 and a prime p. This extends the work of Hadano and, more recently, Ivorra. A key factor in doing this is to have a method for efficiently computing the conductor of an elliptic curve with 2-torsion. We specialize the work of Papadopolous to provide such a method. Next, we determine all the rational points on the hyper-elliptic curves y2 = x5 ± 2 a 3 6 . This information is required in providing the classification mentioned above. We show how the commercial mathematical software pack-age MAGMA can be used in solving this problem. As an application, we turn our attention to the ternary Diophantine equa-tions xn + yn = 2apz2 and x 3 + y 3 = ±pmzn, where p denotes a fixed prime. In the first equation, we show that for p = 5 or p > 7 the equation is unsolvable in integers (x, y, z) for all suitably large primes n. In the second equation, we show the same conclusion holds for an infinite collection of primes p. To do this, we use the connections between Galois representations, modular forms, and elliptic curves which were discovered by Frey, Hellegouarch, Serre, and Wiles. i i Table of Contents Abstract ii Table of Contents iii List of Tables vi Acknowledgement vii Dedication viii Chapter 1. Introduction 1 1.1 Introduction to Diophantine Equations 1 1.2 Generalized Fermat Equations 4 1.3 Statement of Principal Results 6 1.4 Overview of chapters 9 Chapter 2. The Conductor of an Elliptic Curve over Q with 2-torsion 11 2.1 Introduction 11 2.2 Statement of Results 12 2.3 The Proof of Theorem 2.1 17 2.4 The case when v^{a) = 1, v2(b) = 0 24 2.4.1 Proof of Theorem 2.1 part (vii) when u 2( A) = 8 25 2.4.2 Proof of Theorem 2.1 part (vii) when u 2 (A) = 10 ; 26 2.4.3 Proof of Theorem 2.1 part (vii) when v 2 ( A ) - 11 26 2.4.4 Proof of Theorem 2.1 part (vii) when v 2 (A) = 12 27 2.4.5 Proof of Theorem 2.1 part (vii) when u 2( A ) > 13 27 2.5 The Proof of Theorem 2.3 27 2.6 The Proof of Theorem 2.4 29 Chapter 3. Classification of Elliptic Curves over <Q> with 2-torsion and con-ductor 2a3*V 30 3.1 Curves of Conductor 2 a p 2 31 3.1.1 Statement of Results 31 ' 3.1.2 The Proof for Conductor 2 a p 2 45 3.1.3 List of (^isomorphism classes 45 iii Table of Contents iv 3.1.4 The end of the proof 46 3.2 Curves of Conductor 2a3^p 55 3.3 Curves of Conductor 2 a 3 V 90 3.4 Proofs of 2a3^p and 2 a 3 V 146 Chapter 4. Diophantine Lemmata 150 4.1 Useful Results 150 4.2 Diophantine lemmata 152 Chapter 5. Rational points on y2 = a?5 ± 2C*3 /3 177 5.1 Introduction and Statement of Results 177 5.2 Basic Theory of Jacobians of Curves 179 5.2.1 Basic Setup 180 5.2.2 Divisors 180 5.2.3 Principal Divisors and Jacobian 180 5.2.4 Geometric representation of the Jacobian 182 5.2.5 2-torsion in the Jacobian 183 5.2.6 Rational Points 183 5.2.7 Structure of the Jacobian: The Mordell-Weil theorem . . . 184 5.2.8 Computer Representations of Jacobians 185 5.2.9 Some Examples (Using MAGMA) 186 5.2.10 Chabauty's theorem 189 5.3 Data for the curves y2 = x5 ± 2Q3^ 192 5.4 The family of curves y2 = x5 + A 202 5.5 Proof of Theorem 5.1 204 5.5.1 A = 2632 204 5.5.2 A = 2633 206 5.5.3 A = 25 207 5.5.4 Rank > 2 cases 207 Chapter 6. Classification of Elliptic Curves over Q with 2-torsion and con-ductor 2ap2 209 6.1 Statement of Results 209 6.2 The Proof 227 Chapter 7. O n the Classification of Elliptic Curves over <Q> with 2-torsion and conductor 2a32p 229 7.1 Statement of Results 229 iv Table of Contents v 7.2 The Proofs 233 7.2.1 Proof of Theorem 7.1 233 7.2.2 Proof of Theorem 7.2 234 7.2.3 Proof of Theorem 7.3 235 7.2.4 Proof of Corollary 7.4 237 7.2.5 Proof of Lemma 7.5 239 Chapter 8. O n the equation xn + yn = 2apz2 244 8.1 Introduction 244 8.2 Elliptic Curves 245 8.3 Outline of the Proof of the main theorems — 246 8.4 Galois Representations and Modular Forms 247 8.5 Useful Propositions 248 8.6 Elliptic curves with rational 2-torsion 249 8.7 Theorems 8.1 and 8.2 251 8.8 Concluding Remarks 254 Chapter 9. O n the equation x3 + y3 = ± p m z n 255 9.1 Introduction 255 9.2 Frey Curve 256 9.3 The Modular Galois Representation p%b 259 9.4 Proof of Theorem 9.1 261 Bibliography 263 Appendix A . O n the Q-Isomorphism Classes of Elliptic Curves with 2-Torsion and Conductor 2<*&3ps 270 A.l 6>0 270 A. 2 6<0 309 Appendix B. Tables of S-integral Points on Elliptic Curves. 311 B. l 5-integral points on Elliptic Curves 311 B.2 Computing S-integral points on Elliptic Curves — 312 B.3 Tables of 5-integral points on the curves y2 = xz± 2°3 6 314 Appendix C . Tables of Q-Isomorphism Classes of Curves of Conductor 2ap2 with Small p. 317 List of Tables 2.1 Neron type at 2 of y2 = x3 + ax2 + bx 15 2.2 Neron type at 2 of y2 = x3 + ax2 + bx (con't) 16 2.3 Neron type at 3 of y2 = x3 + ax2 + bx 16 2.4 Neron type at p of y2 = x3 + ax2 + bx 16 5.1 Theorem 5.1: A l l points onC :y2 = x5 ± 2a3/3 178 5.2 Data for y2 = x5 + 2a3^ 194 5.3 Data for y2 = x5 + 2<*3/? (con't) 195 5.4 Data for y2 = x5 + 2 a3^ (con't) 196 5.5 Data for y2 = x5 + 2 a 3 / ? (con't) 197 5.6 Data for y2 = x5 - 2 a 3 / 3 198 5.7 Data for y2 = x5 - 2 a3^ (con't) 199 5.8 Data for y2 - x5 - 2 a3^ (con't) 200 5.9 Data for y2 = x5 - 2a313 (con't) 201 B. l ^-integral points on y2 = x3 + 2 a 3 b 315 B. 2 5-integral points on y2 = x3 - 2 ° 3 6 316 C l Extraneous curves of conductor 2p2 318 C. 2 Extraneous curves of conductor 22p2. 318 C.3 Extraneous curves of conductor 2 3 p 2 319 C.4 Extraneous curves of conductor 24p2 320 C.5 Extraneous curves of conductor 2 5 p 2 321 C.6 Extraneous curves of conductor 2 6 p 2 322 C.7 Extraneous curves of conductor 27p2 323 C.8 Extraneous curves of conductor 2 8 p 2 . 324 vi Acknowledgement It gives me great pleasure to thank the many people and organizations who have helped me to get where I am today. I am very grateful for financial support from the University of British Columbia and from NSERC. To the many teachers who have guided me to where I am today - thanks for your knowledge, wisdom, and inspiration. I owe an enormous debt to my supervisor, Dr. Michael Bennett, for having guided me through my PhD, sharing his knowledge, wisdom and experience of mathematics with me along the way. My fiancee, Heather, who has offered love and support through thick and thin - my deepest thanks. And my family, whose love and support, encouragement and guidance, have always been complete, and whose belief in me has enabled me to get to this point, the warmest thanks of all. vii To Heather - my love, my life viii Chapter 1 Introduction 1.1 I n t r o d u c t i o n to D i o p h a n t i n e E q u a t i o n s The study of Diophantine equations has a long and rich history, dating to the "Arithmetica" of Diophantus, written in the middle of the 3rd century, and dealing with the solution of algebraic equations and the theory of numbers. Much of modern number theory, as we know it, stems from tools developed to solve Diophantine equations. By a Diophantine equation, we mean, intuitively, an equation where we are interested only in integer and/or rational solutions. For example the equa-tion 2 , 2 2 x + y• = z has the following solutions in positive integers (x, y, z): (3,4,5), (5,12,13), (8,15,17), (7, 24,25). In fact, there are infinitely many solutions in positive integers to this equation, and they can be parametrized: any solution (with y even, say) is of the form (d(u2 - v2),2uvd, d(u2 + v2)) where u, v. d e Z and gcd(u, v) = 1. On the other hand, the equations xs + y3 = z3, x4 + y4 = z4, and x 5 + y5 - z5 have only the trivial solutions; solutions where one of the values is 0. Fermat1 ^ e o i - u s e s . 1 Chapter 1. Introduction 2 wrote in the margin of his copy of Arithmetica that, in fact, the equation xn + yn = zn has no nontrivial solutions for any n > 3, and commented that he had a mar-velous proof of this fact but the margin was too small to contain it. This be-came known as Fermat's "Last Theorem" 2 . The quest to prove (or disprove, for that matter) Fermat's Last Theorem became the driving force for modern number theory over the last three hundred years. Amateurs and profession-als alike all had their crack at a proof. Their attempts gave birth to many new beautiful ideas and tools that are used in number theory today, though, for more than three centuries, none were enough to resolve Fermat's enigma. Af-ter the work of Godel on "undecidability" in formal systems, many wondered whether the truth of Fermat's Last Theorem was even decidable. Ten years ago, Andrew Wiles announced a proof verifying Fermat's Last Theorem and finally putting to rest Fermat's challenge. Wiles attacked the problem by treat-ing a more general question regarding the connection between elliptic curves and modular forms. We'll say more on this in our final two chapters. Consider the Diophantine equation y2 = x3 + 1. The only integer solutions are ( - 1 , 0 ) , ( 0 , ± 1 ) , ( 2 , ± 3 ) . These are, in fact, the only rational solutions. On the other hand, the Diophan-tine equation y2 = x 3 + 17 has 16 integer solutions (-2, ± 3 ) , (-1, ± 4 ) , (2, ± 5 ) , (4, ± 9 ) , (8, ± 2 3 ) , (43, ± 2 8 2 ) , (52, ± 3 7 5 ) , (5234, ± 3 7 8 6 6 1 ) , '"Last" because it was the remaining conjecture of his that needed resolving. Chapter 1. Introduction 3 and infinitely many rational solutions. Both of these curves are examples of elliptic curves. A curve of the form E : y2 + a\xy + a 3 y = x 3 + a3y = x 3 + a 2 x 2 + a 4 x + ae with ai 6 Z is called an elliptic curve (provided it is nonsingular). For curves in Weierstrass form y2 = x 3 + a 2 £ 2 + a 4 £ + a 6 the condition of being nonsingular is equivalent to the cubic on the right-hand side having distinct roots (i.e. nonzero discriminant). For elliptic curves it is known that the number of integral points is finite (Siegel's theorem, see [69]), but the number of rational points could possibly be infinite. Though the proof of Siegel's theorem was not effective (i.e. did not give a method to find all the integral points) de Weger [30], using Baker's work on bounding linear forms in logarithms, was able to give an algorithm for finding all the integral points on an elliptic curve. The set of rational points E(Q) on an elliptic curve carry an abelian group structure, the identity being the point at infinity which we denote by O (or sometimes oo). That is, there is a natural way to add two rational points P\,P2 G E(Q) to obtain a third rational point P3 — Pi + P2. Geometrically, this is done by taking the (rational) line through Pi and P2 and letting P4 be the third point of intersection of the line with E. Next, take the vertical line through P4 (i.e. the line through P4 and O) and let P3 be the other point of in-tersection with E, and set P\ + P2 = P3. Mordell showed that E(Q) is finitely generated and abelian so it is of the form E(Q) ~ E(Q)lors x Z r where £'(Q) f o r s is a finite group consisting of the torsion elements and r is an integer called the rank of E. E(Q)tors is straightforward to compute; a theo-rem of Nagell and Lutz gives a method for computing its points. Moreover, a general result of Mazur ([50], [51]) states that it can only be one of 15 pos-sible groups (see for example [69], p. 223). However, there is no known al-gorithm for computing the rank of an elliptic curve. There are methods (i.e. a 2-descent) that work on bounding the rank. One can then hope to find enough Chapter 1. Introduction 4 independent points to meet this bound to obtain the rank exactly. In practice this works quite well. For our two examples above we have Ei : y2 = x3 + 1, Ei(Q) ~ Z / 6 Z E2 : y2 = x3 + 17, E2(Q) ~ I2. A hyperelliptic curve is a curve of the form y2 = /(*) where, for our purpose, / e Z[x] of degree 2(7+1. The integer g is called the genus of the curve. For example, an elliptic curve is a genus 1 hyperelliptic curve. However, unlike the situation for elliptic curves, a celebrated theorem of Faltings states that C(Q) is finite when g > 2. Unfortunately, Faltings the-orem is.not effective, but older work of Chabauty has recently been revived and in practice often works very well in determining C(Q). For a hyperelliptic curve C the set of rational points do not form a group, but C(Q) does embed into a finitely generated abelian group called the Jacobian of C, denoted J(Q). The work of Chabauty requires calculation in the Jacobian. In Chapter 5 we occupy ourselves with determining the rational points on curves of the form y2 = x5 ± 2a3 fe. Chapter 5 can be read independently of all other chapters. It provides an introduction to the theory and practice of computing all ratio-nal points on genus 2 curves, with a heavy emphasis on using M A G M A as a computational tool, something the current literature is somewhat lacking. The results of this chapter are used in proofs of the Diophantine lemmata of Chapter 4. 1.2 G e n e r a l i z e d F e r m a t E q u a t i o n s In relation with Fermat's last theorem the equation xp + y'! = zr (1.1) has a long history. For a very fine survey on this topic see [45]. Here, we will provide a very brief outline of what is known. Chapter 1. Introduction 5 The characteristic of equation (1.1) is defined to be x{P,q>r) = j, + \ + f — 1, and the study of these equations has been broken up into three cases: x(p, q, r) > 0 (spherical case), x{p, Q, r) = 0 (euclidean case), and x(p, <7,r) < u (hyperbolic case). Let S(p,q,r) be the set of nontrivial proper solutions to equation (1.1). In the spherical case, S(p, q, r) is infinite and there are in fact parametized solutions. In this case the possible sets of {p, q,r} are {2, 2,r} with r > 2, {2,3, 3}, {2,3,4}, and {2,3,5}, and the proper solutions correspond to ratio-nal points on genus 0 curves. In the euclidean case, possible sets of {p, q,r} are {3,3,3}, {2.4,4}, and {2, 3, 6}, and the points in S(p, q, r) corresponds to rational points on genus 1 curves. It is known that the only proper nontrivial solution corresponds to the equality 1+2 3 = 3 2 . We have already mentioned that 5(3,3, 3) was empty and the fact that 5(2,4,4) is empty was first proven by Fermat using an argument of infinite descent. In the hyperbolic case there are only ten known solutions to date: P + 2 4 = 3 2 , 2 5 + 7 2 = 3 4 , 7 3 + 13 2 = 2 9 , 2 7 + 173 = 71 2 , 3 5 + l l 4 = 1222, 177 + 762713 = 210639282, 14143 + 22134592 = 65 7, 92623 + 153122832 = 1137 , 43 8 + 962223 = 300429072, 33 8 + 15490342 = 156133. Notice that an exponent of 2 appears in each solution. This leads to the fol-lowing conjecture. Conjecture 1.1 Ifmin{p, q, r} > 2 and S(p, g, r) ^ 0 then mhi{p, q, r} = 2. A number of names can be associated with this conjecture, including Beukers, Zagier (who incidently found the five larger solutions above in 1993), Tijde-man, Granville and Beal. The first known result in the hyperbolic case is due to Darmon and Gran-ville [27]. They used Fallings' theorem to show that S(p, q, r) is finite. Next was Wiles' proof of Fermat's last theorem; S(n, n, n) = 0. Since then a num-ber of specific cases have been tackled using the modularity of elliptic curves (Wiles, et al), and Chabauty techniques. Some cases are as follows. Chapter 1. Introduction 6 (p,q,r) (n,n,2) Darmon, Merel (Poonen for n G {5,6,9}) (n,n,3) Darmon, Merel (Lucas n = 4, Poonen for n = 5) (3,3,n) Kraus for 17 < n < 10000, Bruin for n = 4,5 (2,4,") Ellenberg for n > 211, Bruin for n = 5,6, Bennett, Ellenberg, N g for n > 7 (2,n,4) Bennett, Skinner (2,3,7) Poonen, Schaefer, Stoll (2,3,8) Bruin (2,3,9) Bruin (2,2n,3) Chen for 7 < n < 1000, n ^ 31 (5,5,n),(7,7,n) Darmon and Kraus (partial results) (2n,2n,5) Bennett (4,2n,3) Bennett, Chen 1.3 Statement of Principal Results Modularity techniques have since been applied to generalized Fermat equa-tions with coefficients: Axp + By" = CzT. Here, A, B, C, p, q, and r are fixed integers and we are interested in integral solutions for x, y and z. If p = q = r, then results have been obtained by Serre [64] for A = B = 1 and C = Na, a > 1, with JV € {3,5,7,11,13,17,19,23,29,53,59}, N ^ p, p > 11, Kraus [41] for ABC = 15, Darmon and Merel [28] for ABC = 2, and Ribet [61] for ABC = 2°, a > 2. If (p,q,r) ~ (p,p,2) then results have been obtained by Bennett and Skinner [5] for various A, B, C, Ivorra [36] for ABC = 213, and Ivorra and Kraus [38] for various A, B, and C. If (p, q, r) = (p,p, 3) then Bennett, Vatsal and Yazdani [6] have shown Theorem 1.2 (Bennett, Vatsal, Yazdani) Ifp and n are prime, and a is a nonneg-ative integer, then the Diophantine equation xn + yn = paz3 has no solutions in coprime integers x, y and z with \xy\ > 1 and n > p4p2. Chapter 1. Introduction 7 Their proof of this proceeds as follows. Attach to a supposed solution (a, b, c) an elliptic curve E = Ea>btC with a 3-torsion point, and to this a Galois representation pE,n on the n-torsion points. To pE,n there corresponds a cuspi-dal newform of weight 2 and level Nn(E), where Nn(E) can be explicitly determined. It then remains to show that such a newform / cannot exist. In doing this, it is shown that the existence of / implies either n is bounded by pip2 o r that there exists an elliptic curve over Q with rational 3-torsion and conductor 3Tpu. Hence a classification of such curves is needed to finish the argument. In Chapter 8, we apply a similar argument to the equation xn + yn = 2apz2 and prove the following Theorem 1.3 (Bennett, Mulholland) Let p ^ 7 be prime. Then the equation xn + yn = 2apz2 has no solutions in coprime nonzero integers x and y, positive integers z and a, and prime n satisfying n > p27p2. A key ingredient in the proof is a classification of the elliptic curves with conductor 2My2 and possessing a rational 2-torsion point. In Chapter 6, we provide such a classification. In Chapter 9, we study the equation x 3 + y3 = ±pmzn, where p is prime and prove the following, Theorem 1.4 (Mulholland) Let p e T and m > 1 an integer. Then the equation x 3 + y3 = ±pmzn has no solutions in coprime nonzero integers x, y and z, and prime n satisfying n > p8p and n\m. Chapter 1. Introduction 8 Here T denotes the set of primes p for which there does not exist an elliptic curve with rational 2-torsion and conductor 2M32p, 1 < M < 3. Thus, in this case we need a classification of the elliptic curves with conductor 2M32p, 1 < M < 3, and possessing a rational 2-torsion point. In Chapter 7, we provide such a classification. Since we are interested in elliptic curves of conductor 2Mp2 or 2M3Lp and possessing a rational point of order 2 we start by considering the following more general question. P r o b l e m 1 Determine all the ^-isomorphism classes for elliptic curves over Q of conductor 2M3LpN and having at least one rational point of order 2. As is well-known, there do not exist any elliptic curves defined over Q with conductor divisible by 29, 3 6, or q3 for q > 5 prime (see e.g. Papadopou-los [57]). Furthermore, as we show in Chapter 2, the existence of rational 2-torsion implies the conductor is not divisible by 3 3. Therefore, we can suppose in the statement of problem 1 that 0 < M < 8 and 0 < L, N < 2. In addition, a theorem of Shafarevich states that there are only finitely many isomorphism classes, for fixed p (see [69] p. 263). The first work- on Problem 1 appears to have be done by Ogg in 1966, [55], [56]. He determined the elliptic curves defined over Q with conductor of the form 2 M 3 L or 2 M 3. Coghlan in his dissertation [17] also studied the curves of conductor 2 M 3 L independently of Ogg. Velu [78] classified curves of conductor 11, and in general Setzer [66] answers Problem 1 for any prime conductor. He shows that there are two distinct isomorphism classes when p — 64 is a square, and four when p = 17. Hadano [34] begins treatment of conductors pN and 2MpN, and Ivorra, in his dissertion [37], classifies those of conductor 2Mp. There has been other work in classifying elliptic curves with conductors of a particular form and specified torsion structure. Most notable are the works of Hadano [35] and Miyawaki [53]. In Chapter 3, we take up Problem 1 in general. In Section 3.1, we obtain re-sults analogous to those of Ivorra for conductor 2Np2. In Sections 3.2 and 3.3 Chapter 1. Introduction 9 we obtain results for conductor 2NSLp and 2N3Lp2, respectively, thus com-pleting the remaining cases of Problem 1. As seen from glancing at the table of contents, the tables presented account for 120+ pages of this work (not to mention the 30+ pages of refined tables in Chapter 6, and the 40+ pages of technical case by case analysis in Appendix A). We have tried to tidy this work up as best we can and make it readable but, unfortunately, there is no way to fully condense it; the tables are what they are - long and technical. But we believe the determination of these tables provides a useful public service. As seen from glancing at the tables in Chapter 3, one is mainly confronted, as in [66] and [37], with the problem of determining the integer solutions of certain ternary Diophantine equations. In Chapter 4, we take up the problem of resolving these Diophantine equations. We then come back the tables of Chapter 3 with these solutions at hand. This allows us to simplify the tables, these results appear in Chapters 6 and 7. Some of the works mentioned above regarding Problem 1 treat the follow-ing more general problem, which we do not know how to attack in general. Problem 2 Determine all the ^-isomorphism classes for elliptic curves over Q of conductor 2M3LpN. Let us note that Brumer and McGuinness have determined the elliptic curves of conductor p < 108. The definitive web source for tables of all the elliptic curves of conductor < 130000 is John-Cremona's home page 3 . These tables are constantly being expanded so the reader should check the web page to determine their extent at this time. The techniques Cremona uses for con-structing his tables (and, indeed, a fine introduction to the arithmetic of el-liptic curves) can be found in his excellent book [26] which is available for download from his web page. In addition, Cremona has prepared tables for conductor 2km2 with m < 23 prime and also m = 15 and 21. 1.4 O v e r v i e w o f c h a p t e r s A brief outline of the contents of each chapter is as follows. In Chapter 2, we specialize the results of Papadopolous [57] to the problem of computing the conductor of an elliptic curve with a rational 2 torsion point, 3 www.ma ths.nottingham.ac.uk/personal/jee/ Chapter 1. Introduction 10 i.e. curves of the form y2 — x3 + ax2 + bx. There we present an easy criterion for computing the conductor. The results of this section are used throughout the rest of this work. Chapter 3 is the first step toward our classifying problem. Here we present twenty-seven theorems, one for each value of 2M3LpN, listing the Q-isomor-phism classes of the elliptic curves with that conductor. The proof is long and tedious but not that technical, it depends on two main lemmata which are proven in Appendix A. It is in these tables that we are confronted with the problem of determining the integer solutions to certain ternary Diophantine equations. In order to get a useful classification theorem we need to resolve these Diophantine equations. This is taken up in Chapter 4. In order to solve some of the Diophantine equations, it is sufficient to find all {2, 3; oo}-integral points on the genus 1 curves y2 = x3± 2Q3 / ?, and the genus 2 curves y2 = x 5 ± 2a3p. We deal with the former in Appendix B and the latter in Chapter 5. Having these Diophantine results at hand, we come back to the tables of Chapter 3. In Chapter 6, we present nine theorems classifying elliptic curves of conductor 2Mp2 possessing a rational 2-torsion point. These table are anal-ogous to those of Ivorra [37]. In Chapter 7, we investigate the admissible p for which there exist curves of conductor 2M Z2p, 1 < M < 3, with rational 2-torsion. These results will be used in Chapter 9. In Chapters 8 and 9, we look at what can be said about the generalized Fermat equations xn + yn = 2apz2 and x3 + y3 = ±pmzn, respectively. A modified version of Chapter 8 has appeared in print [4]. C h a p t e r 2 The Conductor of an Elliptic Curve over Q with 2-torsion In this chapter, we specialize the work of Papadopolous [57] to elliptic curves over Q with nontrivial 2-torsion: y2 = x3 + ax2 + bx, and show that the exponent of 2 in the conductor of the curve is determined by the values V2(a) and V2(b) and some simple congruences of a and b modulo 2, 4 and 8. Here vp denotes the p-adic valuation on Q. 2.1 Introduction Let E be the elliptic curve over Q defined by with ai G Z . Let b2, 6 4 , b§, bg, C4, CQ, and A be the standard invariants associated with E: E : y2 + a\xy + ayy = x3 + a2x2 + a 4 x + ag 02 = a 2 + 4a 2, 64 = a\a$ + 2a2, be = a | + 4ac 6s = a 2 a 6 - a\asa4 + 4a2a$ + a2a\ — a 2 c 4 = b\ - 2464, c 6 = -b\ + 366264 - 21 666 A = -bjb8 - 86^  - 276| + 9626466. (2.1) (2.2) (2.3) (2.4) The conductor of an elliptic curve over Q is defined to be p 11 Chapter 2. The Conductor of an Elliptic Curve over Q with 2-torsion 12 where fp = vp( A) +1 — np. Here np is the number of irreducible components of the special fibre of the minimal Neron model at the prime p (see [69]). Essen-tially, N is an encoding of the primes for which E has bad reduction and the reduction types at these primes. E has bad reduction at a prime p if and only if p | N, and the reduction type of E at p is multiplicative (E has a node over F p ) or additive (E has a cusp over ¥ p ) depending on whether fp = 1 or > 2, respectively. It is well known that for p 2,3, the value of fp is completely determined by the values of vp(ci), vp(cG) and vp(A). This is not always the case when p = 2 or 3. Papadopolous [57] has determined when the triple (•u2(c4),'U2(c6),f2(A)) (resp. (v3{c4), v3(c6), v3(A))) is not sufficient to deter-mine the value of / 2 (resp. / 3 ) and in these cases he has given supplementary conditions involving the values of a\, a 2, a3, a 4, ae, 62, 64, £>e and b%. In the case of the prime 3 these supplementary conditions involve checking a sin-gle congruence involving c4 and CG modulo 9. However, for the prime 2 the supplementary conditions are a little more complicated. One usually needs to check a number of congruences in sequence for solutions. Furthermore, in the case when (u2(c4),^(cc), ^(A)) = (6, > 9,12) one is unable to decide from Table IV in [57] whether / 2 is 5 or 6 (whereby one is forced to apply Tate's algorithm directly). If E is an elliptic curve over Q with nontrivial 2-torsion then E is isomor-phic to a curve of the form y2 = x3 + ax2 + bx, where a, b e Z are such that vp(a) > 2 and vp(b) > 4 do not both hold for all p. The discriminant in this case is A = 24b2{a2 -46). In this chapter, we show that for such curves the conditions one needs to _ check in [57] simplify greatly. In fact, the value of / 2 is completely determined by the values of u2(a), i>2(6) and the congruence classes of a and 6 modulo 4, with one exception. In this exceptional case, V2{A) = 8, one needs to check a congruence involving a and 6 modulo 8 (see Theorem 2.1). 2.2 Statement of Resul ts . Let p denote a prime > 5. We will prove the following theorems. Chapter 2. The Conductor of an Elliptic Curve over <Q> with 2-torsion 13 Theorem 2.1 If a,b G Z are such that not both V2[a) > 2 and V2{b) > 4 hold, then the Neron type at 2 of the elliptic curve y2 = x3 + ax2 + bx is given by Tables 2.1 and 2.2 on pages 15 and 16. In the cases where / 2 = 0 or 1 the model y2 = xs + ax2 + bx is non-minimal at 2, this is indicated in Table 2.1 by the appearance of "non-minimal" in the corresponding column. In the cases where / 2 ^ 0,1 the model y2 = x 3 + ax2 + bx is minimal at 2. During the course of the proof of Theorem 2.1 we will also deduce the following. Corollary 2.2 In the case that the model E(a, b) : y2 = x3 + ax2 + bx is non-minimal at 2 we have the following: 1. Ifv2(a) = 0, v2(b) > 4 and a = 1 (mod 4) then y2 + xy = x3+ ( ^ p ) x2 + (J£) x is a minimal model for E(a, b) at 2. 2. Ifv2(a) = 1, v2(b) = 0, v2(A) > 12 and § = -1 (mod 4) then is a minimal model for E(a, b) at 2. Theorem 2.3 If a, b G Z are such that not both v$(a) > 2 and vz(b) > 4 hold, then the Neron type at 3 of the elliptic curve y2 = x3 + ax2 + bx is given by Table 2.3 on page 16. In all cases the model y2 — x3 + ax2 + bx is minimal at 3. Theorem 2.4 Let pbea prime > 5. If a, b G Z are such that not both vp(a) > 2 and vp(b) > 4 hold, then the Neron type at p of the elliptic curve y2 = x3 + ax2 + bx is given by Table 2.4 on page 16. ln all cases the model y2 = x3 + ax2 + bx is minimal at p. We have the following corollary to Theorems 2.3 and 2.4. Corollary 2.5 Let q be an odd prime. Ifa,beZ such that not both vq(a) > 2 and vq(b) > 4 hold and N(a^ is the conductor of the elliptic curve y2 = x3 + ax2 + bx then: Chapter 2. The Conductor of an Elliptic Curve over Q with 2-torsion 14 d) Q I N(a,b) if and only ifq\A = 2ib2(o? - 46), (ii) if q || A^(„,6) then q does not divide a, (iii) q2 || N/abj if and only if q divides a and b. "2(a) v2(b) 0 0 0 1 0 2 0 3 Supplementary conditions a E E 1 (4) b = 1 (4) a E E 1 (4) 6 E E -1(4) a = -1 (4) 6 E E 1 (4) a E E -1 (4) 6= -1 (4) a EE 1 (4) a E E -1 (4) a E E 1 (4) a = -1 (4 ) Kodaira symbol II I V III II III 13 III* 15 Case of Tate 3 5 4 3 4 7 6 9 7 Exponent v2{N) of conductor 4 2 3 4 5 3 4 3 4 V2(a) v2(b) 0 4 0 > 5 1 0 1 0 " 2 (A) 7 8 Supplementary conditions a EEE 1 (4) non-minimal a E E - 1 ( 4 ) a EE 1 (4) non-minimal a EE -1 (4) a - b E E 13 (16) a - b E S 5 (16) Kodaira symbol Io I 4 I v 2 ( A ) - 1 2 II 13 I t Case of Tate 1 7 7 7 3 6 7 Exponent v2(N) of conductor 0 4 1 4 7 4 3 Vi{a) v2(b) 1 0 1 0 1 0 1 0 "2(A) 1 8 (con't) 9 10 11 Supplementary conditions a - b = 1 (16) a - b = 9 (16) f = K4) | EE -1(4) f E E 1 (4) § = -1 (4) Kodaira symbol 13 I V * IS III* 13 II* Case of Tate 6 8 6 7 9 7 10 Exponent v2(N) of conductor 4 2 5 4 3 4 3 v2(a) v2(b) 1 0 1 0 •1 1 1 2 1 > 3 > 2 0 > 2 1 "2 (A) 12 > 13 Supplementary conditions f E E 1 (4) f s - 1 (4) non-minimal § = 1 (4) f = -1 (4) non-minimal 6 E E 1 (4) b EE -1(4) Kodaira symbol 1; Io I v 2 ( A ) - 1 2 III 13 ! C 2 ( A ) - 1 0 II III III Case of Tate 7 1 7 2 4 6 7 3 4 4 Exponent v2(N) of conductor 4 0 4 1 7 6 6 6 5 8 Table 2.1: Neron type at 2 of y 2 = x 3 + ax 2 + frc. •"2(a) 2 2 > 3 > 3 vi{b) 2 3 2 3 *2(A) Supplementary conditions ! = 1 (4) H - l (4) i = M<) f = -1(4) Kodaira symbol I v 2 ( A ) - 1 0 15 III* la III* Case of Tate 7 7 9 7 7 9 Exponent V2(N) of conductor 6 7 7 5 6 8 Table 2.2: Neron type at 2 of y2 = x3 + ax1 + bx (con't) V3(a) 0 0 > 1 > 1 1 > 2 > 2 v3(b) 0 > 1 0 1 > 2 2 3 Supplementary conditions bs 1(3) 6= -1 (3) Kodaira symbol WA) lo I,,.., (A) lo III '«.-l(A)-6 III* Case of Tate Exponent V3(N) of conductor 1 0 1 0 2 2 2 2 Table 2.3: Neron type at 3 of yl = x6 + ax1 + bx vp{a) 0 0 > 1 > 1 1 1 > 2 > 2 vp(b) 0 > 1 0 1 2 > 3 2 3 Supplementary conditions a2 £ 46 (p) a2 = 46 (p) a2 £ 46 (p3) a 2 = 46(p3) Kodaira symbol lo! ^ p t A ) l2up(6) lo III 13 * » p ( A ) - 6 I 2 t , „ ( 6 ) - 4 12 nr Case of Tate Exponent vP(N) of conductor 0 1 1 0 2 2 2 2 2 2 Table 2.4: Neron type at p of y 2 = x3 + ax2 + bx. Chapter 2. The Conductor of an Elliptic Curve over <Q> with 2-torsion 17 2.3 The Proof of Theorem 2.1. We prove this theorem using the work of Papadopolous [57]1 except in cases (ix) v2(a) = 1,1*2(6) = 2 and (xiv) u2(a) = 2, i>2(6) = 2 where we will need to apply Tate's algorithm directly. The seventeen cases we consider are labeled as follows: (i) (ii) (iii) (iv) (v) (vi) (vii) (viii) (ix) V2{a) 0 0 0 0 0 0 1 1 1 0 1 2 3 4 > 5 0 1 2 (x) (xi) (xii) (xiii) (xiv) (xv) (xvi) (xvii) v2{a) 1 1 > 2 > 2 2 2 > 3 > 3 Mb) 3 > 4 0 1 2 3 2 3 The standard invariants for the curve y2 — x3 + ax2 + bx are (see (2.1)) ai = 0, a2 — a, 03 = 0, 04 = 6, = 0, b2 = 4a, 64 = 2b, b6 = 0, bs = -b2, c 4 = 2 4 (a 2 - 36), c 6 = 25a(96 - 2a 2), A = 2 4 6 2 (a 2 - 46). Some of the cases immediately follow from Table IV of [57] so we quickly deal with these first. We have the following table. V2{a) V2(b) ^ 2 ( 0 4 ) V2{CQ) « 2 ( A ) Case of Tate Kodaira h (viii) 1 1 5 7 8 4 III 7 (ix) 1 2 > 6 8 10 6 T* 6 (xi)" 1 > 4 6 9 > 14 7 ^ 2 ( A ) - 1 0 6 (xiii) > 2 1 5 > 8 9 4 III • 8 ( X V ) 2 3 7 10 14 9 III* 7 (xvii) > 3 3 7 > 11 15 9 III* 8 Errata: In the column labeled Equation non minimale of table IV in [57] the first column should read [4,6, > 12] not [4,6,12]. Chapter 2. The Conductor of an Elliptic Curve over Q with 2-torsion 18 As for the remaining cases, we must check the supplementary conditions in [57]. (i) When ^ 2 ( 0 ) = 0 and v2(b) = 0 we have / [5 if 6 E E 1 (mod 4), V2 c 4 ) = < w 2 ( c 6 ) = 5, v2(A = 4. [>6 if b= -1 (mod 4), If 6 = 1 (mod 4) then from Table IV of [57] we are in case 3 or 4 of Tate. We use Proposition 1 of loc. cit. with r = t = 1. The congruence f 2 (mod 4) if a = 1 (mod 4), a 4 + a 2 = 6 + a = < [0(mod4) if a = - 1 (mod 4), implies that if a = 1 (mod 4) we are in case 3 of Tate and f2 = 4. So assume a = — 1 (mod 4), whence we are in case > 4 of Tate. Using Proposition 2 of loc. cit. with r — 1 and since 68 + 3r&6 + 3r2t>4 + r3b2 + 3r 4 = 2(1 + 2a) £ 0 (mod 8), we are in case 4 of Tate and J2 — 3. On the other hand, if b = —1 (mod 4) then from Table IV of [57], we are in case 3 or 5 of Tate. Take r — t — I'm. Proposition 1 of loc. cit.. It follows from the congruence [ 0 (mod 4) if a E E 1 (mod 4), a 4 + a 2 = b + a = < { 2 (mod 4) if a E E - 1 (mod 4), that if a E E - 1 (mod 4), we are in case 3 of Tate and J2 = 4, whereas if a E E 1 (mod 4), we are in case 5 of Tate and f2 — 2 . (ii) When ^ 2 ( 0 ) = 0 and 2^(6) = 1 we have V2{c4) = 4, v2{c6) > 7, v2(A) = 6, so, from Table IV of [57], we are in case 3 or 4 of Tate. Using r = t = 0 in Proposition 1 of loc. cit., it follows that we are in case 4 of Tate and J2 = 5. (iii) When v2(a) = 0 and v2(b) = 2 we have v2(c4) = 4., v2{c6) = 6, u2(A) = 8. Chapter 2. The Conductor of an Elliptic Curve over Q with 2-torsion 19 and, from Table IV of [57], we are in case 6, 7 or 8 of Tate. We use Proposition 3 of [57]. The integer r = 2 satisfies the congruence b8 + 3r&6 + 3r2o4 + r3b2 + 3r4 = 0 (mod 32). The integer t = 2 satisfies the congruence Moreover, for r = t = 2 we have the congruence a 6 + ra 4 + r 2 a 2 + r 3 - i a 3 - t2 - r t a i = 2b + 4a + 4 = 0 (mod 16). if and only if a = 1 (mod 4). It follows from Proposition 3 of loc. cit. that if a = — 1 (mod 4) we are in case 6 of Tate and f2 — 4, whereas if a = 1 (mod 4) then we are in case > 7 of Tate. So assume a = 1 (mod 4) and that we are in case > 7 of Tate. Take r = 2 in Proposition 4 of loc. cit.. The congruence 0 = a2 + 3r - ta,] - t2 = 3 - t2 (mod 4) has no solutions for t thus it follows that we are in case 7 of Tate and f2 = 3. (iv) When v2(a) = 0 and v2(b) = 3 we have v2{cA) = 4, v2(c6) = 6, v2(A) = 10, and, from Table IV of [57], we are in case 7 or 9 of Tate. The integer r = 0 satisfies the congruence b8 + 3r66 + 3r2b4 + r3b2 + 3r4 = 0 (mod 32). Moreover, we have the congruence ae + ra4 + r2a2 + r 3 - ta^ - t2 - rta\ = 0 (mod 8). 0 = a2 + 3r — ta\ — t 2 _ 1 - t2 (mod 4) if a = 1 (mod 4), 3 -t2 (mod 4) if a = - 1 (mod 4), has a solution for t if and only if a = 1 (mod 4). It follows from Proposition 4 of loc. cit. that if a = — 1 (mod 4), we are in case 7 of Tate and f2 = 4, whereas if a = 1 (mod 4), we are in case 9 of Tate and / 2 = 3. Chapter 2. The Conductor of an Elliptic Curve over Q with 2-torsion 20 (v) When 1*2(0) = 0 and V2{b) = 4 we have ^ 2(c 4) = 4, v2(cG) = 6, v2{A) = 12, and, from Table IV of [57], we are in case 7 of Tate or the model is non-minimal. The integer r — 0 satisfies the congruence b8 + Srb6 + 3r2b4 + r3b2 + 3r 4 = 0 (mod 32). Moreover, the congruence 2 (1 - t2 (mod 4) if a = 1 (mod 4), 0 = a2 + 3r - tai - t = < [3 -t2 (mod 4) if a = - 1 (mod 4), has a solution for t if and only if a = 1 (mod 4). It follows from Proposition 4 of loc. cit. that if a = — 1 (mod 4), we are in case 7 of Tate and f2 = 4, whereas if a = 1 (mod 4) the model is non-minimal. In the latter case, consider the change of variables x = 4X, y = 8Y + 4X. We obtain the new model with integer coefficients (ai,a'2,a3,a4,a6) = (1, — ^ , 0 , ^ ° ) > and such that v2(c'A) = 0, v2(c'6) = 0, and v2(A') = 0. Hence we are in case 1 of Tate and f2 = 0. (vi) When v2(o) = 0 and v2(b) > 5 we have v2{c4) = 4, v2{cG) = 6, u2(A) > 14, and, from Table IV of [57], we are in case 7 of Tate or the model is non-minimal. The integer r = 0 satisfies the congruence bs + 3rb6 + 3r2b4 + r3b2 + 3r4 = 0 (mod 32). Moreover, the congruence „ f 1 - t2 (mod 4) if a = 1 (mod 4), 0 = a2 + 3r - tai - t = < 1 3 - t2 (mod 4) if a = -1 (mod 4), Chapter 2. The Conductor of an Elliptic Curve over Q with 2-torsion 21 has a solution for t if and only if a = 1 (mod 4). It follows from Proposition 4 of loc. cit. that if a = —1 (mod 4), we are in case 7 of Tate and f2 = 4, whereas if a = 1 (mod 4) the model is non-minimal. In the latter case, take the change of variables x = 4X, y = 8Y + 4X to obtain the new model with integer coefficients {a[,a2,a'3,a'A,a'6) = (1, ,0, j ^ O ) . Then v2(c'4) = 0, i>2(c6) = 0, and v2(A') > 2, whence we are in case 2 of Tate and f2 = 1. (vii) When v2(a) = 1 and v2(b) — 0 we have v2(c4) = 3, v2(c6) = 6, u 2(A) > 7. We consider the cases v2(A) = 7, 8, 9, 10, 11, 12, > 13 separately. If v2(A) = 7 then from Table IV of [57] we are in case 3 of Tate and f2 = 7. If v2(A) = 9 then from Table IV of [57] we are in case 6 of Tate and f2 = 5. In the remaining cases; v2(A) — 8, 10, 11, 12, > 13, some work is required to determine f2. We defer the proof for these cases until Section 2.4. (x) When v2(a) = 1 and v2(b) = 3 we have ^ 2(c 4) = 6, « 2 ( C 6 ) > 9 , «2(A) = 12, and from Table IV of [57] we are in case 7 of Tate. There are, however, two possibilities for f2. We need to apply Tate's algorithm directly in this case. We will use the pseudocode for Tate's algorithm given in [26]. It is straight-forward to check that we may pass directly to line 42 in loc. cit. without hav-ing to make any changes to our model. Furthermore, in the notation of loc. cit. since xa3 = (Jf- = 0 is even, xa6 = f | = 0 is even, and xa4 = ^ = | is odd we exit the loop after line 54, with m = 2. Thus f2 = v2(A) — 6 = 6 and the Kodaira symbol is Z* • (xii) When v2(a) > 2 and v2(b) = 0 we have ^2(c4) = 4, v2(c6) > 7, v2(A) = 6, Chapter 2. The Conductor of an Elliptic Curve over Q with 2-torsion 22 so, from Table IV of [57], we are in case 3 or 4 of Tate. Take r — 1 and t = 0 in Proposition 1 of loc. cit.. It follows from the congruence „ , 2 [2 (mod 4) if b = 1 (mod 4), aG+ra4+r a2+r —ta3—t —rtai = b+a+1 = < ( 0 ( m o d 4 ) if b = -1 (mod 4), that if 6 EE 1 (mod 4), we are in case 3 of Tate and f2 = 6, whereas if 6 EE -1 (mod 4), we are in case 4 of Tate and f2 — 5. (xiv) When v2(a) = 2 and v2(b) = 2 we have v2(c4) = 6, v2(c6) = 9, v2(A) > 13, so, from Table IV of [57], we are in case 7 of Tate. There are, however, two possibilities for f2 depending on whether v2(A) = 13 or v2(A) > 14. We claim v2{A) = 13 if and only if 6/4 EE -1 (mod 16). Indeed, since A = 1662(a2 - 46), the hypothesis on a and 6 imply - ( A ) = . 3 « * ( ( ! ) ' - ! ) = - . But (a/4)2 = 1 (mod 4), from which the claim follows. Thus, f2 = 7 if 6/4 = -1 (mod 4) and f2 == 6 if 6/4 = 1 (mod 4). (xvi) When v2(a) > 3 and 1*2(6) = 2 we have v2(c4) = 6, u2(c6)-> 10, v2(-A) = 12, and, from Table IV of [57], we are in case 7 of Tate. There are,-however, two possibilities for f2. We need to apply Tate's algorithm directly in this case. Again, we will use the pseudocode for Tate's algorithm given in [26]. We consider the cases 6/4 EE -1 (mod 4) and 6/4=1 (mod 4) separately. Suppose 6/4 EE —1 (mod 4). Before starting the algorithm let us first make the change of variables x = X + 2, y = Y so our new model has coefficients ai = 0, a 2 = a + 6, a 3 = 0, a 4 = 6 + 4a + 12, a 6 = 26 + 4a + 8. It follows that v2(a2) = 1, v2(a4) = 3, v2(a6) > 5, Chapter 2. The Conductor of an Elliptic Curve over Q with 2-torsion 23 where we've used the fact that 6 / 4 = —1 (mod 4) . It is straight forward to check that we may pass directly to line 42 in loc. cit. without having to make any changes to our model. Furthermore, in the notation of loc. cit. since xa3 = = 0 is even, xa6 = y | is even, and xa4 = ^ is odd, we exit the loop after line 54, with m = 2. Thus f2 — v2(A) - 6 = 6 and the Kodaira symbol is I\. Suppose 6 / 4 = 1 (mod 4) . Similar to above, we first make the change of variables x = X + 6, y = Y + 4 to obtain a new model with coefficients ai = 0, a2 = a+ 18, a 3 = 8, a 4 = 6 + 12a + 108, a 6 = 66 + 36a + 200, and find v2{a2) = l, i*2(a3) = 3, 1*2(04) > 4, v2(a6) > 5, (here we've used the fact that 6 / 4 = 1 (mod 4)) . Moreover, ^2(62) = 3, 1*2(64) > 5, i » 2 ( 6 6 ) = 3, v2{b8) > 7. It is straightforward to check that we may pass directly to line 42 in loc. cit. without having to make any changes to our model. Furthermore, in the nota-tion of loc. cit. since xa3 = ^ = 0, xa6 = f|, and xa4 = ^ are all even, we have from line 56 that _ ( 4 if § is odd, [0 if §§ is even. We then must apply the change of variables transcoord(r, 0,0,1) at line 59. In either case the change of variables leads to a curve (a[,a'2,a'3, a'A, a'c) such that o'i = 0, v2{a'2) = 1, i*2(o3) = 3, v2(a'A) > 4, v2(a'6) > 5. We have now reached the end of the loop and return back to line 45. Since xa3 = ^ is odd we exit the loop after line 47 with rn — 3. Thus f2 = v2(A) — 7 = 5 and the Kodaira symbol is 7 3 . To finish the proof it remains to verify the cases when v2(a) = 1,1*2(6) = 0, and v2{A) = 8,10,11,12, and > 13. We do this in the next section. Chapter 2. The Conductor of an Elliptic Curve over Q with 2-torsion 24 2.4 T h e case w h e n v2(a) = 1, ^2(6) = 0. We have already determined in part (iv) of the proof of Theorem 2.1 the values of / 2 when v2(A) = 7 or 9. In this section we determine the value of f2 for the remaining cases: ^ ( A ) = 8,10, 11, 12, > 13. First we make two observations. Lemma 2.6 I/a, 6 G Z such that v2{a) = 1, v2(b) = 0 and v2(A) — v2(l6b2(a2 -46)) > 8 then 6=1 (mod 4). furthermore, if v2(A) = 8 then 6 = 5 (mod 8). Proof. If v2(A) = v2(16b2(a2 - 46)) > 8 then w2((§ ) 2 - 6)) > 2. It follows that 6 = 1 (mod 4) since a/2 is odd. Moreover, if i; 2(A) = 8 then t>2((|)2 - 6) = 2 thus 6 ^ 1 (mod 8). • We will use the next lemma when applying Proposition 4 of [57]. Lemma 2.7 For a, 6 G Z swcfo f/zaf i>2(a) = 1 fl^d ^ 2(6) = 0 f/ie congruence - 6 2 + 6r26 + 4r 3a + 3r4 = 0 (mod 32) has no solutions for r ifb = —1 (mod 4), whereas for 6=1 (mod 4) it has solutions Proof. If 6 = —1 (mod 4) then the congruence has no solutions mod 8 (when a is even). So, it certainly can't have any solutions mod 32. Assume 6 = 1 (mod 4) and write 6 = 4k + 1 for some k G Z. If a = 2 (mod 8) then we may write a — 8£ + 2 for some I G Z. Taking r = 3 we have 3 if a = 2 (mod 8), 1 if a = 6 (mod 8). - 6 2 + 6r26 + 4r 3 a + 3r4 = 16fc(fc + 1) = 0 (mod 32). Similarly, one can easily show r = 1 is a solution when a = 6 (mod 8). • Chapter 2. The Conductor of an Elliptic Curve over Q with 2-torsion 25 2.4.1 Proof of Theorem 2.1 part (vii) when v2(A) = 8 It follows from Lemma 2.6 that 6 = 5 (mod 8). Since v2(c4) = 4, v2(c§) = 6 and v2(A) = 8 it follows from Table IV of [57] that we are in case 6, 7 or 8 of Tate. We use Proposition 3 of loc. cit.. By Lemma 2.7 the congruence b8 + 3r6 6 + 3 r 2 6 4 + r 3 6 2 + 3 r 4 = 0 (mod 32) has solutions _ J 3 if a EE 2 (mod 8), \ l if a EE 6 (mod 8). In either case the integer t — 2 satisfies the congruence a 6 + r a 4 + r 2 a 2 + r 3 - i a 3 - t 2 - rtax = rb + r2a + r3 - t2 = 4 - t2 = 0 (mod 8). Fix t = 2 and r as above. Suppose a = 2 (mod 8). We have the congruence a 6 + ra4 + r2a2 + r 3 - ta3 - l2 - rtax EE 36 + 9a + 7 EE 0 (mod 16) if and only if a — 6 EE 5 (mod 16). Thus, we are in case 6 of Tate (and f2 = 4) if a — 6 EE 13 (mod 16), and in case > 7 of Tate if a — b = 5 (mod 16). So, suppose the latter holds. Taking r = 3 in Proposition 4 of loc. cit. the congruence a 2 + 3r - sai - s2 EE 3 - s2 = 0 (mod 4) has no solution for s, whereby we are in case 7 of Tate and f2 = 3. In the state-ment of the theorem we do not need to include the condition a EE 2 (mod 8) since this automatically follows from the congruences 6 EE 5 (mod 8) and a — b EE 5 or 13 (mod 16). Now suppose a EE 6 (mod 8). We have the congruence a 6 + r a 4 + r2a2 + r3 - ta3 - t2 - rta\ = b+ a - 3 = 0 (mod 16) if and only if a — 6 EE 9 (mod 16). Thus, we are in case 6 of Tate (and f2 = 4) if a — 6 EE 1 (mod 16) and in case > 7 of Tate if a — 6 EE 9 (mod 16). So, suppose the latter holds. Taking r = 1 in Proposition 4 of loc. cit. the congruence a2 + 3r - sai - s2 = 1 - s 2 EE 0 (mod 4) Chapter 2. The Conductor of an Elliptic Curve over Q with 2-torsion 26 has solution s = 1, whereby we are in case 8 of Tate and f2 = 2. Again, we do not need to include the condition a = 6 (mod 8) in the statement of the theorem since it follows automatically from 6 = 5 (mod 8) and a — b = 1 or 9 (mod 16). 2.4.2 Proof of Theorem 2.1 part (vii) when v2(A) = 10 In this case we have v2(c4) = 4, v2(c6) = 6, v2{A) = 10, so from Table IV of [57] we are in case 7 or 9 of Tate. We use Proposition 4 of loc. cit. to distinguish between these two cases. By Lemma 2.7, the congruence 68 + 3r66 + 3r 26 4 + r3b2 + 3r4 = 0 (mod 32) has solutions _ f3 if a = 2 (mod 8)), \ l i fa = 6 (mod 8)). Furthermore, the congruence 2 f 3 - t2 (mod 4) if a EE 2 (mod 8), 0 EE a2 + 3r t EE < [ 1 - t 2 (mod 4) if a EE 6 (mod 8), has solution t = 1 if a = 6 (mod 8) and no solution for t otherwise. Thus, we are in case 9 of Tate if a EE 6 (mod 8) and in case 7 of Tate if a EE 2 (mod 8). The assertion follows. 2.4.3 Proof of Theorem 2.1 part (vii) when v2(A) = 11 In this case v2{c4) = 4, v2(c6) = 6, v2{A) = 11, so, from Table IV of [57], we are in case 7 or 10 of Tate. By exactly the same argument as in Section 2.4.2, if a = 6 (mod 8), we are in case 10 of Tate and if a EE 2 (mod 8), we are in case 7 of Tate. The assertion follows. Chapter 2. The Conductor of an Elliptic Curve over Q with 2-torsion 27 2.4.4 Proof of Theorem 2.1 part (vii) when v2{A) = 12 In this case v2(c4)=4, v2(c6) = 6, v2(A) = l2, so, from Table IV of [57], we are in case 7 of Tate or the model is non-minimal. By exactly the same argument as in section 2.4.2, if a EE 2 (mod 8), we are in case 7 of Tate and if a EE 6 (mod 8), the model is non-minimal. In the case that the model is non-minimal, we make the change of variables x = 4X-a/2, y = SY + AX. (2.5) The new model has coefficients i i i i i i\ ( a,+ 2 a? - 46 a(a2 - 46) \ (ai, a 2 , a 3 ) a 4 ) a 6 ) = I I , — ,0, ^ — , — I , (2.6) which are all integers (by assumptions on a and 6). Also, v2(c'4) = 0, v2{c'(i) = 0, and v2(A') = 0, whence we are in case 1 of Tate and f2 = 0. 2.4.5 Proof of Theorem 2.1 part (vii) when v2(A) > 13 In this case v2(c4) = 4, v2(c6) = 6, v2(A) > 13, so, from Table IV of [57], we are in case 7 of Tate or the model is non-minimal. By exactly the same argument as in section 2.4.2, if a EE 2 (mod 8), we are in case 7 of Tate and if a EE 6 (mod 8), the model is non-minimal. In the case that the model is non-minimal we take the change of variables (2.5) which gives us a new integral model with coefficients as in (2.6). Since v2(c'A) = 0, v2(c'6) = 0, and v2(A') > 1, we are in case 2 of Tate and f2 = 1. This completes the proof of Theorem 2.1. 2.5 T h e P r o o f o f T h e o r e m 2.3 . We can quickly deal with the following cases by using Table II of [57]. Chapter 2. The Conductor of an Elliptic Curve over Q with 2-torsion 28 vj, (a) v3(b) V3(C4) V3{C6) « 3(A)' Case of Tate Kodaira /3 0 > 1 0 0 > 2 2 IU 3(A) 1 > 1 0 1 > 3 0 1 lo 0 1 > 2 2 3 > 6 6 or 7 ^ 3 (A)-6 2 > 2 1 2 > 5 3 4 in 2 > 2 2 3 > 6 6 6 T* 2 > 3 3 4 > 8 9 9 III* 2 There are only three remaining cases to check: (1) 1*2(0.) = 0, v2(b) = 0; (2) v2{o) = 1, v2{b) = 1; (3) v2{a) = 2, v2(b) = 3. (1) Suppose 1/3(0) = 0 and 1/3(6) = 0. Then v2(c4) = 0, 1/3(00) = 0, and 3 divides A if and only if b = 1 (mod 3). It follows that 1 i f 6 = l ( m o d 3 ) , 0 if 6 E E -1 (mod 3), and the Neron type at 3 is I„3(A) if 6 EE 1 (mod 3) and In if b = —1 (mod 3). (2) Suppose vs(a) = 1 and 1/3(6) = 1. Then v2(cA) > 2, VS{CQ) = 3, and i>3(A) = 3. We consider the intervening condition P2 in Table II of [57]. P2 is decided if we have Hi) H (D 1))' + 2 s s- 2 4((i)a-i)^ 9'-or equivalently (l)'(i) + (i)a+2(i)-ls0(mod9)-Since ^ 3 ( 0 ) = vs(b) = 1, this is certainly the case. Therefore / 3 = 2 and the Neron type at 3 is III. (3) Suppose 1/3(0) = 2 and w3(6) = 3. Then v 2(c 4) > 4, ^ (co) = 6, and i>3(A) = 9. We consider the intervening condition P 5 in Table II of [57]. P5 is decided if we have Chapter 2. The Conductor of an Elliptic Curve over Q with 2-torsion 29 or equivalently Since 1^3(0) = 2 and 1*3(6) = 3, this is the case. Therefore f$ — 2 and the Neron type at 3 is III*. 2.6 T h e P r o o f o f T h e o r e m 2.4. We can quickly deal with the following cases by using Table I of [57]. vp(a) vp(b) vp(c4) vP{c6) vp{A) Case of Tate Kodaira fP 0 > 1 0 0 > 2 2 hvp{b) 1 > 1 0 0 > 1 0 1 Io 0 > 1 1 1 > 2 3 4 III 2 1 > 3 2 3 > 8 7 T* 2 > 2 2 2 > 4 6 6 T* 1o 2 > 2 3 3 > 5 9 9 III* 2 There are only two remaining cases to check: (1) 1*2(0) = 0, 1*2(6) = 0; (2) v2(a) = 1,1-2 (6) = 2. (1) Suppose i*2(a) = 0, ^2(6) = 0. In this case, p can divide at most one of C4, CG and A . If p does not divide A then fp = 0. If p | A then p does not divide C4 or C6 , whence / 2 = 1 and the Neron type at p is I„ (A) • (2) Suppose 1*2(0) = 1,1*2(6) = 2. Then i*p(c4) > 2, vp(ce) > 3, and w p(A) > 6. Moreover, in this case, p3 can divide at most one of a2 — 36, 96 — 2a 2 and a2 - 46. If vp(A) > 7, i.e. a 2 — 46 = 0 (mod p3), then we are in case 7 of Tate, fp — 2, and the Neron type at p is I* (A)-6 - ^ n t n e o t n e r hand, if"i*p(A) = 6, i.e. a2 — 46 5= 0 (mod p3), then we are in case 6 of Tate, / p = 2, and the Neron type at pis IQ. This proves Theorem 2.4. Chapter 3 Classification of Elliptic Curves over Q with 2-torsion and conductor 2 a 3 V Let p be a prime number and L, M and iV integers satisfying the inequalities p > 5, 0 < M < 8, and 0 < L , N < 2. In what follows we announce twenty-seven theorems which describe, up to Q-isomorphism, all the elliptic curves over Q, of conductor 2M3LpN, having a rational point of order 2 over Q. The first nine theorems list curves of con-ductor 2Mp2. The next nine list curves of conductor 2M3Lp, and the last nine list those of conductor 2M3Lp2. Together, with the work of Ogg on conductor 2M, Coghlan on conductor 2 M 3 L , Setzer on prime conductor, and Ivorra on conductor 2Mp, this completes the classification problem of curves with bad reduction at 2, 3, and p > 5, and having rational 2-torsion. The results which are obtained are presented in the form of tables anal-ogous to those of [26] and [37]. Each row consists of an elliptic curve of Q realizing the desired conditions. The columns of the table consist of the fol-lowing properties of E: i. A minimal model of E of the form y2 + a\xy = x3 + ci2X2 + a4x + a$, where the a.j are in Z; except in the cases when N < 2, in these cases minimal models could be found using Corollary 2.2 but we choose not to do this here. A l l models listed are chosen such that a\ = a 3 — a 6 = 0, so in the statements of these theorems we omit the columns correspond-ing to these coefficients. 30 Chapter 3. Elliptic Curves with 2-torsion and conductor 2c'3l3p5 31 ii. The factorization of the discriminant A of E. Also appearing in the table are letters of identification (A,B,...) for each elliptic curve. Our usage of such letters is inspired by the tables of Cremona (and [37]) but one should not attempt to assign any meaning to our labeling other than the following. The curves which are labeled by the same letter are linked by an isogeny over Q of degree 2 or a composition of two such isogenies. For example if two curves are labeled A l and A2 then they are linked by a degree 2 isogeny, whereas if four curves are labeled A l , A2, A3, and A4 then A l is linked to each of the other three by a two isogeny and A2, A3, A4, are linked to each other by degree 4 isogenies. Moreover, they are numbered in the order of how they are to be determined. Notations a. For each elliptic curve E over Q, we denote by E' the elliptic curve over Q obtained from E by a twist by <J~—1. b. Given an integer n which is a square in Z we denote, in the rest of this work, by y/n the square root of n satisfying the following condition: 3.1 C u r v e s o f C o n d u c t o r 2 a p 2 The tables presented here are an intermediate step in the classification prob-lem for curves of conductor 2ap2. In Chapter 6, we refine these tables by using the Diophantine lemmata of Chapter 4 to resolve the Diophantine equations in the tables below. If the reader is interested in a classification of curves of conductor 2Np2 then it would be best to look at the results in Chapter 6 for the "polished" tables. The results here are strictly transitional. 3.1.1 Statement of Results Theorem 3.1 The elliptic curves E defined over Q, of conductor p2, and having at least one rational point of order 2, are the ones such that one of the following conditions is satisfied: = 1 mod 4 if n is odd if n is even . (3.1) Chapter 3. Elliptic Curves with 2-torsion and conductor 2 ° 3 / V 5 32 1. p = 7 and E is Q-isomorphic to one of the elliptic curves: a-2 0-4 minimal model T2 A A l 7-3 2 4 -7 [ 1 , - 1 , 0 , - 2 , - 1 ] 2 7 3 A2 - 7 2 - 3 2 4 . ? 3 [1,-1,0,-107,552] 2 7 9 BI - 2 - 7 - 3 - 7 [1 , -1 ,0 , -37 , -78] 2 7 3 B2 2 • 7 2 • 3 - 7 3 [1,-1,0,-1822, 30393] 2 7 9 2. p = 17 and E 1 is Q-isomorphic to one of the elliptic curves: a2 C14 minimal model T2 A C l 17-33 2 4 • 17 3 [1,-1,1,-1644,-24922] 4 1 7 8 C2 - 2 - 1 7 - 3 3 17 2 [1,-1,1,-26209,-1626560] 2 17 7 C3 17-9 2 4 - 172 [1,-1,1,-199,510] 4 17 7 C4 2-17-15 1 7 4 [1,-1,1,-199,-68272] 2 1 7 i o 3. p — 64 is fl square and E is Q-isomorphic to one of the elliptic curves: ai a2 0 4 a 6 \T2\ A A l 1 P\/p—64— 1 4 0 2 v1 A2 1 PVP~ 64— 1 4 4p2 p - V P - 64 2 - P 8 Theorem 3.2 The elliptic curves E defined over Q, of conductor 2p2, and having at least one rational point of order 2, are the ones such that one of the following conditions is satisfied: 1. there exist integers m>7 and n > 0 such that 2mpn + 1 is fl square and E is Q-isomorphic to one of the elliptic curves: a2 a.4 A A l epy/2mpn + 1 2 m - 2 p 7 i + 2 22mp2n+6 A2 -e2py/2mpn + 1 P2 where e e {±1} is the residue ofp modulo 4. 2. there exist integers m > 7 and n > 0 such that 2m + pn is a square and E is Q-isomorphic to one of the elliptic curves: Chapter 3. Elliptic Curves with 2-torsion and conductor 2a3!3pd 33 a2 a4 A Bl epsJ2m + pn 2m-2p2 22m,pn+6 B2 -e2p v /2m + p n pn+2 2"i-|-6p2n+6 where e G {±1} is the residue ofp modulo 4. 3. there exist integers rn > 7 and n > 0 such that 2rn — pn is a square and E is Q-isomorphic to one of the elliptic curves: a2 0,4 A CI epy/2m - pn 2 m - 2 p 2 22rripn+6 C2 -e2px/2m - pn -pn+2 2"i+6p2n-h6 where e G {±1} is the residue ofp modulo 4. 4. there exist integers m > 7 and n > 0 SMCTI that pn — 2rn is a square and E is Q-isomorphic to one of the elliptic curves: 0,2 0,4 A DI tV\lpn - 2 m - 2 m - V 22mpn+B D2 -e2p^pn - 2m pn+2 _2m+&p2n+ 6 where e £ {±1} is the residue ofp modulo 4. 5. £h<?r<? exist integers m > 7 and £ G {0,1} such that 2*"p+1 is a square and E is Q-isomorphic to one of the elliptic curves: a2 0 4 A E l 2m-2p2t+l 22rrip3+6i E2 p 2 t + l 2?n-f 6^3+6£ w/zere e G {±1} is the residue ofpw modulo 4. 6. rftere exist integers m > 7 and t G {0,1} such that 2 m ~ l is a square and E is Q-isomorphic to one of the elliptic curves: 0,2 a4 A FI 2m-2p2t+l 22rrip3+6t F2 _p2t+l 2^+6^3+6t Chapter 3. Elliptic Curves with 2-torsion and conductor 2a3l3p' 34 where e e {±1} is the residue ofp+ modulo 4. Theorem 3.3 The elliptic curves E defined over Q, of conductor 4p2, and having at least one rational point of order 2, are the ones such that one of the following conditions is satisfied: 1. there exists an integer n > 0 such that pn — 4 is a square and E is Q-isomorphic to one of the elliptic curves: 0-2 0,4 A A l epy/pn — 4 -P2 24pn+G A2 - e 2 p v > n - 4 pn+2 2 8 p 2 n + 6 where e e {±1} is the residue of p modulo 4. Theorem 3.4 The elliptic curves E defined over Q, of conductor 8p2, and having at least one rational point of order 2, are the ones such that one of the following conditions is satisfied: 1. there exist integers m e {4, 5} and n > 0 such that 2mpn + 1 is a square and E is Q-isomorphic to one of the elliptic curves: a.2 CI4 A A l tpy/2mpn^-\r 1 2m-2pn+2 2 2 77). p2n-\-6 A2 - e 2 p v / 2 7 > n + 1 P2 where e £ {±1} is the residue ofp modulo 4. 2. there exists an integer n > 0 such that 4+pn is a square and E is Q-isomorphic to one of the elliptic curves: a2 0,4 A BI -epJ4+pn v2 24pn+6 B2 e2py/4 + pn pn+2 where e € {±1} is the residue ofp modulo 4. 3. there exist integers m 6 {4,5} and n > 0 such that 2rn + pn is a square and E is Q-isomorphic to one of the elliptic curves: Chapter 3. Elliptic Curves with 2-torsion and conductor 2a3l3ps 35 0-2 CL4 A C l epy/2m + pn 2 m - 2 p 2 22mpn+& C2 -e2p^/2m + pn pn+2 2m+6p2n+6 where t € { ± 1 } is the residue ofp modulo 4. 4. there exist integers m e {4,5} and n > 0 such that 2rn — pn is a square and E is Q-isomorphic to one of the elliptic curves: a2 A D I ep^/2m - pn 2 ™ - 2 p 2 __22mpn+d D2 - e 2 p v / 2 m - Pn _pn+2 2"i.+6p2n+6 where e € { ± 1 } is the residue ofp modulo 4. 5. there exists an integer n > 1 such that pn — 4 is a square and E is Q-isomorphic to one of the elliptic curves: a2 a 4 ' A E l epy/pn — 4 -P2 2 4 p n+6 E2 -t2py/pn - 4 pn+2 _ 2 8 p 2n+C where e e {±1} is the residue ofp modulo 4. 6. there exist integers rn e {4, 5} and n > 0 such that pn — 2rn is a square and E is Q-isomorphic to one of the elliptic curves: a 2 0-4 A F l epy/pn - 2 m _ 2 m - 2 p 2 22mpn+6 F2 -t2psjpn - 2m pn+2 _2 m + 6 p2n+6 where e £ {±1} is the residue ofp modulo 4. 7. there exists an integer t € {0,1} such that ^ is a square and E is Q-isomorphic to one of the elliptic curves: a2 a.4 A G I p2t+l 24p3+6t G2 e2pt+\/±±l y V V p2t+l 2 V + 6 T Chapter 3. Elliptic Curves with 2-torsion and conductor 2a3l3ps 36 where e € {±1} is the residue ofpt+1 modulo 4. 8. there exist integers m € {4,5} and t € {0,1} such that 2 m p + 1 is a square and E is Q-isomorphic to one of the elliptic curves: a2 0,4 A HI 2m-2p2t+l 2 2 m ^ 3 + 6 t H2 p2t+l 2 m+6 ^ 3 + 61 where e € {±1} is the residue ofpt+1 modulo 4. 9. there exist integers m e {4,5} and t e {0,1} such that 2 m ~ x is a square and E is Q-isomorphic to one of the elliptic curves: « 2 0,4 A 11 2m-2p2t+l 22rrip3+6t 12 _p2t+l 2^71+6^3+6^ where e e {±1} zs ffoe residue ofpl+l modulo 4. Theorem 3.5 Tfte elliptic curves E defined over Q, of conductor 16p2, and having at least one rational point of order 2, are the ones such that one of the following conditions is satisfied: 1. there exist integers m > 4 and n > 0 such that 2rnpn + 1 is a square and E is Q-isomorphic to one of the elliptic curves: a-2 04 A A l -tpy/2mpn + 1 2m—2pn+2 22m.p2n+6 A2 t2px/2mpn + 1 P2 2m+6pn+6 where e e {±1} is the residue ofp modulo 4. 2. there exists an integer n > 0 such that 4+pn is a square and E is Q-isomorphic to one of the elliptic curves: a 2 a 4 A B l epy/4 + pn P2 24pn+6 B2 -e2py/4+pn pn+2 2Sp2n+6 Chapter 3. Elliptic Curves with 2-torsion and conductor 2a3f3ps 37 where e G { ± 1 } zs the residue ofp modulo 4. 3. there exist integers m > 4 and n > 0 such that 2m + pn is a square and E is Q-isomorphic to one of the elliptic curves: a 2 (14 A C l - q V 2 ™ + p n 2m~2p2 C2 e 2 p v / 2 m + pn pn+2 2"T-+6p2n+6 where e G { ± 1 } is the residue ofp modulo 4. 4. there exist integers m > 4 and n > 0 suc/z that 2rn - pn is a square and E is Q-isomorphic to one of the elliptic curves: a 2 0,4 A DI -ep^/2m - pn 2m~2p2 D2 e 2 p v / 2 m - Pn _pn+2 2m+6p2n+6 where e G { ± 1 } is the residue ofp modulo 4. 5. there exists an integer n> I such that pn -4 is a square and E is Q-isomorphic to one of the elliptic curves: a-2 a 4 A E l epy/pn — 4 -p2 2 4 pn + 6 E2 -e2p^/pn - 4 pn+2 _ 2 8 p 2 n + 6 where e G { ± 1 } is the residue ofp modulo 4. 6. f/iere exz'sf integers m > 4 and n > 0 swdz f/zaf p n — 2 m zs a square and E is Q-isomorphic to one of the elliptic curves: a-2 a 4 A F l -epy/pn - 2m - 2 m - 2 p 2 22m,pn+6 F2 e2p v /P™ - 2 m pn+2 _2"i+6p2n+6 w/zere e G { ± 1 } zs £«e residue ofp modulo 4. 7. £«ere exz'sfs an integer t G {0,1} SWCH that is a square and E is Q-isomorphic to one of the elliptic curves: Chapter 3. Elliptic Curves with 2-torsion and conductor 2a3t3ps 38 a2 ct4 A G I t+ i / 4 + 1 ' V p 2 m p 2 t + l 2 4 p 3+6t G2 p 2 t + l 28 p3+6t where e £ {±1} is the residue ofp1 modulo 4. <§. there exist integers m > 4 and t e {0,1} swdz f/zfl£ 2 ' " + 1 zs a square and E is Q-isomorphic to one of the elliptic curves: a2 a 4 A H I 2rrip2£+l 22?n.p3+6£ H2 p2t+l 2m-f-6p3+6t wfaere e € { ± 1 } is the residue of pt+1 modulo 4. 9. there exist integers m > 4 ana* t £ {0,1} such that 2 m ~ x is a square and E is Q-isomorphic to one of the elliptic curves: a2 a 4 A 11 2 m p 2 t + l 22m^3-|-6£ 12 _ p - i + i 2m4-6p3+6£ i f^ere e e { ± 1 } is the residue of pi+l modulo 4. Theorem 3.6 The elliptic curves E defined over Q, of conductor 2>2p2, and having at least one rational point of order 2, are the ones such that one of the following conditions is satisfied: 1. there exists an integer n > 0 such that pn -lis a square and E is Q-isomorphic to one of the elliptic curves: a2 a 4 A A l 2py/pn - 1 -P2 2 6 p n+6 A 2 -4py/pn - 1 4pn+2 2 1 2 p 2 « + 6 A l ' - 2 p v / P n - 1 -P2 26 p 7 i + 6 A 2 ' 4py/pn - 1 4 p n + 2 212 p2n+6 Chapter 3. Elliptic Curves with 2-torsion and conductor 2Q3 /V 5 39 2. there exists an integer t e {0,1} such that E is Q-isomorphic to one of the elliptic curves: (a) p= 1 (mod 4); (b) p = -1 (mod 4); a-2 C14 A BI 0 -p2t+1 2<y+6t B2 0 4p2t+l - 2 l 2 p 3 + 6 t a2 0,4 A C l 0 p2t+l - 2 V + 6 t C2 0 - 4 p 2 i + 1 2 ] V + 6 t 3. there exists an integer n > 0 such that 8pn + 1 'is a square and E is Q-isomorphic to one of the elliptic curves: a2 0,4 A D I p^8pn + 1 2pn+2 26p2n+6 D2 - 2 p v / 8 p n + 1 P2 29pn+6 D I ' -p^8pn + 1 2pn+2 <2&p2n+Q D2' 2p^8pn + 1 p2 29pn+6 4. there exists an integer n > 0 such that 8+pn is a square and E is Q-isomorphic to one of the elliptic curves: a2 0,4 A E l + Pn 2p2 26pn+li E2 -2Py/8 + pn pn+2 2 9 p 2n+6 E l ' -p^/8 + pn 2p2 2 6 p n+6 E2' 2 P x /8 + pn pn+2 2 9p 2 7i+6 5. f/tere exists an integer n > 1 SWCH 8—p" fs a square and E is Q-isomorphic to one of the elliptic curves: a2 a4 A F l - P" 2p2 - 2 6 p n + 6 F2 - 2 P v / 8 -pn _ p n+2 2 9 p 2n+6 F l ' - P \ / 8 - p n 2p 2 _ 2 6 p n + 6 F2' 2 ^ 8 - p" _pn+2 29p2n+6 Chapter 3. Elliptic Curves with 2-torsion and conductor 2a3^ps 40 6. there exists an integer n > 1 such that pn -Sis a square and E is Q-isomorphic to one of the elliptic curves: 0-2 a 4 A G I Ps/pn - 8 -2p2 26pn+& G2 -2Py/pn - 8 pn+2 - 2 c V n + 6 G I ' -pVPn - 8 ~2p2 2 6 p n+6 G2' 2p^pn - 8 pn+2 7. there exists an integer t € {0,1} such that is a square and E is Q-isomorphic to one of the elliptic curves: a2 a 4 A H I 2p2t+1 26 p3+6t H2 _p2t+l 29 p3+6t H I ' 2p2t+l 26 p3+6( H 2 ' „ p 2 t + l 29p3+6t T h e o r e m 3.7 The elliptic curves E defined over Q, of conductor 64p2, and having at least one rational point of order 2, are the ones such that one of the following conditions is satisfied: 1. there exists an integer n > 0 such that pn -lis a square and E is Q-isomorphic to one of the elliptic curves: a 2 a 4 A A l W P " - 1 pn+2 - 2 V N + G A 2 -4ps/pn - 1 -4p2 2 12 p n+6 A l ' -2 ?Vp n - 1 pn+2 - 2 V N + 6 A 2 ' 4pyjpn - 1 ^4p2 212pn+6 2. there exists an integer t e {0,1} such that E is Q-isomorphic to one of the elliptic curves: (a) p = 1 (mod 4); 02 a 4 A BI 0 p2t + X _ 2 6 p 3+6t B2 0 _4 p2t+l 212 p3+6t Chapter 3. Elliptic Curves with 2-torsion and conductor 2c"3l3p5 41 (b) p = -1 (mod 4); a 2 a 4 A Cl 0 _ p 2 t + l 26 p3+6t C2 0 4p2t+l _ 2 l 2 p 3 + 6 £ 3. f/tere exist integers m > 3 and n > 0 swc/z i/zaf 2T Oj/1 + 1 is a square and E is Q-isomorphic to one of the elliptic curves: a 2 a 4 A D I 2 p v / 2 m p n + 1 2 m p n + 2 22m+6p2n+6 D2 ~4p^/2rnpn + 1 4p 2 2m+12p?i+6 D I ' - 2 p v / 2 m P n + 1 2m p n + 2 22m+6 p27i+6 D2' 4 p x / 2 m p n + 1 4p 2 2m+12pn+6 4. i/iere exz'st integers m > 2 and n > 0 swc/t f«af 2 m + is a square and E is Q-isomorphic to one of the elliptic curves: a 2 a 4 A E l 2py/2m + pn 2mp2 22m+6pn+6 E2 ~ 4 p v / 2 m +pn 4pn+2 2m+12p2n+6 E l ' - 2 p v / 2 m + p n 2 m p 2 22m+6p7i+6 E2' 4py/2m + p" 4p n + 2 . 2m+12p2?i+6 5. fHere exzsf integers m > 2 and n > 0 stzc/z 2 m — pn is a square and E is Q-isomorphic to one of the elliptic curves: a 2 a 4 A F l 2py/2m - pn 2 m p 2 22m+6pn+6 F2 - 4 p v / 2 m - p n - 4 p n + 2 2m+12p2n+6 F l ' - 2 p v / 2 m - p n 2"'p2 22"T-+6pn+6 F2' 4p v / 2 T O - p n - 4 p n + 2 2m+12p2n+6 6. fnere exz'st' integers m > 2 and n > 0 swc/z f«flf — 2m is a square and E is Q-isomorphic to one of the elliptic curves: a 2 a 4 A G I 2p^3epn - 2 m - 2 m p 2 22m+6pn+6 G2 - 4 p v / 3 f p n - 2 m 4 p n + 2 _2m+12p2n+6 G I ' - 2 p > / 3 V - 2 m - 2 m p 2 22m+6pn+6 G2' 4 P V / 3 V 1 - 2 m 4 p n + 2 _ 2 m+12 p 2n+6 Chapter 3. Elliptic Curves with 2-torsion and conductor 20c3l3ps 42 7. there exist integers m > 2 and t £ {0,1} such that 2 m p + 1 is a square and E is Q-isomorphic to one of the elliptic curves: 0 4 A H I 2mp2t+l 22m+6p3+6t H2 -v+V2^ 2m+12p3+6t H I ' 2 m p 2 t + l 22m+6p3+6t H 2 ' 4p2t+l 2m+12p3+6t 8. f/tere exz'st integers m > 2 and £ £ {0,1} swdz f^ iaf 2—p-^ is a square and E is Q-isomorphic to one of the elliptic curves: «2 a 4 A 11 2 m p 2 t + l 22m+6p3+6t 12 - 4 p 2 t + 1 _ 2 m+12 3+6t 11' 2 m p 2 t + l 22m+6p3+6t 12' v+V2^ _4 p 2(+l _2m+12p3+6( T h e o r e m 3.8 Tfte elliptic curves E defined over Q, of conductor 128p2, rmd having at least one rational point of order 2, are the ones such that one of the following conditions is satisfied: 1. there exists an integer n > 0 such that 2p" — 1 is a square and E is Q-isomorphic to one of the elliptic curves: a 2 a 4 A A l 2py/2pn - 1 2p n + 2 22m+6p2n+6 A2 -Apy/2pn - 1 - 4 p 2 2m+12pn+6 A l ' -2py/2pn - 1 2p"+2 22m+6p2n+6 A 2 ' 4p v /2p n - 1 - 4 p 2 2m+12pn+6 B l 2pv /2p™ - 1 - p 2 2Tn+6pn+6 B2 - 4 p v / 2 p " - 1 8 p n + 2 22m+12p2n+e B l ' -2py/2pn - 1 - p 2 2m+6pn+6 B2' 4p v / 2p" - 1 8 p n + 2 22m+12p2n+C Chapter 3. Elliptic Curves with 2-torsion and conductor 2a30p5 43 2. there exists an integer n > 0 such that 2+pn is a square and E is Q-isomorphic to one of the elliptic curves: «2 (I4 A C l 2Py/2 + pn 2p2 22m+6pn+6 C2 -4pv/2 + pn i p n + 2 2m+12p2n+6 C l ' -2p^/2 + p" 2p2 <22m+6 p7i+6 C2' 4Py/2 + pn 4p"+2 2m+12p2n+6 D I 2psJ2 + pn pn+2 2"i+6p2n+6 D2 -4pv/2 + pn 8p 2 22m+12pTi+6 D I ' -2ps/2 + pn pn+2 2m+6p2?i+6 D2' 4py/2 + pn 8p 2 22m+12pn+6 3. there exists an integer n > 0 such that 2-pn is a square and E is Q-isomorphic to one of the elliptic curves: a2 0,4 A E l 2Py/2 - pn 2p 2 22m+6pn+6 E2 -4pv/2 - p n - 4 p n + 2 2m+\2p2n+G E l ' -2Ps/2 - pn 2p 2 22m+G pn+G E2' 4pV2 - p n - 4 p n + 2 2m,+ 12p2n+6 F l 2 P v / 2 - Pn -p"+2 2m+Gp2n+6 F2 -4pv/2 - P n 8p 2 _22m.+12pn+6 F l ' - 2 p V 2 - p n —pn+2 2m+6 p2n+6 F2' 4PV2 - p n 8p 2 _ 2 2m+12 p n+6 4. there exists an integer n > 0 such that pn -2 is a square and E is Q-isomorphic to one of the elliptic curves: a.2 a 4 A G I 2Py/pn - 2 p"+ 2 22m+6pn+6 G2 - 4 p v / p n - 2 - 8 p2 _2'Ti+12 p2n+6 G I ' - 2 p v / p " - 2 p"+ 2 22777.+6pn+6 G2' 4 p v / p n - 2 - 8 p2 _ 2 " i . + 12p2n+6 H I 2pv/p™ - 2 - 2 p 2 _ 2m+6p2n+6 H2 -4pv/p n - 2 4 p n + 2 22m+12pn+6 H I ' - 2 p V P n - 2 - 2 p 2 _2m+6p2n+6 H 2 ' 4pv/p n - 2 22m+12pn+6 Chapter 3. Elliptic Curves with 2-torsion and conductor 2a3l3p5 44 T h e o r e m 3.9 The elliptic curves E defined over Q , of conductor 256p 2 , and having at least one rational point of order 2, are the ones such that one of the following conditions is satisfied: 1. there exists an integer and n > 0 such that p £ l is a square and E is Q-isomorphic to one of the elliptic curves: a-2 a 4 A A l 29p2n+6 A 2 8P \ / P "2 + 1 8 p n + 2 215pn+6 A l ' 2p2 2 9 p 2n+6 A 2 ' 8pn+2 2 15 p n+6 B l 2pn+2 2 9 p n+6 B2 8p2 215p2n+6 B l ' -*PXJ^ 2pn+2 2 9 p n+6 B2' 8p 2 215p2n+6 2. there exists an integer n > 0 SMC/J fTzaf is a square and E is Q-isomorphic to one of the elliptic curves: 0.2 a 4 A C I 2p n + 2 - 2 9 p 2 n + 6 C2 - 8 p 2 2 15 p n+6 c r 2p"+2 _ 2 9 p 2n+6 C2' - 8 p 2 2 1 5 p n + 6 D I - 2 p 2 2 9 p™ + 6 D2 g pn+2 _ 2 15 p 2n+6 D I ' - 2 p 2 2 9 p n+6 D2' g pn+2 _ 2 15 p 2n+6 3. E 1 zs Q-isomorphic to one of the elliptic curves: Chapter 3. Elliptic Curves with 2-torsion and conductor 2a3l3ps 45 a 2 CZ4 A E l 0 2p2 - 2 9 p 6 E2 0 -8p2 2 1 5 p 6 FI 0 -2p2 2 9 p 6 F2 0 8p2 - 2 1 5 p 6 4. t/*zere exz'sts an integer t e {0,1} size/1! f/zaf Z? zs Q-isomorphic to one of the elliptic curves: 0 2 0 4 A GI 0 2p2t+l 29p3+6t G2 0 -8p2t+1 2 l5 p 3+6 t H I 0 -2p2t+1 2V + 6 T H2 0 8p2t+l 2 l5 p 3+6 t 3.1.2 The Proof for Conductor 2ap2 3.1.3 List of Q-isomorphism classes Let E be an elliptic curve over Q of conductor 2Mp2 with 0 <. M < 8 and having at least one rational point of order 2. We may assume that E is given by a model of the form y2 = x3 + ax2 + bx, where a and 6 are integers both divisible by p, a and b have no other common odd divisors, and that this model is minimal outside of 2. From the hypothesis on the conductor of E, there exist two natural numbers a and 5, with S > 2, such that b2(a2 - 4 6 ) = ± 2 V - (3-2) It follows that 6 / 0 and its only possible divisors are 2 and p. We consider the two cases: (i) 6 > 0, (ii) 6 < 0. Lemma 3.10 Suppose 6 > 0. Then there exists an integer d, and non-negative integers m and n satisfying one of the equations in the first column and E is Q-isomorphic to the corresponding curve in the second column, for some r\,r$ 6 {0,1}; except in cases 1,2 and 5, ivhere ifm = \ then r\ e {1,2}. Chapter 3. Elliptic Curves with 2-torsion and conductor 2 ° 3 / V 5 46 Diophantine Equation y2 = x3 + a2x2 + C14X a2 0,4 1 6? - 2mpn = ± 1 2ripd 2 m + 2 r ! - 2 p n + 2 2 d2 - 2m = ±pn 2ripd 2 m + 2 r a - 2 p 2 5 pd2 - 2m = ± 1 2ripr3+1d 10 d2 -pn = ±2m 2r^+1pd 2 2 r l p n + 2 11 d2 - 1 = 2mpn 2ri+1pd 2 2 l V 14 pd2 - 1 = 2m 2ri+1pr3+1d 2 2 r l p 2 r 3 + l 19 2d2 -pn = ± 1 2ri+2pd 2 2 r i + l p n + 2 20 2d2 - 1 = pn 2ri+2pd 2 2 r - 1 + l p 2 23 2pd2 - 1 = ± 1 2 2 r i + l p 2 r 3 + l Proof. This lemma follows immediately from Lemma A . l in Appendix A by removing the prime factor 3 from all places and setting r% = 1. Of course doing this makes a number of the rows identical, so ignoring the redundant rows we end up with the table above. The numbers in the first column of the table above are included to indicate which row of the table in Lemma A . l these rows correspond. • Similarly, from Lemma A.2 we obtain the following. Lemma 3.11 Suppose b < 0. Then there exists an integer d, and non-negative integers m and n satisfying one of the equations in the first column and E is Q-isomorphic to the corresponding curve in the second column, for some r\,r$ € {0,1}; except in case 2, where if m = 1 then r\ e {1,2}. Diophantine Equation y2 = x3 + a2x2 + C14X 0-2 2 d2 + 2m = pn 2ripd _ 2 m + 2 r i - 2 p 2 10 d2 +pn = 2m 2ri+1pd _ 2 2 r l p n + 2 11 d2 + 1 = 2mpn 2ri+1pd - 2 2 V 14 pd2 + l = 2m 2 r ' + V 3 + 1 d _ 2 2 r l p 2 r 3 + l 20 2d2 + l= pn 2 r i + V _ 2 2 r 1 + l p 2 24 2pd2 + 1 = 1 2 n + 2 p r 3 + i d _ 2 2 r ! + l p 2 r 3 + l 3.1.4 The end of the proof In this section, we verify that the elliptic curves appearing in Theorems 3.1-3.9 are the only curves, up to Q isomorphism, having the stated properties. Chapter 3. Elliptic Curves with 2-torsion and conductor 2a3/3p5 47 Our method of proof is similar to that of Ivorra [37]. It is sufficient to prove the following. (*) Let F be an elliptic curve appearing in one of the Lemmata 3.10 or 3.11. Then, F is Q-isomorphic to one of the elliptic curves appearing in Theorems 1 through 9. In fact, let N be an integer such that 0 < N < 8 and E and elliptic curve over Q of conductor 2Np2, having at least one rational point of order 2. Ac-cording to the work done in the previous section (and Appendix A), E is Q-isomorphic to an elliptic curve F appearing in Lemmas 3.10 or 3.11. It fol-lows from assertion (*) that F is thus Q-isomorphic to one of the curves in Theorems 1 through 9. Furthermore, such is also the case for E. Since E is of conductor 2Np2, it follows that E is Q-isomorphic to one of the curves in the tables of the theorem corresponding to the value of N. This finishes the proof of the theorems. Assertion (*) is a consequence of the following assertion: (**) Let F be an elliptic curve appearing in one of the lemmata 3.10 or 3.11. Let F' be the quadratic twist of F by yf—1. Then, one of the curves F and F' is Q-isomorphic to one of the elliptic curves appearing in Theorems 1 through 9. In fact, consider an elliptic curve F referenced in Lemma 3.10 or 3.11. From (**), we can suppose that F' is Q-isomorphic to one of the elliptic curves in Theorems 3.1 through 3.9. a) If F' is isomorphic to a curve in theorem 3.1, then F is isomorphic to a curve in theorem 3.4. b) Suppose that F' is Q-isomorphic to a curve in Theorems 3.2 through 3.9. b.l) If F' is isomorphic to a curve in Theorems 3.6 through 3.9, we see that the same must be true of F. b.2) If F' is isomorphic to a curve in Theorems 3.3 or 3.4, then F is isomor-phic to a curve in Theorem 3.5. b.3) If F' is isomorphic to a curve in Theorem 3.2, then F is isomorphic to a curve in Theorem 3.5. b.4) Suppose now that F' is Q-isomorphic to an elliptic curve appearing in Theorem 3.5. Chapter 3. Elliptic Curves with 2-torsion and conductor 2a3^ps 48 If F' is isomorphic to one of the curves A l or A2: if m £ {4,5}, then F is isomorphic to one of the curves A l or A2 in Theorem 3.4; if m = 6, then p = 17 and F is isomorphic to one of the curves in 3.1; if m > 7 the curve F is isomorphic to one to the curves A l or A2 of Theorem 3.2. If F' is isomorphic to one of the curves B l or B2 then the curve F is iso-morphic to one of the curves B l or B2 in Theorem 3.4. If F' is isomorphic to one of the curves CI or C2 of Theorem 3.5; if m 6 {4,5}, then F is isomorphic to one of the curves CI or C2 in Theorem 3.4; if m = 6, then p = 17 and F is isomorphic to one of the curves in 3.1; if m > 7, then F is isomorphic to one of the curves B l or B2 in Theorem 3.2. If F' is isomorphic to the curve DI or D2; if m G {4, 5}, then F is isomor-phic to the curve DI or D2 of Theorem 3.4; if m > 7, then F is isomorphic to one of the curves CI or C2 in Theorem 3.2. If F' is isomorphic to one of the curves E l or E2 then the curve F is iso-morphic to one of the curves E l or E2 in Theorem 3.4. If F' is isomorphic to one of the curves FI or F2 of Theorem 3.5; if m G {4,5}, then F is isomorphic to one of the curves FI or F2 in Theorem 3.4; if m = 6, then either p = 17 or p = d2 + 64 and F is isomorphic to one of the curves in 3.1; if m > 7, then F is isomorphic to one of the curves DI or D2 in Theorem 3.2. If F' is isomorphic to one of the curves GI or G2 then the curve F is iso-morphic to one of the curves GI or G2 in Theorem 3.4. If F' is isomorphic to one of the curves H I or H2 of Theorem 3.5; if m G {4,5}, then F is isomorphic to one of the curves H I or H2 in Theorem 3.4; if m > 7, then F is isomorphic to one of the curves E l or E2 in Theorem 3.2. If F' is isomorphic to one of the curves II or 12 of Theorem 3.5; if rn G {4,5}, then F is isomorphic to one of the curves II or 12 in Theorem 3.4; if m = 6, then p = 7 and F is isomorphic to one of the curves in 3.1; if m > 7, then F is isomorphic to one of the curves FI or F2 in Theorem 3.2. This proves assertion (*) in this case. Al l that remains now is to show that assertion (**) holds for Lemmata 3.10 and 3.11. Chapter 3. Elliptic Curves with 2-torsion and conductor 2a3l3p5 49 Assertion (**) holds for Lemma 3.10: Since assertion (**) is concerned only with the curves up to quadratic twist we may choose the sign of a2 which makes calculations most convenient. This usually involves specifying the congruence class of pd, a factor of a2, modulo 4. We will make extensive use of the tables in Chapter 2 for computing con-ductors. In what follows we will refer to the curves appearing in Lemma 3.10 by their numbers in the first column. In particular, for the Diophantine equations involving " ± " we would like to consider the curves corresponding to the "+" equation separately from the curves corresponding to the " - " equation. In the former case, we put a superscript of "+" on the curve number, and in the latter, a superscript of "—". This is made clear in the first two cases below. 1+) Suppose that (p,d,m,n)'i satisfy d2 = 2mpn + 1, and E is the elliptic curve with coefficients a2 — 2Tlpd and a4 — 2 m + 2 r i ~ 2 p n + 2 . We may assume d is such that pd = — 1 (mod 4). Thus, using the tables in Chapter 2, the conduc-tor of E is 2^2p2 where (Observe how the assumption pd = -1 (mod 4) reduced the number of possibilities for the value of f2 in the case when r\ = 0 and m > 4.) Now we can easily see that E is curve DI, A l , or DI in Theorems 3.6, 3.5, 3.7, respectively. 1") Suppose that (p, d, m, n) 2 satisfy d2 = 2mpn - 1, and E is the elliptic curve with coefficients a2 = 2T1pd and a 4 = 2 m + 2 r i ~ 2 p n + 2 . We may assume d is such that pd = — 1 (mod 4). The conductor of E is 27p2 and so E is curve A l , if r\ = 0, and curve B2', if r\ = 2, of Theorem 3.8. Notice we could have just written "E is curve A l , if r\ — 0, and curve B2', if r\ — 2" from which it should be clear that the curve A l and B2' to which we refer are the ones in Theorem 3.8, since E is of conductor 27p2. In what 'Then m > 3. 2 T h e n m 6 {0; 1}. if r\ i f n i f n 0, m = 3; 0, m > 4; l,m > 2. Chapter 3. Elliptic Curves with 2-torsion and conductor 2a3f3p6 50 follows, we will not explicitly note which of Theorems 3.1 through 3.9 we are referring; this is clear from the conductors under consideration. 2 +) Suppose that (p, d, m, n) satisfy d? = 2m + pn, and E is the elliptic curve with coefficients a2 = 2T1pd and a 4 = 2m+2ri~2p2. We may assume d is such that pd = — 1 (mod 4). Thus, from Theorem 2.1, the conductor of E is 2Ep2 where h = 3 if r\ = o, m = 2; 5 if ri = 0, m = 3; 4 if r\ = 0, m > 4; < 7 if T*I = 1, rn = 1; 6 if r\ = 1. m > 2; 7 if r\ -2, m = 1. Thus E is curve BI, E l , C l , C l , E l or B2', respectively. 2~) Suppose that (p,d,m,n)3 satisfy d2 — 2m — pn, and E is the elliptic curve with coefficients a2 — 2ripd and a 4 = 2rn+2ri~2p2. We may assume d is such that pd=-l (mod 4). The conductor of E is 2^2p2 where {5 if r\ = 0, rn, = 3; 4 if n = 0, m > 4; 6 if n = l , m > 2; Thus £ is curve F l , DI, or F l , respectively. 5 +) Suppose that (p,d,m,n)4 satisfy pd2 — 2rn + 1, and E is the elliptic curve with coefficients a2 — 2Tlpr'i+id and o,4 = 2 m + 2 r i _ 2 p 2 r ' 3 + 1 . We may as-sume d is such that pT'i+1d = —1 (mod 4). The conductor of E is 2^ 2 p 2 where {3 i f n =0,TO = 2; 4 if n = 0, rn > 4; 6 if n = 1, 777 > 2. Thus, E is curve GI (if n = 0, m = 2), HI (if n = 0, m > 4) and HI or HI' (if n = 1, 777 > 2). 5~) Suppose that (p,d,m,n)5 satisfy pd2 = 2m - 1, and E is the elliptic 3 Then m > 3. 4 T h e n m ^ 1,3. 5 Then m > 3. Chapter 3. Elliptic Curves with 2-torsion and conductor 2a3l3p5 51 curve with coefficients a2 — 2 n p r 3 + 1 d and a4 = 27n+2ri~~2p2r3+1. We may as-sume d is such that p7'3+ld = — 1 (mod 4). The conductor of E is 2^ 2p 2 where {5 if n = 0, m = 3; 4 if n = 0,m > 4; 6 i f r i = l , m > 2 . Thus £ is curve HI or H I ' (if n = 0, m = 3), II (if n = 0, m > 4) and II or II' (if r1 = l,rn> 2). 10+) Suppose that (p, cf, 777, n) 6 satisfy d 2 = p" + 2 m , and E is the elliptic curve with coefficients a2 = 2Tl+1pd and a 4 = 2 2 n p r , + 2 . We may assume d is such that pd = 1 (mod 4). The conductor of E is 2^ 2p 2 where /2 = 7 if r\ = 0. m = 1 4 if r i = 0 m = 2 5 if r i = 0 m = 3 < 4 if r\ = 0 rn > 4 7 if rj = 1 m = 1 6 if r\ = 1 m > 2 Thus E is curve DI (or DI'), BI, E2 (or E2'), C2, E2 (or E2'), and C2 (or CT), respectively. 10") Suppose that (p,d,m,n) satisfy d 2 = pn - 2rn, and E is the elliptic curve with coefficients a,2 — 2 n + 1 p d and a4 = 2 2 n p n + 2 . We may assume d is such that pd = 1 (mod 4). The conductor of E is 2^ 2p 2 where 6 if r i = o, 777 = 0; 7 if r\ = o, 777 = i ; 4 if rj = 0, 777 = 2; 5 if 7-j = 0, 777 = 3; 4 if r\ 777 > 4; 5 if r i = 1, 777 = 0; 7 if r\ = 1, 777 = 1; 6 if r i = 1, 777 > 2. 6Then m ^ 0. Chapter 3. Elliptic Curves with 2-torsion and conductor 2a3l3p5 52 Thus E is curve A l (or Al ' ) , GI (or GI'), E2, G2 (or G2'), F2, A2 (or A2'), G2 (G2'), and H2 (or H2'). 11) Suppose that (p, d, ?n, ra)7 satisfy d2 = 2mpn + 1, and E is the elliptic curve with coefficients a2 = 2Tl+lpd and 0,4 = 2 2 n p 2 . We may assume d is such that pd = 1 (mod 4). The conductor of E is 2^2p2 where Thus E is curve D2 (or D2'), A2, and D2 (or D2'), respectively. 14) Suppose that (p, d, m. n) 8 satisfy pd2 = 2rn + 1, and E is the elliptic curve with coefficients a2 = 2 r i + 1 p ' 3 + 1 d and a 4 = 2 2 n p 2 r 3 + 1 . We may assume d is such that p 7 ' 3 + 1 d = 1 (mod 4). The conductor of E is 2^2p2 where Thus E is curve G2, H2, H2 (or H2'), respectively. 19+) Suppose that (p,d,m,n) satisfy 2d2 = pn + 1, and E is the elliptic curve with coefficients a2 = 2 n + 2 p d and a 4 = 2 2 r i + 1 p n + 2 . The conductor of 1? is 2 V / thus £" is curve B l (or Bl') if n = 0, and A2 (or A2') if r a = 1. 19~) Suppose that (p,d,rn,ri) satisfy 2d2 = pn — 1, and E is the elliptic curve with coefficients a2 = 2ri+2pd and a 4 = 22ri+lpn+2. The conductor of E is 2 V / * u s £ is curve CI (or CI') if n = 0, and D2 (or D2') if n = 1. 20) Suppose that (p, d, m, ra) satisfy 2d2 = p" +1, and I? is the elliptic curve with coefficients 0 2 = 2 r i + 2 p d and a 4 = 2 2 r i + 1 p 2 . The conductor of E is 28p2, thus £ is curve A l (or Al ' ) if n = 0, and B2 (or B2') if n = 1. 23+) There are no solutions to 2pd2 = 1 + 1 so we have no curves corre-sponding to this case. 23~) Suppose that (p, d, m, n) satisfy 2pd2 = 1 — 1, then d = 0, and E is the elliptic curve with coefficients a2 = 0 and a 4 = 2 2 r i p 2 r 3 + 1 . The conductor of E is 2 8p 2 and E is the curve Gl if r\ = 0 or the curve H2 if r\ = 1. 7Then m > 3. 8Thenm ^ 0,1,3. 0,m = 2; 0, m > 4; 1, m > 2. Chapter 3. Elliptic Curves with 2-torsion and conductor 2a3l3p5 53 This completes the proof that assertion (**) is satisfied for all curves in lemma 3.10. Assertion (*•) holds for Lemma 3.11: 2) Suppose that (p, d,m,n) satisfy d 2 = pn — 2m, and E is the elliptic curve with coefficients a2 = 2Tlpd and aA = —2m+2ri~2p2. We may assume d is such that pd = — 1 (mod 4). Thus, using the tables in chapter 2, the conductor of E is 2^2p2 where f 4 if r i = 0, m = 2; 5 if r\ = 0, m = 3; 4 if r\ = 0,77i > 4; 7 if r i = 1, m = 1; 6 if n = l ,m > 2; 7 if r i = 2,777 = 1. Thus E is curve E l , GI , F l , H I (or HI'), GI (or GI') and GI (or GI'), respec-tively. 10) Suppose that (p, d, rn, rif satisfy d2 = 2rn — pn, and E is the elliptic curve with coefficients a2 — 2 r i + 1 p d and a4 = - 2 2 r i p n + 2 . We may assume d is such that pd = 1 (mod 4). The conductor of E is 2^2p2 where 5 if r i = 0, 777 = 0 (i.e. 77 = 0) 7 if 7*i = 0,777 = 1; 5 if 7*1 = 0,777 = 3; 4 if 7*1 = 0,777 > 4; 6 if ?*i = l , m = 0 (i.e. 77 = 0) 7 if r i = 1, 777 = 1; 6 if 7*1 = 1, 777 > 2. Thus E is curve A l (or Al ' ) , F l (or Fl'), F2 (or F2'), D2, A2 (or A2'), and F2 (or F2'), E2 (or E2') respectively. 9Then m ^ 2. Chapter 3. Elliptic Curves with 2-torsion and conductor 2a3l3p5 54 11) Suppose that (p, d, m, n)w satisfy d2 = 2mpn - 1, and E is the elliptic curve with coefficients a2 = 2ri+1pd and 0,4 = —2 2 r ip 2. We may assume d is such that pd = 1 (mod 4). The conductor of E is 2^2p2 where 5 if r\ — 0, m - 0; 7 if r i — 0, m = i ; 6 if r*i = 1, m = 0; 7 if r\ = l , m = i ; Thus E is curve A l (or Al ' ) , BI (or BI'), A2 (or A2') and A2 (or A2'), respec-tively. 14) Suppose that (p, d, m, n)n satisfy pd2 = 2m — 1, and E is the ellip-tic curve with coefficients a2 — 2 n + 1 p r 3 + 1 d and 0,4 = - 2 2 r i p 2 r 3 + 1 . We may assume d is such that pT'i+1d = 1 (mod 4). The conductor of E is 2-^2p2 where 6 if r\ = 0, rn = 0,77 = -1 (mod 4) 5 if 7*1 = 0, rn = 0,p = 1 (mod 4); 5 if r\ — 0, m = 3; 4 if r\ = 0,777 > 4; 5 if r\ = 1,777 = 0,p = -1 (mod 4) 6 if r\ = 1, 777 = 0,p = 1 (mod 4); 6 v. if r ] = 1, 777 > 2. Thus E is curve C l , BI, H2 (or H2'), 12, C2, B2,12 (or H2'), respectively. 20) Suppose that (p, d, m, 77) satisfy 2d2 = pn — 1, and E is the elliptic curve with coefficients a2 = 2 r i + 2 p d a n d 0.4 = - 2 2 r i + 1 p 2 - The conductor of E is 2 8p 2, thus E is curve DI (or DI') if n = 0, and C2 (or C2') if n = 1. 24) Suppose that (p, d, m, n) satisfy 2pd2 = 1 - 1 , then d = 0, and E is the elliptic curve with coefficients a2 = 0 and 0 4 = — 2 2 r i p 2 7 ' 3 + 1 . The conductor of E is 2 8p 2 and E is the curve HI if r\ = 0 or the curve G2 if 7*1 = 1. This completes the proof that assertion (•*) is satisfied for all curves in lemma 3.11. This completes the proof of Theorems 3.1 through 3.9. 'Then 777 < 1. •Then m ^ 1,2. Chapter 3. Elliptic Curves with 2-torsion and conductor 2a3l3ps 55 3.2 C u r v e s o f C o n d u c t o r 2a3pp As we mentioned in the introduction to this chapter, the models presented in the following table are minimal except in the case when the conductor is not divisible by 4. In these cases (i.e. Theorems 3.12 and 3.13) the model is minimal except at 2, and a minimal model can be found using Corollary 2.2. We choose not to do this here. T h e o r e m 3.12 The elliptic curves E defined over Q, of conductor 3bp, and having at least one rational point of order 2, are the ones such that one of the following conditions is satisfied: 1. there exist integers £ > 2 — b and n > 1 such that 2&3epn + 1 is a square and E is Q-isomorphic to one of the elliptic curves: a 2 a 4 A A l e • 3 f t" V 2 6 3 V + 1 243^+2(fc-l)pn 2l232f+6(b-l)p27J A2 -e • 2 • 3 b" V 2 < 5 3 V + 1 32(6-1) 2 1 2 3 f + 6 ( 6 - l ) n where e G {±1} is the residue of3b 1 modulo 4. 2. there exist integers £ > 2 — b and n > 1 such that 263e + pn is a square and E is Q-isomorphic to one of the elliptic curves: 0.2 1I4 A B l e-3 f a -V 2 6 3 f +Pn 243<?+2(b-l) 21232«?+6(b-l) n B2 - e - 2 - 3 6 - V 2 6 ^ + P " 32(b-iy 2123^+6(6-l )p2n where e G {±1} is the residue of 3b 1 modulo 4. 3. there exist integers £ > 2 - b and n > 1 such that 263e — pn is a square and E • is Q-isomorphic to one of the elliptic curves: 0.2 04 A CI e ^ - V 2 ^ ~Pn 243<M-2(fe-l) _ 21232<*+6(6-l)pn C2 -c-2-3h-1y/263i -pn _ 3 2 ( b - l ) p n 2 1 2 3 « + 6 ( b - l ) p 2 7 i where e G {±1} is the residue of3b 1 modulo 4. Chapter 3. Elliptic Curves with 2-torsion and conductor 2a3l5p5 56 4. there exist integers £>2 — b and n > 1 such that 26pn + 3l is a square and E is Q-isomorphic to one of the elliptic curves: a2 0 4 A DI e • 3 b -V2V + 3* 2 4 3 2 ( 6 - i y 2 1 2 3 ^ + 6 ( b - l ) p 2 n D2 - e - 2 - 3 b - V 2 V + 3 ^ 3^+2 (b - l ) 2 1 2 3 2 ' H - 6 ( b - l ) ? : ) n where e e {±1} is the residue of3b 1 modulo 4. 5. there exist integers £>2-b and n > 1 such that 2 6 + 3epn is a square and E is Q-isomorphic to one of the elliptic curves: a2 a.4 A E l £-3 6 -V2 6 + 3 V 2 4 3 2 ( 6 - l ) 2 123«?+6(b-l)pn E2 -e-2-3 ! ,- 1v /2 6 + 3 V 3<'+2(6-])pn 2 1 2 3 2 € + 6 ( b - l ) p 2 n where e e {±1} is the residue of3b 1 modulo 4. 6. there exist integers £ > 2 — b and n > 1 such that 2 6 — 3epn is a square and E is Q-isomorphic to one of the elliptic curves: a2 0,4 A F l e - 3 6 - V 2 6 - 3 V 2 4 3 2 ( 6 - l ) _ 2 l 2 g - ! + 6 ( 6 - l ) n F2 -t • 2 • 3h~'iyj2fi - 3epn _^t+2(b-l)pn 2 1 2 3 2 ^ + 6 ( f t - I ) p 2 n where e € {±1} is the residue of3b 1 modulo 4. 7. there exist integers £ > 2 — b and n > 1 such that 3e — 26pn is a square and E is Q-isomorphic to one of the elliptic curves: 0,2 0,4 A GI _ 2 4 3 2 ( 6 - l ) p „ 2 1 2 3 ( ; + 6 ( b - l ) p 2 n G2 - e - 2 - 3 b - V 3 ( - 2 y 3^+2 (6 -1) _ 2 1232/;+6(b-l)pn where e € {±1} is the residue of3b 1 modulo 4. 8. there exist integers £ > 2 — b and n > 1 such that pn — 263e is a square and E is Q-isomorphic to one of the elliptic curves: Chapter 3. Elliptic Curves with 2-torsion and conductor 2a3f3p6 57 «2 a 4 A H I e • 36" VP™ " 263£ -2 4 3 £ + 2 ( B _ 1 ) 21232r?+6(6-l) pn H2 -e • 2 • 3FT" V P " - 2 6 3 ^ 32(i.-l)pn _2l23t»+6 (b - l ) ? J 2r t w/zere e G {±1} zs r/ze residue of3b 1 modulo 4. In fne rase iTzaf b = 2, i.e. N = 2 • 3 2p 2, rx>e furthermore could have one of the following conditions satisfied: 9. there exist integers n > 1 and s £ {0,1} such that 2 is a square and E is Q-isomorphic to one of the elliptic curves: 0-2 a 4 A 11 24 32.s+l 2 1 23 3 + 6> N 12 3 2 s + V 2 1 23 3 + 6> 2 N where e £ {±1} is the residue o /3 s + 1 modulo 4. 20. f/zere exz'sf integers n > 1 and s G {0,1} sizc/z zTzaf 2 ~ p zs a square and E is Q-isomorphic to one of the elliptic curves: a 2 (I4 A n 2432'5+1 _ 2 12 3 3+6. 5 p n J2 - E . 2 . 3^V^ -32s+1pn 21233+6Spn where e £ {±1} is the residue of3s+1 modulo 4. 11. there exist integers n > 1 and s 6 {0,1} szicn tttflf p ~ 2 zs fl square and E is Q-isomorphic to one of the elliptic curves: 02 a 4 A K I £ . 3 ^ V ^ -243 2 S + 1 21233+6spn K2 _ e. 2 . 3 - + ^ ^ 32s+1pn _ 2 12 3 3+6 S p 2n where e G {±1} is the residue of 3 S + 1 modulo 4. Chapter 3. Elliptic Curves with 2-torsion and conductor 2 a 3 / V 5 58 T h e o r e m 3.13 The elliptic curves E defined over Q, of conductor 2 • 3bp, and having at least one rational point of order 2, are the ones such that one of the following conditions is satisfied: 1. there exist integers m>7,£>2-b and n > 1 such that 2m3epn + 1 is a square and E is Q-isomorphic to one of the elliptic curves: (12 0,4 A A l € - 3 b - V 2 m 3 V + l 2rn-2^e+2{b-l)pn 22m32€+6(6-l)p2n A2 - e - 2 - 3 b - 1 v / 2 m 3 y i + l 3 2(6-l) 2m+6^+6(t-l)pn where e G {±1} zs the residue of3b 1 modulo 4. 2. there exist integers m > 7, £ > 2 — b and n > 1 such that 2m3e + pn is a square and E is Q-isomorphic to one of the elliptic curves: a2 0,4 A BI t • 3h-1y/2m3e- +pn 2m-2<^t+2{b-l) 22m32/?+6(i)-l)pn B2 - e - 2 • 3h~1^2m3e+pn 3 2 ( 6 - l ) p n 2m+63<?+6(6-l)p2n where e G {±1} is the residue of3b 1 modulo 4. 3. there exist integers m > 7, £ > 2 — b and n > 1 such that 2m3i - pn is a square and E is Q-isomorphic to one of the elliptic curves: a2 0,4 A C l e • 3b~l y/2m3e -pn 2?n-23^+2(')-l) _22m 32/!+6(b-l)pn C2 -e-2-3b-1y/2m3e -pn _ 3 2 ( 6 - i y 2^1+6^1+6(6-\)p2n where e G {±1} zs the residue of3b 1 modulo 4. 4. there exist integers m > 7, £ > 2 — b and n > 1 such that 2mpn + 3e is a square and E is Q-isomorphic to one of the elliptic curves: a.2 0,4 A DI e • 3h~1^2mpn + 3e 2m-2 3 2(6-l)p7i 22m3^+6(6-l)p2n D2 -t-2-3h-1^/2mpn + 3i 3<+2(6-l) 2m+632<;+6(b-l)pn where e G {±1} is the residue of3b 1 modulo 4. Chapter 3. Elliptic Curves with 2-torsion and conductor 2a30ps 59 5. there exist integers m > 7 , £ > 2 - b and n > 1 such that 2m + 3 pn is a square and E is Q-isomorphic to one of the elliptic curves: a 2 <z4 A E l e- 3 b - V 2 m + 3 V 2 1 1 7 , - 2 3 2 ( 6 - 1 ) 2 2 m 3 £ + 6 ( b - l ) p n E2 -e • 2 - 3 6 - 1 v / 2 m + 3 V 3 < ? + 2 ( 6 - l ) p n 2 7 7 7 + 6 3 2 ^ 6 ( 6 - 1 ) ^ 2 7 7 where t e {±1} zs zTze residue of3b 1 modulo 4. 6. t/zere exz'sf integers m > 7, £ > 2 — b and n > 1 swc« z7zaf 2 m — 3^>n zs fl square and E is Q-isomorphic to one of the elliptic curves: a 2 a 4 A FI « - 3 b - 1 v / 2 m - 3 V 2 7 7 1 - 2 3 2 ( 6 - 1 ) _ 2 2 " 7 . 3^ + 6 ( 6 - l ) p 7 l F2 - e - 2 - 3 b - V 2 m - 3 V _ 3 « + 2 ( 6 - l ) p 7 7 2 7 7 i + 6 3 2 » ? + 6 ( b - l ) p 2 n where e € {±1} is the residue of3b 1 modulo 4. 7. f/zere exz'sf integers m > 7, £ > 2 — b and n > 1 swdz t/taf 3epn — 2 m zs a square and E zs Q-isomorphic to one of the elliptic curves: a 2 a 4 A G I f • ^ V ^ " - 2m _ 2 7 7 7 - 2 3 2 ( 6 - l ) 2277737? + 6 ( 6 - l ) p 7 1 G2 -e • 2 • 3 6 - V 3 V - 2m 3«+2(6 - l ) n _ 2 7 7 7.+632<?+6(6 - l)p2n where e £ {±1} is the residue of3b 1 modulo 4. 8. there exist integers m > 7, £ > 2 — b and n > 1 such that 3e - 2rnpn is a square and E is Q-isomorphic to one of the elliptic curves: « 2 a 4 A H I e -3b-^3e - 2mpn _ 2 m ~ 2 3 2 ( 6 - 1) n 2 2 m 3 < ? + 6 ( 6 - l ) p 2 n • H2 -e • 2 • 3h-1^/3e - 2mpn 3 < + 2 ( 6 - l ) —2m+632e+6(b-1)pn where e e {±1} is the residue of3b 1 modulo 4. 9. there exist integers m > 7, £ > 2 — b and n > 1 such that pn — 2m3e is a square and E is Q-isomorphic to one of the elliptic curves: Chapter 3. Elliptic Curves with 2-torsion and conductor 2a3l3ps 60 0,2 aA A 11 e • 3b~ V P " " 2 M ^ — 2 T O - 2 3 ^ + 2 ( h - 1 ) 2 m32£+6(b-l)pn 12 - e • 2 • 3 6 - 1 yV - 2m3* 32(b-l)pn _2*n+63<+6(b-l)p2n wnere e e {±1} is the residue of3b 1 modulo 4. In fne case i«af b — 2, i.e. N = 2 • 32-p, we furthermore could have one of the following conditions satisfied: 10. there exist integers m > 7, n > 1 and s € {0,1} such that 2 g P is a square and E is Q-isomorphic to one of the elliptic curves: 02 04 A JI 2m-232.s+l 2 2 ?n 3+6 S p T i J2 e - 2 . 3 s + y ^ ± ^ 3 2 ,+ lpn 2"?-+6^3+6sp2n wnere e G {±1} is the residue of3s+l modulo 4. 11. there exist integers m > 7, n > 1 and s G {0,1} swctt fnaf 2 r " ~ ? / l fs a square and E is Q-isomorphic to one of the elliptic curves: a2 a 4 A KI ^2"»-232s+l 22TnQ3+6spn K2 _ e . 2 . 3 - 5 + y ^ _32»+lpn 2 rn,+6 2 3-(-6-f p 2 77. wner-? e G {±1} fs fne residue o / 3 s + 1 modulo 4. 22. inere em'sf integers m > 7, n > 1 and s G {0,1} swcn fnaf -""g2 1 is a square and E is Q-isomorphic to one of the elliptic curves: 02 04 A LI _2m-232.s+l 2 2m 2 3+6 n L2 3 2 s + 1 p n 2rn+^33+6-5p2n wnere e € {±1} is the residue of3s+1 modulo 4. Chapter 3. Elliptic Curves with 2-torsion and conductor 2a3l3ps 61 Theorem 3.14 The elliptic curves E defined over Q, of conductor 223bp, and having at least one rational point of order 2, are the ones such that one of the following conditions is satisfied: 1. there exist integers £ > 2 - b and n > 1 such that 4 • 3^  + pn is a square, 3e = -1 (mod 4) and E is Q-isomorphic to one of the elliptic curves: 0-2 04 A A l 3^+2(fa-l) 2432/J+6(6-l)pn A2 -e-2-3 b-V4-3^+p n 3 2 ( 6-l) pn 283»M-6(b-l) p2n where e G { ± 1 } is the residue of3b 1 modulo 4. 2. there exist integers £ > 2 — b and n > 1 such that 4 • 3e - pn is a square, 3e = — 1 (mod 4) and E is Q-isomorphic to one of the elliptic curves: a2 a 4 A Bl e-3 b -V 4-3^-p n 3^+2(6-1) -2 43 M + 6 ( f t ~ 1 ) p n B2 -e • 2 • 3 b " V 4 - 3' -p" _ 3 2 ( 6 - l ) p n 283<'+6(6-l)p2n where e £ {±1} is the residue of3b 1 modulo 4. 3. there exist integers £ > 2 — b and n > 1 such that 4pn — 3e is a square, pn = — 1 (mod 4), and E is Q-isomorphic to one of the elliptic curves: «2 a 4 A C I e • 3 b " V 4 P n ~ 3^ 3 2 ( 6 - l ) p n _ 2 4 3 « + 6 ( f c - l ) p 2 n C2 -e • 2 - 36" V 4 P " ~ 3 ' _3^+2(b-l) 2 8 3 2 f + 6(b-1 ) n where e G {±1} is the residue of3b 1 modulo 4. 4. there exist integers £ > 2 — b and n > 1 such that pn — 4 • 3e is a square, 3e = 1 (mod 4), and E is Q-isomorphic to one of the elliptic curves: a 2 0 4 A DI e-3b~1y/pn - 4-3e ' _ 3 « + 2 ( 6-l) 2 4 3 2 ^ + 6 ( 6 - 1) n D2 -e-2-3b-1^pn - 4-3e 3 2 ( b - l ) p n _ 2 8 3 £ + 6 ( 6 - l ) p 2 n Chapter 3. Elliptic Curves with 2-torsion and conductor 2ct3l3p5 62 where e £ { ± 1 } is the residue of3b 1 modulo 4. In the case that 6 = 2, i.e. N = 2 2 3 2 p , we furthermore could have one of the following conditions satisfied: 5. there exist integers n > 1 and s £ {0,1} such that A p n ~ l is a square, pn = 1 (mod 4), and E is Q-isomorphic to one of the elliptic curves: a-2 a 4 A E l 3 2 s + 1 p n - 2 4 3 3 + 6 V E2 e - 2 - 3 ' 5 + 1 v / 4 ^ i - 3 2 s + 1 2 8 3 3 + 6 s p " where e £ { ± 1 } is the residue of3s+1 modulo 4. 6. there exists an integers n > 1 and s £ {0,1} swcft f/zaf p 2f4 is a square and E is Q-isomorphic to one of the elliptic curves: a-2 0,4 A FI 3 2 s + l 2 4 3 3 + 6 s p n F2 32s+1pn 2 8 3 3 + 6 V n where e £ { ± 1 } is the residue o / 3 s + 1 modulo 4. Theorem 3.15 The elliptic curves E defined over Q, of conductor 2z3bp, and having at least one rational point of order 2, are the ones such that one of the following conditions is satisfied: 1. there exist integers m £ {4,5}, t > 2 — 6 and n > 1 such that 2m3epn + 1 is a square and E is Q-isomorphic to one of the elliptic curves: a2 a4 A A l e-3h-1y/2m3epn + 1 2m~2^e+2{b-l)pn 22m32(!+6(b-l) p2n A2 -e • 2 • 3 f c - V 2 m 3 V + 1 32(6-l) 2m+6^(+6(b-l)pTi where e £ { ± 1 } is the residue of3b 1 modulo 4. 2. there exist integers £ > 2 — b and n > 1 such that 4 • 3^ + pn is a square, 3e = 1 (mod 4), and E is Q-isomorphic to one of the elliptic curves: Chapter 3. Elliptic Curves with 2-torsion and conductor 2a3@ps 63 a2 a.4 A BI -e-3h-1y/4-3e+pn 3 £ + 2(b-l) 24 3 2/?+6(b-l) pn B2 e - 2 - 3 b - V 4 ' 3 * +pn 3 2 ( b - l ) p n 2 8 3 ^ + 6 ( b - l ) p 2 n where e e {±1} is the residue of3b 1 modulo 4. 3. there exist integers m e { 4 , 5 } , ! > 2 - [ i and n > 1 swc/t tTzat 2m3< ? + -p™ zs a square and E is Q-isomorphic to one of the elliptic curves: 0,2 a 4 A C l e • 3h~l •s/2m3t + pn 2m - 2 ^ + 2(t-l) 22m 32^+6(b-l) pn . C2 - e - 2 - 3 ' , - 1 v / 2 " 1 ' 3 * + p " 3 2'b-l-yi 2m+63,;+6(6-l)p2n where e £ {±1} is the residue of3b 1 modulo 4. 4. i/zere exzsi integers £ > 2 — 6 and n > 1 swc« rTzaf 4 • 3e — pn is a square, 3e = 1 (mod 4), and JE zs Q-isomorphic to one of the elliptic curves: a 2 0 4 A DI - e - 3 ( , - 1 v / 4 - 3 < - - p n 3f+ 2(b-l) _ _ 24 3 2 f + 6(b-l)pn D2 e - 2 - 3 f c - V 4 - 3 f - p n _ 3 2 ( 6 - i y 2 8 3 ^ + 6 ( b - l ) p 2 n where e 6 {±1} zs the residue of3b 1 modulo 4. 5. there exist integers m £ {4,5}, £ > 2 — b and n > 1 SMC/? r7za£ 2m3< ? — p n is a square and E is Q-isomorphic to one of the elliptic curves: a 2 114 A E l £ - 3 b - V 2 m 3 ^ - p " 2m.-2 3'!+2(b-l) E2 - e - 2 - 3 b - V 2 m 3 f - p n „ 3 2 ( b - i y 2m+63<,+6(6-l) p2n where e 6 {±1} zs the residue of3b 1 modulo 4. 6. frtere exz'st integers m £ {4,5}, £ > 2 — b and n > 1 swdz tTzflt 2 m p n + 3£ zs a square and E is Q-isomorphic to one of the elliptic curves: a 2 0 4 A F l e - 3 b - 1 v / 2 m P n + 3 £ 2 m - 2 3 2(b-l) pn 22m3<?+6(b-l)p2n F2 3^+2(6-1) 2Tn+632£+6(b-l)p« Chapter 3. Elliptic Curves with 2-torsion and conductor 2a3l3ps 64 where e £ {±1} is the residue o/3 b 1 modulo 4. 7. there exist integers £ > 2 — b and n > 1 such that 4pn - 3e is a square, pn = 1 (mod 4), and E is Q-isomorphic to one of the elliptic curves: « 2 0 4 A GI -e • 3 b - J v^p™ - 3£ 3 2 ( 6 - l ) p n G2 e • 2 • 3 b" V 4 P n - 3^  _ 3 ^ + 2 ( 6 - ] ) 2832f+6(6-l) pn zy/zere £ £ {±1} is the residue o/3 h 1 modulo 4. 8. frtere exist integers £ > 2 — b and n > 1 such that 4 + 3*pn is a square and E is Q-isotnorphic to one of the elliptic curves: a-2 o 4 A HI -e • 3 6 " 1 v / 4 + 3 V 3 2 ( 6 - 1 ) 2 4 3^+6(6-1) n H2 £-2 -3 ( , -V 4 + 3 V 3«+2(b-l) n 2 8 3 2 « + 6 ( 6 - l ) p 2 n w/tere E G { ± 1 } is r«e residue o/3 b 1 modulo 4. 9. frtere exzsf integers m G {4,5}, ^ > 2 — b and n>\ such that 2rn + 3epn is a square and E is Q-isomorphic to one of the elliptic curves: a-2 04 A 11 £ - 3 b - V 2 m + 3 V 2 m-232(b - l ) 2 2m 3 < ?+6())-l)pTi 12 - £ - 2 - 3 b " V 2 m + 3 V 3M-2(fc-l) « 2 m+632f+6(6 - l )p2n where c G {±1} is the residue o/3 b 1 modulo 4. 10. there exist integers m G {4,5}, £ > 2 — b and n > 1 such that 2rn — 3epn is a square and E is Q-isomorphic to one of the elliptic curves: a2 04 A Jl £ • 3 6 - 1 y/2m - 3 V 2m-232(fc-l) _22m3f+6(b-l)pn J2 -e • 2 • 3 b _ 1 yj2m - 3V _3f+2(fe-l)pn 2?n+632(!+6(6-l)p2n where t G {±1} is the residue ofZh 1 modulo 4. 11. there exist integers rn G {4,5}, £ > 2 — b and n > 1 such that 3epn — 2m is a square and E is Q-isomorphic to one of the elliptic curves: Chapter 3. Elliptic Curves with 2-torsion and conductor 2a3l3p5 65 a-2 a4 A K I e • 31"1 y/3epn - 2m _2m-2g2(6 - l ) 22m^e+6(b-l)pn K2 -e - 2 • 3 b _ V 3 V - 2 m 3<?+2(b-l)pn _2m+6g2e+6(b-l) p2n where e G {±1} zs the residue of3b 1 modulo 4. 22. fnere exfsf integers m G {4,5}, £ > 2 — b and n > 1 swc« t«ar 3^  - 2mpn is a square and E is Q-isomorphic to one of the elliptic curves: a-2 a.4 A L I e • 3 f a" V 3 * - 2mpn -2m~ 23 2( 6 _ 1)p n 22m^e+6(b- 1) p2n L2 -e-2-3b-1s/3e -2mpn %e+2(b-i) _2m+632t'+6(fe-l)pn w/zer.? e G {±1} zs tTze residue o/3b 1 modulo 4. 23. fnere exz'sf integers £ > 2 — b and n > 1 swc/z (7zaf pn — 4 • 3e is a square, 3e = — 1 (mod 4), and £ zs Q-isomorphic to one of the elliptic curves: a 2 a.4 A M l _ 3 « + 2 ( 6 - l ) 2432*!+6(f>-l)pn M 2 e-2-3h-1y/pn - 4-3( 3 2 ( 6 - l ) p n _ 2 83<f+6(6-l)p2n zw/zere e G {±1} is the residue of3b 1 modulo 4. 14. there exist integers m G {4, 5}, £ > 2 — b and n > 1 szzrfz that pn — 2rn3e is a square and E is Q-isomorphic to one of the elliptic curves: a-2 CJ4 A N I e-3h~1y/pn - 2m3l _ 2m-23«"+2(()-l) 2 2m 3 2£+6 ( fc - l )pn N 2 £ • 2 • 3 b " V P " - 2 M 3 ^ 3 2 ( 6 - l ) p n _ 2 m+63<?+6(6- l)p2n w/zere e G {±1} is the residue of3b 1 modulo 4. In the case that 6 = 2, i.e. N = 2332p, we furthermore could have one of the following conditions satisfied: 15. there exist integers m G {4,5}, n > 1, and s G {0,1} such that 2 "+P" is a square and E is Q-isomorphic to one of the elliptic curves: Chapter 3. Elliptic Curves with 2-torsion and conductor 2"2>^p5 66 0.2 0,4 A O l 2m-2g2s+l 22m -^3+6Spn 0 2 e - 2 - 3 a + 1 ^ ± 2 l 3 2 s + V 2 m + 6 2 3+6 s p 2 n where e 6 {±1} is the residue o /3 s + 1 modulo 4. 26. f/zere exz'sf integers m G {4, 5}, n > 1, and s € {0,1} swcn tTztzi 2 "3-0 is a square and E is Q-isomorphic to one of the elliptic curves: a2 a 4 A PI 2m-232s+l _22m-^3+6,Spn P2 _ 3 2 5 + l p n 2^71+6 ^3+6.Sp2n i^ nere e € {±1} is the residue o /3 s + 1 modulo 4. 27. f«ere exz'sf integers n > 1 and s G {0,1} such that 4p"3~1 is a square and E is Q-isomorphic to one of the elliptic curves: a2 0,4 A Q i 32s+1pn - 2 4 3 3 + 6 s p 2 n Q2 - 3 2 s + 1 2 8 3 3 + 6 s p n where e € {±1} is the residue o /3 s + 1 modulo 4. 28. f/tere exz'sf integers n > 1 and s 6 {0,1} such that is a square and E is Q-isomorphic to one of the elliptic curves: a2 0,4 A Rl — 3 2 s + 1 2 4 3 3 + 6 ' 5 p n Pv2 t-2-3S+1yJ^ 3 2 s + 1 p ™ - 2 8 3 3 + 6 V n where e € {±1} zs residue o / 3 s + 1 modulo 4. 29. f/zere exz'sr; integers m G {4, 5}, n > 1, and s G {0,1} swcn z7za£ p 3 2 zs a square and E is Q-isomorphic to one of the elliptic curves: Chapter 3. Elliptic Curves with 2-torsion and conductor 2a3l3ps 67 a-2 a4 A SI _2m-2g2s+l S2 32s+lpn 2m+6^3-f-6Sp2n where e e {±1} is the residue of 3 S + 1 modulo 4. T h e o r e m 3.16 The elliptic curves E defined over Q, of conductor 243bp, and having at least one rational point of order 2, are the ones such that one of the following conditions is satisfied: 1. there exist integers m > 4, £ > 2 - b, and n > 1 such that 2rn3ipn + 1 is a square and E is Q-isomorphic to one of the elliptic curves: a2 a4 A A l - e • 3b-1y/2m3epn + 1 22Tn32<?+6(6-l)p2n A2 £ • 2 • 3 b - 1 y / 2 m 3 e p n + 1 3 2(b-l) 2m+6^i+6(b-l) n where e e. {±1} is the residue of3b ] modulo 4. 2. there exist integers £ > 2 — b and n > 1 such that 4 -3e + pn is a square and E is Q-isomorphic to one of the elliptic curves: 02 0,4 A Bl e-3b-1y/4-3e + pn 3*+2(b-l) 2432<!+6(b~l)pn B2 - £ - 2 - 3 6 - 1 v / 4 - 3 f + p n . 3 2(6- l ) p n 28 3(!+6(b-l)p2n where £ 6 { ± 1 } is the residue of3e+b 1 modulo 4. 3. there exist integers rn > 4, £ > 2 — b, and n > 1 such that 2m3e + pn is a square and E is Q-isomorphic to one of the elliptic curves: 02 04 A CI -t-3h-x sj2m3l +pn 2m-23«f+2(6-l) 22?n32i?+6(b-l)pn C2 £ • 2 • 3b" V2 m 3 £ + p n 3 2 (b- l ) p n 2m+63<?+6(6-l)p2ri w/iere e € {±1} is rTze residue of3b 1 modulo 4. 4. there exist integers £ > 2 — b and n > 1 such that 4 • 3e — pn is a square and E is Q-isomorphic to one of the elliptic curves: Chapter 3. Elliptic Curves with 2-torsion and conductor 2a3l3ps 68 0-2 a.4 A D I e-3b~l^4-3l -pn - 2 4 3 M + 6 ( f t " 1 ) p n D2 - e - 2 - 3 6 - V 4 - 3 ^ - P n _ 3 2 ( b - l ) p n 2S^e+6(b-l)p2n where e € {±1} is the residue of3e+b 1 modulo 4. 5. there exist integers m > 4, £ > 2 — b, and n > 1 such that 2rn3e — pn is a square and E is Q-isomorphic to one of the elliptic curves: a2 0,4 A E l -e-3h~1^2m3e -pn 2m-2g-?+2(b-l) —22 m 3 2 ^+ 6 ( f e _ 1 )p n E2 e • 2 • 3 6 - 1 y/2m3* - pn _ 3 2 ( 6 - l ) p n 2m+63<?+6(b-l)p2n where e <E {±1} is the residue of3b 1 modulo 4. 6. f«ere ex/sf integers m > 4, £ > 2 — b, and n > 1 such that 2mpn + 3e is a square and E is Q-isomorphic to one of the elliptic curves: a2 0 4 A F l - e - 3 b - 1 v / 2 ' T l P n + 3 ^ 2 m ~ 2 3 2 ( b - l ) p n 2 2 m 3 , ' + 6 ( b - l ) p 2 T i F2 e - 2 - 3 b - V 2 m P n + 3 ( ? 3 ^ + 2 ( b - l ) 2 m + 6 3 2 ^ + 6 ( b - ] ) p 7 i where eg {±1} is the residue of3b 1 modulo 4. 7. there exist integers £ > 2 — b and n > 1 such that 4pn — 3e is a square and E is Q-isomorphic to one of the elliptic curves: a.2 0,4 A G I e - 3 b - V 4 P n - 3 * 32 ( b - i y _ 2 4 3 ^ + 6 ( b - l ) p 2 n G2 - e - 2 - 3 b - V 4 P n - 3 < ? _ 3 f + 2 ( b - l ) 2 8 3 2 ^ + 6 ( b - l ) p n • where e G {±1} is the residue of 3b 1pn modulo 4. 8. there exist integers £ > 2 — b and n > 1 such that 4 + 3epn is a square and E is Q-isomorphic to one of the elliptic curves: a2 (34 A H I e - 3 b - V 4 + 3 V 32(b-l) 24^e+6(b-l)pn H2 -e • 2 • 3 6 - 1 s/4 + 3 V 3i+2(b-l)pn 28 3 2 ' ! + 6 ( 6 - l ) p 2 r l Chapter 3. Elliptic Curves with 2-torsion and conductor 2°3l3ps 69 where e £ {±1} is the residue of3b 1 modulo 4. 9. there exist integers rn > 4, £ > 2 - b, and n > 1 such that 2rn + 3epn is a square and E is Q-isomorphic to one of the elliptic curves: a.2 aA A 11 -E • 3 b - V 2 m + 3 V 2m.-2g2(t.-l) 22m3-?+6(6-l) n 12 e - 2 - 3 6 - 1 v / 2 m + 3 V 3<+2(b-l) n 2m+6 3 2£+6 (b- 1) 2n where e G {±1} is the residue of3b 1 modulo 4. 10. there exist integers m > 4, £ > 2 — b, and n > 1 such that 2m — 3epn is a square and E is Q-isomorphic to one of the elliptic curves: a2 C14 A JI -e • 3b-1^2m - 3 V 2m-232(b-l) _22m3^ +6(b-l)pn J2 t • 2 • 3h~1s/2m - 3epn „3^+2(b-l)pn where e G {±1} z's the residue of3b 1 modulo 4. 11. there exist integers m > 4, £ > 2 — b, and n > 1 such that 3ep'n — 2rn is a square and E is Q-isomorphic to one of the elliptic curves: «2 0,4 A KI - £ - 3 b - V 3 V - 2 " ' _2m-2g2(b-l) 22m3^+6(b-l)p7i K2 e • 2 • 3b-V3V l - 2 m 3£+2(b-l)pn — 2 m + 6 3 2 ^+ 6 C ' - 1 )p2n where e G {±1} is the residue of3b 1 modulo 4. 12. there exist integers m > 4, £ > 2 — b, and n > 1 swc« tTzai 3^  — 2mpn is a square and E is Q-isomorphic to one of the elliptic curves: a 2 ' a4 A LI -e-3b-1s/3e -2mpn —2m~~232(b~1)pn 22m.3^+6(b-l)p2n L2 e - 2 - 3 b - 1 v / 3 € - 2 m p " 3<M-2(6-l) _2TO+632^+6(b-l)pn where e G {±1} is the residue of3b 1 modulo 4. 13. there exist integers £ > 2 — b and n > 1 SMC/Z that pn - 4 • 3£ zs a square and E is Q-isomorphic to one of the elliptic curves: Chapter 3. Elliptic Curves with 2-torsion and conductor 2a3l3ps 70 a2 a.4 A M l £ ' 3 B - V P N - 4 - 3 ' 2432*?+6(6-l)pn M 2 - e • 2 • 3 6 " V P " - 4 • 3* 3 2 ( 6 - l ) p n _ 2 8 3 f + 6 ( ( > - l ) p 2 7 i where e g { ± 1 } zs (Tie residue ofSe+b modulo 4. 24. f/iere exz'sf integers m > A, t > 2 — b, and n > 1 swc/z f/zaf p" — 2 m 3 ^ is a square and E is Q-isomorphic to one of the elliptic curves: 0,2 a.4 A N I -e • 3h-Vpn ~ 2 m 3 f _ 2 m - 2 3 < ' + 2 ( b - l ) 22m32t?+6(b-l) pn N2 e • 2 • 3 b -VP™ - 2 m 3^ 3 2 ( 6 - l ) p n _ 2 m + 6 3 f + 6 ( 6 - l ) p 2 n where e G { ± 1 } is the residue o / 3 6 1 modulo A. In the case that 6 = 2, i.e. N = 2 4 3 2 p , we furthermore could have one of the following conditions satisfied: 15. there exist integers n > 1 and s G {0,1} such that Ap'i~l is a square and E is Q-isomorphic to one of the elliptic curves: a 2 04 A O l 3 2 s + V - 2 4 3 3 + 6 s p 2 " 0 2 e • 2 • 3 s + 1 v / 4 p , ; - 1 - 3 2 s + 1 2833 + 6 s p ™ where e € { ± 1 } zs the residue ofZsp'1 modulo A. 16. there exist integers n > 1 and s G {0,1} such that is a square and E is Q-isomorphic to one of the elliptic curves: a2 a 4 A PI 3 2 a + 1 2 4 3 3 + 6 s p " P2 £-2-3 i + 1 v/Z£4 3 2 s + 1 p n where e G { ± 1 } is the residue of 3 s modulo A. 17. there exist integers m > A, n > 1 and s G {0,1} such that 2 ' " + P " is a square and E is Q-isomorphic to one of the elliptic curves: Chapter 3. Elliptic Curves with 2-torsion and conductor 2a3@p6 71 a2 a 4 A Ql 2 m - 2 g 2 s+l 22m ^3+6spTi Q2 £ . 2 . 3 3 + 1 ^ 2 ^ 32s+1pn where e G {±1} Z'S f/ze residue of 3s modulo A. 28. z7zere exz'sf integers m > 4, n > 1 azzd s G {0,1} szzc/z that 2 " ~ p ' zs a square and E is Q-isomorphic to one of the elliptic curves: a-2 a 4 A Rl 2 ^ . - 2 3 2 ^ + 1 R2 _ _ 3 2 S+l p n 2m+Gg3+65p2n where e G {±1} z's f/ze residue of 3s modulo 4. 29. fTzere exz'sf integer n > 1 a n d s G {0,1} stzcTz f/zaf z's a square and E is Q-isomorphic to one of the elliptic curves: a-2 0 4 A SI _ 3 2 5 + l 2 4 3 3 + 6 s p n S2 e-2-3"+1y/p7'-4 3 2 5 + 1 p ™ _ 2 8 3 3 + 6 5 p 2 n if/zere e G {±1} is the residue of 3 s modulo 4. 20. tTzere exz'sf integers rn > A, n > 1 a n d s G {0,1} szzc/z that p ' g 2 z's a square and E is Q-isomorphic to one of the elliptic curves: 0 2 a 4 A TI 2 m ~ 2 ^ 2 s + l 2 2 m ^ 3 + 6 S p n T2 -e-2.3'+V^ 3 2 s + V 2 " i + 6 ^ 3 + 6 s p 2 n w/Vre e G {±1} z's f/ie residue of 3s modulo A. T h e o r e m 3.17 The elliptic curves E defined over Q, of conductor 2b3bp, and having at least one rational point of order 2, are the ones such that one of the following conditions is satisfied: Chapter 3. Elliptic Curves with 2-torsion and conductor 2a3@ps 72 1. there exist integers £ > 2 — b and n > 1 such that 3epn + 1 is a square and E is Q-isomorphic to one of the elliptic curves: a 2 0 4 A A l 2-3 f t-V3V + l 2 632f+6 (b - l ) p 2n A 2 -4-3 b -V3V + l 4 . 32(b-l) 2 12 3 M -6 (b - l ) p n A l ' - 2 • 3 6 " 1 v / 3 V + 1 ^e+2(b-l)pn 2 632-?+6(6- l) p 2n A 2 ' 4-3 b -V3V + l 4 . 2123<;+6(b-l)pn 2. there exist integers £ > 1 and n > 1 swc/z that 3 + p™ is a square and E is Q-isomorphic to one of the elliptic curves: (a) £ is even; a 2 0,4 A BI 2 • ^-V^'+p" 32(b-l)pn 263^+6(b-l)p2n B2 - 4 - 3 b - V 3 f +pn 4 . 3<?+2(6-l) 21232f+6(b-l)pn B I ' - 2 - 3 b " V 3 f +Pn 32(b-l)pn 26^+G{b~l)p2n B2' 4 • 3 b - V 3 ^ +Pn 4 . 3^ +2(b-i) 21232«+6(b-l)pn (b) £ is odd; a2 04 A C l 2 • 3b~'[y/3e+pn 3(?+ 2(b-l) 26 32<!+6(b-l) pn -C2 - 4 - 3 t " V 3 ' + P n 4 . 3 2 ( b - l ) p r l 2123<!+6(b-l) p2n C l ' - 2 - 3 b ~ V 3^+P n 3 « + 2 ( 6-l) 2 6 3 2 ' ! + 6 ( 6 - l ) p n C2' 4 - 3 6 - 1 v^+p™ 4 - . 3 2 ( b - l ) p n 2l23<:+6(6-:)p2n 3. there exist integers £ > 1 and n > 1 such that 3 — p™ zs a square and E is Q-isomorphic to one of the elliptic curves: (a) £ is even; 02 04 A D I 2 - 3 b - 1 v / 3 ( ' - p " _ 3 2 ( 6 - l ) p n 263<;+6(6-l) p2n D2 - 4 - 3 b - - <f3e -pn 4 . 3<?+2(6-l) _ 2 1 2 3 2 £ + 6(b-l) pn D I ' -2-3b~lsj3l -pn _ 3 2 ( f c - l ) p n 2 63/?+6(b-l) p 2n D2' 4 - 3 ^ V 3 ' - P n 4 . 3^+2(6-1) _ 21232<!+6(b-l) pn (b) £ is odd; Chapter 3. Elliptic Curves with 2-torsion and conductor 2a3l3ps 73 a2 0,4 A E l 2-3 f e - 1 v / 3 < ! - p n 3«+2(6-l) - 26 3 2 < , + 6 ( b ~ 1 ^ ™ E2 - 4 - 3h~1y/3t -pn —4.32(fc-1)pn 2l23''+6(b-l)JD2n E l ' -2-3 f e-V3^ -Pn 3^+2(6-1) _2632*;+6(b-l)pn E2' 4 - 3 b - V 3 f - p " - 4 • Z^-Vp" 2123£+6(b-l)p2n 4. f«ere ex/sf integers £>2 — b and n > 1 swc/z frtflf p™ - 3 is a square and E is Q-isomorphic to one of the elliptic curves: (a) I is even; a2 a.4 A F l 2 - 3 b " V P n - 3 e _3<?+2(6-l) 2632(!+6(b-l)pn F2 -4-3 b -Vp n - 3* 4 • S2 ^ - 1 ^ " - 2 1 2 3 ' ? + 6 ( b _ 1 ^ p 2 n F l ' -2-3 b -Vp n - 3 £ _3'!+2(b-l) 2632":+6(b-l)pn F2' 4 • 3 b " 1 y/p" - 3f- 4 • S 2 ^ " 1 V _2123"J+6(b-l)p2n (b) lis odd; a2 a 4 A GI 2-3h~ls/pn -3f 3 2 ( 6 - l ) p n _ 2 6 3 ^ + 6 ( 6 - l ) p 2 n G2 - 4 - 3 b " V P n " 3^ _4 . 3-2+2(6-1) 2 1232«+6 (6 - l ) p n G I ' - 2 - 3 b ~ V P n - 3 f 32(6-l)pn _ 2 6 3 « + 6 ( 6 - l ) p 2 n G2' 4 - 3 b - 1 v / P n - 3 £ _ 4 . 3^+2(6-1) 2 1232£+6 (6 - l ) p n 5. f«f?re exist integers I > 2 — b and n > 1 swcn 8 • 3 pn + 1 is a square and E is Q-isomorphic to one of the elliptic curves: a2 0,4 A H I 3 b - V 8 - 3 V + l 2 • 3«+2(b-l)pn 26 32«+6(b-l)p2?i H2 -2-3 b -V 8 -3V l + l 32(b-l) 293<!+6(b-l)pn H I ' -3b-V8-3V + l 2 • 3«+2(6-l) pn 2 6 3 2 f + 6 ( 6 - l ) p 2 n H2' 2-3b-V8-3V + l 32(b-l) 293<!+6(6-l)pn 6. there exist integers £ > 2 — b and n > 1 such that 8 • 3e + pn is a square and E is Q-isomorphic to one of the elliptic curves: Chapter 3. Elliptic Curves with 2-torsion and conductor 2cc3l3ps 74 a4 A 11 3h~W8 • 3e + Pn 2 • 3e+2(b-V 2632^+6(6-1)^1 12 - 2 - 3 b - 1 v / 8 - 3 ^ + p T l 32(b-l)pn 2 9 3 ^ + 6 ( 6 - l ) p 2 n 11' -3b-1 y/8 • 3e + pn 2 . 3<+2(6-D 2632£+6(ft-l)pn 12' 2 - 3 b - V 8 - 3 * +Pn 32(6-l) p n 293t"+6(6-l)p2n 7. there exist integers £ > 2 — b and n > 1 swcfr f/zaf 8 • 3e — pn is a square and E is Q-isomorphic to one of the elliptic curves: a2 0 4 A J l 3b~xyJ% -3e-pn 2 . 3 /+2(b-l) _ 2 6 3 2 £ + 6 ( b - l ) p n J2 - 2 - 3 6 - V 8 - 3 € -pn _ 3 2 ( b - l ) p n 293(?+6(b-l) p2n J l ' - 3 b - V 8 - 3 € - p " 2 . 3 / + 2 ( 6 - l ) _2632f+6(b-l) p n J2' 2 - 3 b " V 8 - 3 ^ - p n _ 3 2 ( 6 - l ) p n 293<?+6(6-l)p2n 8. there exist integers £ > 2 — b and n > 1 such that 8pn + 3e is a square and E is Q-isomorphic to one of the elliptic curves: ai a4 A K I 3b~lx/8pn + 3t 2 . 3 2 ( 6 - l ) p n 2 6 3 ^ + 6 ( 6 - l )p2n K2 - 2 - 3 b - V 8 p n + 3^ 3<?+2(6-l) 2 932^+6(b-l)pn K I ' - 3 b - V % > n + 3 < r 2 . 3 2 ( b ^ l ) p n 263f+6(b-l) p 2n K2 ' 2 - 3 b - V 8 p n + 3 £ 3 « ? + 2 ( 6 - l ) - 2932<"+6(b-l)pn 9. there exist integers £ > 2 — b and n > 1 such that 3epn - 8 zs a square and E is Q-isomorphic to one of the elliptic curves: ai a 4 • A LI 3b"V3V - 8 - 2 • 3 2 ( b - x ) 26 3 <M-6(b-l) p n L2 - 2 - 3 b - V 3 V - 8 3 ^ + 2 ( b - l ) p n _ 2 9 3 2 ^ + 6 ( b - l ) p 2 n L I ' - 3 b ~ V 3 V - 8 _ 2 . 32(6-1) 2 6 3 » ' + 6 ( b - l ) p n L2' 2 - 3 b - V 3 V - 8 3e+i{b-i)pn _ 2 9 3 2 « + 6 ( b - l ) p 2 n 10. there exist integers £ > 2 — b and n > 1 such that 3e - 8pn is a square and E is Q-isomorphic to one of the elliptic curves: Chapter 3. Elliptic Curves with 2-torsion and conductor 2a3@ps 75 0.2 a4 A M l Zb-\^JZ1 _ 8 p n -2 • 3 2 ( b " V 263f+6(b-l)p2n M 2 - 2 - 3 b - V 3 ^ - 8 P n 3^+2(6-1) M l ' - 3 h - V 3 * - 8p n - 2 • 3 2 < b - V 2 6 3 £ + 6 ( b - l ) p 2 n M 2 ' 2 - 3 b - 1 v / 3 f - 8 P n 3 « + 2 ( b - l ) _ 2 9 3 2 » M - 6 ( 6 - l ) p n 11. there exist integers I > 2 — b and n > 1 such that pn - 8 • 3 fs a square and E is Q-isomorphic to one of the elliptic curves: a 2 a4 A N I 3 f c - V p n - 8 - 3 ^ -2 • 3«+ 2 ( b -D 2&^2l+6(b-\)pn N2 - 2 - 3 b - V P n - 8 - 3 « 3 2 ( b - l ) p n _ 2 93^+6(6 - l ) p 2n N I ' - 3 b " V P " - 8 - 3 ^ - 2 • ^+2(b-l) 2 6 3 2 « + 6 ( b - l ) p n N 2 ' 2 - 3 b - V p n - 8 - 3 ' 3 2 ( b - l ) p n - 2 9 3 € + e ( b _ 1 ) p 2 n In the case that b — 2, i.e. N — 2532p, we furthermore could have one of the following conditions satisfied: 12. there exist integers n > 1 and s e {0,1} such that is a square and E is Q-isomorphic to one of the elliptic curves: a4 A O l 2 • 3 S + 1 32s+l 2 6 3 3 + 6 s p n 02 4 • 3 2 s + V 212 3a+63 p2n o r 2 - 3 s + 1 v / p " 3 + 1 3 2 s + 1 2 6 3 3+6. 5 p n 0 2 ' 4 • 3 2 s + V 2l2 3 3+6s p 2n 13. there exist integers n > 1 and s € {0,1} such that 2^—^ is a square and E is Q-isomorphic to one of the elliptic curves: a2 a4 A PI 2 • 3 S + 1 325 + l p n _ 2 6 3 3+6.s p 2n P2 - 4 • 3 2 s + 1 2 12 3 3+6s p n PI ' _ 2 . 3 ' ' + y ^ 32s + l p n _ 2 6 3 3 + 6 S p 2 n P2' 4 - 3 ' 5 + 1 v / S l 1 - 4 - 32 s + 1 212 33+6.s pn Chapter 3. Elliptic Curves with 2-torsion and conductor 2°3l3p5 76 14. there exist integers n > 1 and s G {0,1} such that p-^— is a square and E is Q-isomorphic to one of the elliptic curves: a 2 0,4 A Ql 2 • 3 2 5 + 1 2633+6spn Q2 2 - 3 ' 5 + 1 V P " 3 + 8 32s+1pn 2 9 3 3 + 6 . 5 p 2 n Ql' - 3 S + 1 V ^ 2•3 2 s + 1 2 6 3 3 + 6 . 5 p n Q2' 2 - 3 s + 1 V P " 3 + 8 32.s + l p n 2 9 3 3 + 6 s p 2 n 15. there exist integers n > 1 and s e {0,1} such that — g 2 - is a square and E is Q-isomorphic to one of the elliptic curves: 02 a 4 A Rl 2 • 32s+- _ 2 6 3 3 + 6 s p 7 i R2 -2'3 s +V^F _32.,+y,. 2 9 3 3 + 6.s p 2 n Rl' - 3 S + 1 v/V 2 • 3 2 5 + 1 -2633+6spn R2' 2-3s+V 8 T _ 3 2 , + 1 p n 2 9 3 3 + 6 . S p 2 n 16. there exists an integer n > 1 and s G {0,1} such that 2-3 - is a square and E is Q-isomorphic to one of the elliptic curves: o,2 o 4 A SI 3' 5 + 1\/^? - 2 • 3 2 s + 1 2 6 3 3 + 6 s p n S2 2 • 3 - 5 + i 3 2 .s+l p , i SI' -3'S +V^ - 2 • 3 2 s + 1 2633+6spn S2' 2 • 3 S + 1 i / ^ V 8 32s+1p" _ 2 9 3 3 + 6 S p 2 n Theorem 3.18 The elliptic curves E defined over Q, of conductor 2&3bp, cmd having at least one rational point of order 2, are the ones such that one of the following conditions is satisfied: 1. there exist integers £ > 2 - b and n > 1 such that 3epn + 1 is a square and E is Q-isomorphic to one of the elliptic curves: Chapter 3. Elliptic Curves with 2-torsion and conductor 2oc3l3ps 77 a-2 a4 A A l 2-3 b-V3V l + l 32 (6 -1) 2 63 ' '+6 (6 - l )pn A2 -4-3 b -V3V + l 4 . 3<-+2(6-l) p n 2l232/?+6(b-l)p2n A l ' - 2 - 3 b - V 3 V + l 3 2(b- l ) 2 63f+6(b- l )pn A 2 ' 4 - 3 b - V 3 V + l 4 . 3 « + 2 ( f e - l ) p n 2 1232<+6(b-l)p2n 2. there exist integers £ > 1 and n > 1 such that 3 + p n is a square and E is Q-isomorphic to one of the elliptic curves: (a) £ is even; a 2 a.4 A BI 2 - 3 b - 1 v / 3 - ? + P n 3 «+2(b - l ) B2 - 4 - 3 b - 1 v / 3' ?+P" 4 • 3 2 ( b " V 2 1 2 3 « + 6 ( 6 - l ) p 2 n B I ' - 2 - 3 b - V 3 £ + P 7 1 32(b-l) 2 C32'?+0(b- l) p r7, B2' 4 - 3 b - V 3 ^ + P n 4 . 3 C+2(b - l ) p n 2123(!+6(b-l) p27i •fW £ is odd; 02 a 4 A C l 2 - 3 b - V 3 ^ + P n 3 2 ( b - i y 2 63f+6(b-l)p2n C2 - 4 • 3 I , - V 3'+P' , 4 . 3 ^ 2 ( 6 - 1 ) 2l232-;+6(6-l) n C l ' - 2 - 3 b - V 3 ' + P n 3 2 ( b - i y 26 3':+6(b-l)p27i C2' 4 • 3 b " V 3 ^ +Pn 4 . 3f+ 2 ( 6-l) 21232^+6(6-1)^71 3. t/iere exisf integers £ > 1 and n > 1 swdi 3e — p™ is fl square and E is Q-isomorphic to one of the elliptic curves: (a) £ is even; «2 a 4 A D I 2 - 3 f c - 1 v / 3 * - p n 3<?+2(b-l) _2632/?+6(b-l)pn D2 - 4 - 3 b - V 3 f - p n —4 • 32 ( 6 - 1 )p n 2l23f+6(b-l)p2n D I ' - 2 - 3 b - V 3 f - P n 3^+2(6-1) -2 6 3 2 ( ! + 6 ( b ~ 1 )p n D2' 4-3b~W^ ~Pn -4 • S 2' 6- 1^" 2l23-?+6(6-l)p2n (b) £ is odd; Chapter 3. Elliptic Curves with 2-torsion and conductor 2a3l3ps 78 a-2 a 4 A E l 2 • 3h'1y/3e~pn _ 3 2 ( 6 - l ) p n 2 6 3 « + 6 ( b - l ) p 2 n E2 - 4 - 3 b - V ^ ~Pn 4 . 3€+2(6-l) _2 1 2 3M + 6 ( b - 1 )p™ E l ' -2 • 3b~1^3i -pn _ 3 2 ( 6 - l ) p n 2 6 3 £ + 6 ( b - l ) p 2 n E2' A-3b~ly/3i -pn 4 . 3<?+2(b-i) _ 2 1 2 3 2 t ' + 6 ( 6 - l ) p n 4. f/ter<? exist integers £ > 2 — b and n > 1 swc/z zTzzzf p™ — 3e is a square and E is Q-isomorphic to one of the elliptic curves: (a) £ is even; <22 0 4 A FI 2 - 3 b - 1 v V - 3 * 3 2 ( 6 - l ) p n _ 2 6 3 « + 0 ( b - l ) p 2 n F2 - 4 - 3 b - V p n - 3 £ _ 4 . 3^+2(6-1) 2 1 2 3 2 f + 6 ( 6 - l ) p n F I ' - 2 - 3 b - V p " - 3 f 3 2 ( 6 - l ) p n _ 2 63«+6 ( fc- l) p 2n F2' 4 - 3 b - V p n - 3 £ _ 4 . 3^+2(6-1) 2 1 2 6 3 2 ^ + 6 ( b - l ) p n (b) £ is odd; a2 a 4 A G I 2 - 3 b " V p n - 3 ^ _ 3 ^+2 (b - l ) 2 6 3 2 £ + 6 ( b - l ) n G2 - 4 • a 6 - 1 s/pn - 3e 4 . 3 2 ( b - i ) p « _ 2 1 2 3 ( ' + 6 ( b - l ) p 2 T i G I ' - 2 • 3 b " V P " - 3* _3«+2(fc-l) 2 6 3 2 / ! + 6 ( 6 - l ) p n G2' 4- 3 b - V p n - 3 £ 4 • S 2 ' 6 - 1 ) ^ _ 2 1 2 3 « + 6 ( b - l ) 2n 5. there exist integers m > 3, £ > 2 — b and n > 1 such that 2m3 p™ + 1 zs square and E is Q-isomorphic to one of the elliptic curves: a 2 a 4 A H I 2- 3 b _ 1 y/2m3epn + 1 2 2m+632<?+6 (b- l) p 2n H2 - 4 - 3 b - V 2 m 3 V + l 4 . 3 2 ( b - l ) 2 m + 1 2 3 * + 6 ( b - l ) p n H I ' - 2 - 3 b - 1 v / S ^ V + l 2 m 3^+2(6-1) TI 22777.+632i"+6(b-1 ) p 2 n H2 ' 4 - 3 b - V 2 m 3 V + l 4 . 3 2 ( b - l ) 2 m + 1 2 3 « + 6 ( b - l ) p n 6. there exist integers m > 2, £ > 2 - b and n > 1 such that 2T"3 + p" is a square and E is Q-isomorphic to one of the elliptic curves: Chapter 3. Elliptic Curves with 2-torsion and conductor 2a3,3p5 79 a2 0,4 A 11 2 -3 6 -V2 m 3*+p n 22m+6^2e+6(b-l)pn 12 - 4 • 3b~1^2m3e +pn 4 . 3 2 ( f c - i ) p « 2TO+122<?+6(b-l)p27i 11' -2 • 3b~1y/2m3e +pn 2m 3^+2 ( 6 - 1 ) 22rn+632i?+6 ( t ) - l) pn 12' 4 • 3b~1 yf2m3t + pn 4 : 3 2( b-V 2m+123£+6(b-l)p2n 7. £/W<? exzsf integers m > 2, £ > 2 — b and n > 1 t^af 2 m 3 — pn is a square and E is Q-isomorphic to one of the elliptic curves: 0-2 04 A Jl 2 • 3b~1 y/2m3e - p n 2m3t?+2(b-l) _ 2 2 m + 6 3 2 £ + 6 ( b - l ) p 7 i J2 - 4 • 3h~1 y/2m3e - pn - 4 • 32(b" V 277i+123t?+6(b-l)p2n J l ' -2-3h~1y/2m3l - pn 27713/2+2(6-1) _22m+632/?+6 (b - l )p7 i J2' 4 • 3h~l y/2m3e - p n _ 4 . 3 2 ( b - l ) p n 2777.+ 123(?+6(b-l)p27i 8. there exist integers m > 3, £ > 2 — b and n > 1 such that 2rnpn + 3 is a square and E is Q-isomorphic to one of the elliptic curves: 0.2 a4 A KI 2 • 3 b - 1 ^ 2 m p n + 3e 277132 ( 6—1 ) ^ 7 7 2 2 m+63>?+6 ( 6 - l)p2n K2 - 4 • 3b~1y/2mpn + 3e 4 . 3^+2 (6-1) 2m+1232<?+6 (6 - l)p77 K I ' - 2 - 3h-ly/2mpn + 3e 2 m 3 2 ( b - l ) p n 2 2 m+63<?+6 ( b - l)p2n K2' 4 • 3b-1y/2mpn + 3e 4 .3^+2 ( 6-1) 2 7 7 7+1232/^+6 ( 6-1 ) ^ 7 7 9. there exist integers £ > 2 - b and'n > 1 such that 4pn — 3 is a square and E is Q-isomorphic to one of the elliptic curves: a2 a4 A LI 2 • 3 6 - V 4 P n - 3* 4 . 32(6-l)pn _2l03«"+6(6 - l ) p 27i L2 - 4 • 3 b - V 4 P n - 3* _ 4 . 3t+2(b-l) 21432/?+6(6-l) pn L I ' -2 • 3 b - V 4 P " - 3£ 4 . 3 2 ( 6 - l ) p r 7 _ 2 i 0 3 f+6(6 - i) p2n L2' 4 • 3 b - V 4 P " - 3 ^ _ 4. 3/:+2(6-i) 21432^+6(6-l)pn 10. there exist integers m > 2, £ > 2 — b and n > 1 such that 2m + 3epn is a square and E is Q-isomorphic to one of the elliptic curves: Chapter 3. Elliptic Curves with 2-torsion and conductor 2a3l3ps 80 a-2 a.4 A Ml 2 - 3 6 - 1 v / 2 m + 3 V 2mg2(6-l) 22m+63*!+6(6-l)pn M 2 - 4 • 3 b - V 2 m + 3 V 1 4 . 3 « + 2(b - l) pn 2m+1232^+6(b-l)p2n M l ' - 2 • 3 b _ 1 ^2m + 3 V 2m.32(fc-l) 22771+63^+6(6-1 )^71 M 2 ' 4 - 3 b - 1 v / 2 m + 3 V 4 . 3^+2(6- l) pn 2m+1232^+6(b-l)p2n 11. there exist integers m > 2, £ > 2 — b and n > 1 such that 2m — 3epn is a square and E is Q-isomorphic to one of the elliptic curves: a-2 04 A N I 2 • 3h~1yj2m - 3 V l 277132(6-1) _227n+63*?+6(6-l)p7i N2 - 4 • 3h~1y/2m - 3epn _ 4 . 3 « + 2 ( 6 - l ) p n 277i+1232£+6(b-l)p27i N I ' -2 • 3h~^2m - 3 V 277132(6-1) _2277i+63i°+6 (6- l)p7i N 2 ' 4 • 3 b _ 1 yj2m - 3epn _ 4 . 3 « + 2 ( 6 - l ) p 7 i 2m+1232("+6(6-l)p2n 12. there exist integers m > 2, £ > 2 — b and n > 1 such that 3epn - 2m is a square and E is Q-isomorphic to one of the elliptic curves: a-2 0,4 A 0 1 2 • 3 b - V 3 V - 2 m _277i32(6-l) 2277i,+63i'+6(6-l)p7i 0 2 - 4 • 3 6 " 1 y/3epn - 2m 4 . 3 « + 2 ( 6 - l ) p n _ 2771+1232^+6 (6-1 )^271 o r -2 • 3h~lyj3tpn - 2m — 2 7 7 l 3 2 ( b _ 1 ) 2 2 7 7 7 + 6 3 ^ + 6 ( 6 - 1 ) n 0 2 ' 4 • 3h-x^/3ipn - 2m 4 . 3 ^ + 2 ( 6 - l ) p n _ 2T O + l 2 3 2 ^ + 6 ( 6 - l ) p 2 n 13. there exist integers rn > 2, £ > 2 — b and n > 1 such that 3 - 2mpn is a square and E is Q-isomorphic to one of the elliptic curves: a-2 a.4 A PI 2 • 3 b _ 1 y/3e - 2 m p n _ 2 r r a 3 2C'-l ) p n 22771+63^+6 (6 -1)^271 P2 - 4 • 3 b _ 1 y/3e - 2 m p n 4 . 3^+2(6-1) _277i+1232(!+6(b-l)pn P I ' -2-3h~1y/3i - 2 m p n - 2 m 3 2 ( b - 1 ) p n 22m+63f+6(6-l) p2n P2' 4 • S ^ V 3 * - 2mpn 4 . 3^+2(6-1) _ 2771+1232^+6(6-l)p71. 14. there exist integers rn > 2, £ > 2 — b and n > 1 such that pn — 2m3e is a square and E is Q-isomorphic to one of the elliptic curves: Chapter 3. Elliptic Curves with 2-torsion and conductor 2a3@ps 81 a2 a4 A Q i 2- 3 6 - V p n - 2 m 3 f —2 m 3^+ 2 ( f c - 1 ) 22m+622^+6(b-l)pn Q2 - 4 - 3 6 - V P " -2 m 3* 4 . 3 2 ( f t - i ) p n _2m+123<?+6(fc-l)p2n Q l ' -2 • 3 b _ 1 v V - 2 m 3 ^ _2TO3«?+2(b-l) 22m+632i?+6(6-l)pn Q2' 4 - 3 6 - 1 v V - 2 m 3 * 4 . 32 (b- l )p i _27n+123«?+6(6-l)p2n In f/te case that b = 2, i.e. N = 2632p, we furthermore could have one of the following conditions satisfied: 15. there exist integers n > 1 and s £ {0,1} such that p '3+1 is a square and E is Q-isomorphic to one of the elliptic curves: a 2 a 4 A Rl 2 ' 3 S + 1 X / p " 3 + 1 3 2 s + V 2 6 3 3 + 6 s p 2 " R2 - 4 - 3 s + 1 v ^ P 4 • 3 2 s + 1 2 12 3 3+6.s p 7i Rl' -2-3^ +V^ 3 2 s + V n 2 6 3 3 + 6 ' 5 p 2 n R2' 4-3 s +V^l 4 • 3 2 5 + 1 2 1 2 3 3 + c V l 16. there exist integers n > 1 and s € {0,1} swc« that p 3 1 fs a square and E is Q-isomorphic to one of the elliptic curves: a 2 0.4 A SI 2-3s + 1 v />+ 1 - 3 2 s + 1 2 6 3 3 + G s p ™ S2 4 • 3 2 s + 1 p n _ 2 12 3 3+6. S p 2n SI' 2 - 3 s + 1 V ^ _ 32»+l 2 6 3 3 + 6 s p n S2' 4 . 3 ^ ^ 4 • 3 2 s + 1 p n - 2 1 2 3 3 + 6 s p 2 n 17. there exist integers n > 1 and s G {0,1} swdz f/taf 4 p 3 1 fs a square and E is Q-isojnorphic to one of the elliptic curves: Chapter 3. Elliptic Curves with 2-torsion and conductor 2°c3l}p& 82 0-2 a.4 A TI 2-3s+1x['K=± 4 • 3 2 s + 1 )p n —21 03 3 + 6 sp 2 n T2 4 • 3 S + 1 y ^ r i _ 4 . 3 2s+l ) 2 1 43 3 + 6 s p n TI ' 4 • 3 2 s + 1 V _ 2 i o 3 3 + 6 . y n T2' 4 . 3 » + i v / 4 £ ^ - j . - 4 . 32S+1) 21433+6spn 18. there exist integers m > 2, n > 1 and s e {0,1} such that p *2" is a square and E is Q-isomorphic to one of the elliptic curves: a2 0 4 A U I 2 . 3 , + i < y E ! l ± 2 ^ 2 m 3 2 s + l 22m+6^3+6s^n U2 4 • 3 2 s + V 2m+12^3-f 6sp2n U I ' 2 • 3 S + 1 x j ^ 1 22m+6^3+6Spn U2' 4 • 3 2 s + V 2m+12^3+6.Sp2Ti 19. there exist integers m > 2, n > 1 and s 6 {0,1} swc/z that 2 3 P zs a square and E is Q-isomorphic to one of the elliptic curves: 0 2 04 A V I 2nJ'32.S+l 22m+6 |^3+6s pn V2 - 4 • 3 2 ' 5 + V n 2^+12^3+65^ 271 V I ' - 2 - 3 s + V ^ 2 ^ 2 2 5 + 1 22"'+ 6^3+65,^71 V2' 4 " 3 S + V ^ - 4 • 32-9+1pn 2m+12g3+6.Sp2n 20. there exist integers m > 2, n > 1 and s e {0,1} such that p 3 2 z's a square and E is Q-isomorphic to one of the elliptic curves: 02 0,4 A WI 2 m 3 2 s + i 22m+623+6Spn W2 4 • 3 2 s + V _2"i+12^ 3+6Sp2n W I ' - 2 - 3 5 + 1 2 m 3 2 ^ + i 22m+6^3+6.Spn W2' 4 • 3 * + 1 sJv^F- 4 • 3 2 ' 5 + V _2m+1233+6-5p2n Chapter 3. Elliptic Curves with 2-torsion and conductor 2a3f3ps 83 Theorem 3.19 The elliptic curves E defined over Q, of conductor 273bp, and having at least one rational point of order 2, are the ones such that one of the following conditions is satisfied: 1. there exist integers £ > 2 — b and n > 1 such that 2 • 3e + p11 is a square and E is Q-isomorphic to one of the elliptic curves: a2 a.4 A A l 2 - 3 b - V 2 - 3 f +pn 2 . 3^+2(6-1) A 2 . - 4 - S 6 " 1 A / 2 -3e+pn 4-3 2 ( b "V 213 3»2+6(b-l)p2n A l ' -2 • 3 b - V 2 -3e+Pn 2 . 3<+2(6-l) 2832/J+6(b-l)pn A 2 ' 4 - 3 b - V 2 - 3 * + P n 4 • 32(-b-Vpn 213 3<;+6(6-l)p2n BI 2 - 3 b - V 2 - 3 * + P n 32(b-l)pn 2 7 3 M -6 ( 6 - l ) p 2 n B2 - 4 - 3 b - V 2 - 3 £ + P n 8 . 3<+2(k-i) 2 1 4 3 2«+6 ( 6 - l ) p n B I ' - 2 - 3 b - V 2 - 3 f +Pn 32 ( 6 - l ) p n 2 7 3 « + 6 ( 6 - l ) p 2 n B2' 4 - 3 b - V 2 - 3 * + P n 8 . 3«+2(b-l) 214 32^+G(b-l) pn 2. there exist integers £ > 2 — b and n > 1 such that 2 • 3 — pn is a square and E is Q-isomorphic to one of the elliptic curves: a2 0,4 A C l 2 • 3h~l y/2 - 3 e - p n 2 . 3<+2(b-i) _ 2 8 3 2 £ + 6 ( b - l ) p n C2 - 4 - 3 b - V 2 - 3 ^ - p n _ 4 . 3 2 ( 6 - l ) p n 213 3»«+6(b-l) p2n C l ' -2 -3 b ~V 2 -3^-p n 2 . 3<+2(6-l) _ 2 8 3 2 £ + 6 ( 6 - l ) p n - C 2 ' 4 - 3 b - V 2 - 3 £ - p n - 4 • 32<b"V 213 3i+6(b-l) p2n D I 2 - 3 ^ V 2 - 3 f - 7 7 n _ 3 2 ( b - l ) p n 2 7 3 ^ + 6 ( b - l ) p 2 n D2 -4-3b~'1y/2-3e -pn 8 . 3<+2(b-l) _ 2 1 4 3 2 ( ! + 6 ( b - l ) p n D I ' - 2 - 3 b - V 2 - 3 ^ - p n _32(b-l)pn 2 7 3 / ? + 6 ( b - l ) p 2 n D2' 4-3b~l^2-3i -Pn 8 . 3/+2(6- l ) _ 2 1 4 3 2 M - 6 ( b - l ) p n 3. there exist integers £> 2 - b and n > 1 such that 2pn + 3e is a square and E is Q-isomorphic to one of the elliptic curves: Chapter 3. Elliptic Curves with 2-torsion and conductor 2°3 / V 5 84 a2 a 4 A E l 2 - 3 ^ 1 v / 2 p n + 3« 2 • 3 2 ( b " V 283«+6(b-l)p2n E2 - 4 - 3 b - V 2 P n + 3* 4 . 3 ^ + 2 ( 6 - 1 ) 2l332/i+6(b-l)pTi E l ' - 2 - 3 b - V 2 P " + 3 f 2 • 3 2( b-V 2 83«+6(b - l) p 2n E2' 4 - 3 b - V 2 P " + 3 £ . . 4 . 3 ^ + 2 ( 6 - 1 ) 2l332f+6(6-l)pn F l 2 - 3 b - V 2 P n + 3* 3^+2(6-1) 2 7 3 2 » ? + 6 ( 6 - l ) p n F2 - 4 - 3 b - V 2 P n + 3 < ; 8 • 32( b- a>p n 2l43^+6(b-l)p2n F l ' - 2 - 3 b ~ V 2 P n + 3 < ! 3<M-2(b-l) 2 732«+6(b - l )p7i F2' 4- 3 b - J y/2pn + 3e 8 • 32^h-^pn 2l43f2+6(b-l)p2ra 4. there exist integers t > 2 — b and n > 1 such that 2pn - 3 is a square and E is Q-isomorphic to one of the elliptic curves: (12 a 4 A GI 2 - 3 b - 1 v V 1 - 3 < ! 2 . 3 2(b- l) p 77 _ 28 3*+6(&-i) 2 n G2 - 4 • 3 b " 1 s/2pn - 3e _ 4 . 3<?+2(b-i) 21332f+6(6-l) pn G I ' -2-3b-1^2pn - 3e 2 . 3 2 ( 6 - l ) p n _ 2 8 3 t " + 6 ( 6 - l ) p 2 n G2' 4 • 3 6 " 1 sj2pn - 3e _4 . 3^+2(6-1) 2l332f+6(b-l)pn H I 2 • 3b~l sj2pn - 3l- _3«+2(6 - l ) 2732^+6(6-l)pn H2 - 4 - 3 b - V 2 P n - 3 £ 8 . 3 2 ( 6 - l ) p n _2l4 3«+6(b-l)p2n H I ' - 2 - 3 b - V 2 P n - 3 f _ 3<?+2(6-l) 2732e+6(b-l)pn H 2 ' 4- 3 b - V 2 P n - 3* 8 . 3 2 ( b - l ) p n _ 2 l 4 3f+6(b- l) p 2n 5. there exist integers I > 2 — b and n > 1 such that 2 + 3 pn is a square and E is Q-isomorphic to one of the elliptic curves: (12 a 4 A 11 2 - 3 b - V 2 + 3 V l 2 . 3 2 ( 6 - 1 ) 283^+6(6-1) n 12 - 4 - 3 b - V 2 + 3 V 4 . 3£+2(6-l) pn 21 3 3 2 t " + 6 ( 6 - l ) p 2 7 7 11' - 2 - 3 b - V 2 + 3 V 2 • S 2^" 1) • 283^+6(b-l)pn 12' 4 - 3 b - V 2 + 3 V 4 . 3^+2(6-l)pn 2l332£+6 (6 - l ) p 2n JI 2 - 3 b - V 2 + 3 V 3«?+2(b-l)pn 2732£+6(b-l)p2n J2 - 4 - 3 b - V 2 + 3 V 8.32(b-l) 21437?+6(b-l)pn JI ' - 2 - 3 b - V 2 + 3 V 3^ +2(b-l)pn 27 3 2 £ + 6 ( 6 - l ) p 2 n J 2 ' 4 - 3 6 - 1 v / 2 + 3 y i 8 • S 2 ^- 1 ) 2143*?+6(b-l)pn Chapter 3. Elliptic Curves with 2-torsion and conductor 2a3^p5 85 6. there exist integers £ > 2 — b and n > 1 such that 3 pn — 2 is a square and E is Q-isomorphic to one of the elliptic curves: a 2 a 4 A KI 2 • 3 6 " V 3 V - 2 -2 • 32^ 2 8 3 ^ + 6 ( 6 - 1 ) ^ K2 - 4 - 3 b " V 3 V - 2 4 . 3 « + 2 ( 6 - l ) p n _ 21332^+6(b-l)p2n KI' -2 • 3 b - V 3 V - 2 _ 2 . 32(6-1) 2 8 3 ^ + 6 ( b - l ) p n K2' 4 - 3 b - V 3 V - 2 4 . 3 ^+2(b - l ) p n — 2 1 3 3 2 ^ + 6 ( b - 1 . ) p 2 r l LI 2 - 3 f c - V 3 V - 2 3^+2(b-l) pn _ 2732^+6 (6 - l ) p 2n L2 -4 • 3b-1y/3epn - 2 _ 8 . 3 2 ( 6 - D 2143*3+6(6-l)pn LI' - 2 - 3 b - V 3 V - 2 3 « + 2 ( b - l ) p n _ 2 7 3 2 ^ + 6 ( 6 - l ) p 2 n L2' 4 • 3 b " V 3 V - 2 _ 8 . 3 2 ( b - l ) 2 1 4 3 f : + 6 ( 6 - l ) p n 7. there exist integers £ > 2 — b and n > 1 swc/z tTzaf 3^  - 2pn is a square and E is Q-isomorphic to one of the elliptic curves: a 4 A M l 2 • 3 b ~ V 3 £ - 2 p n -2 • S 2 ' 6 - 1 ) ^ " 2 83<'+6 (6 - l ) p 2n M2 -4 • 3 b- V 3 € _ 2 P « 4 . 3«+2(b-l) _21332*3+6(b-l)pn M l ' -2 • 3 b" V 3 ^ - 2 p n -2 • S 2^- 1)?" 2 8 3 ^ + 6 ( 6 - 1 ) ^272 M2' 4 - 3 b - V 3 ^ - 2 p n 4 . 3^+2(6-1) _ 2 l 3 3 2 « + 6 ( 6 - l ) p n NI 2 • 3 b " V 3 ^ - 2 p n 3^+2(6-1) - 2 7 3 2 £ + 6 ( b _ 1 ) p n N2 -4 • 3 6" 1 \/3« - 2pn -8 • 32<b"1 V 2l43f+6(b-l)p2n NI' -2 • 3h~ V 3 * - 2 p " 3*3+2(6-1) - 2 7 3 2 £ + 6 ( b ~ 1 ) p n N2' 4 - 3 b ~ V 3 ^ - 2 p " _8- 3 2(6-l)p" 2 l 4 3 f + 6 ( 6 - l ) p 2 n 8. there exist integers £ > 2 — b and n > 1 such that pn — 2 • 3e is a square and E is Q-isomorphic to one of the elliptic curves: Chapter 3. Elliptic Curves with 2-torsion and conductor 2a3l3ps 86 a 4 A O l 2 - 3 6 - V P " - 2 - 3 * _ 2. 3£+2(b-l) 2 8 3 2f+6(6-l)pn 0 2 -A-3b~1y/pn - 2- 3e 4 . 3 2 ( 6 - l ) p n _2l33f+6(b-l)p2n 0 1 ' -2 • 3h~ V P " - 2 • 3^  - 2 • 3e+2V>-V 2832f+6(b-l)pn 0 2 ' 4 • 3 f t" V P ' 1 - 2 ' 3* 4 • 3 2 < B -V _ 213 3/:+6(6-l)p2n PI 2 - 3 ^ V P " - 2 - 3 ' 3 2 ( 6 - l ) p n _ 27 3«+6(b-l) p2n P2 -4 • 3 1 -VP" - 2 • 3£ - 8 • S^2^-1) 214 32/J+6(6-l)pTi PI' - 2 • 3 b - V P " - 2 • 3 f 32(b-l)pn _273^+6(b-l)p2n P2' 4-3 6-VpN-2-3< _8.3/+2(b-l) 214 32«+6(b-l) pn In the case that 6 = 2, i.e. N = 2732p, we furthermore could have one of the following conditions satisfied: 9. there exist integers n > 1 and s € {0,1} such that 2 p 3 + 1 i s a square and E is Q-isomorphic to one of the elliptic curves: 0.2 a 4 A Q l 2 - 3 s + 1 - v / 2 p " 3 + 1 2 • 3 2 s + V Q2 4 - 3 s + 1 A / 2 = 3 ± i 4 • 32 s + 1 21333+6spn Q l ' 2-3s+1]/2pn+1 2 • 3 2*+V 2 8 3 3+6s p 2n Q2' 4-3s+1yJ2pn+l 4-,-32*+l 2 13 3 3+6. S p n R l 2-3s+1yJ2pn3+1 3 2 s + 1 2 7 3 3 + 6 s p n R2 4 - 3 s + i v / 2 p r ; + 1 8 • 32 ' 5 + V l 2 I4 3 3+6, p 2n R l ' 2-3s+1yJ2pn3+1 3 2 s + 1 2733+6spn R2' 4-3s+iv/2p;+i 8 • 3 2 - 5 + 1 p n 2 14 3 3+6 S p 2n 10. there exist integers n > 1 and s e {0,1} such that P-^L- is a square and E is Q-isomorphic to one of the elliptic curves: Chapter 3. Elliptic Curves with 2-torsion and conductor 2a3l3p5 87 a-2 a 4 A SI 2-3 s + 1 A y p " 3 + 2 2 • 3 2 5 + 1 2833+6spn S2 4 • 32s+1pn 2 l 3 3 3 4 - 6 s p 2 n SI' 2 - 3s + y z ^ 3 ± 2 2 • 3 2 s + 1 2833+6sp™ S2' 4 • 32s+1pn 2 l 3 3 3 + 6 5 p 2 n TI 2 . 3 , + i y P » + 2 32s+1pn 2733+6-5p2n T2 8 • 3 2 s + 1 2l4 33+6* pn TI' 2-3- 5 + 1 x / p " 3 + 2 32s+1p™ 2733+6sp2™ T2' 4.3'+V^ 8 • 3 2 s + 1 21433+6spn 11. there exist integers n > 1 and s £ {0,1} swcn that ^ - g - 2 - is a square and E is Q-isomorphic to one of the elliptic curves: a2 a 4 A U l - 2 • 3 2 s + 1 2 8 3 3 + 6 s p 7 i U2 - 4 . 3 * + y ^ 4 • 32s+1pn - 2 1 3 3 3 + 6 V U l ' - 2 • 3 2 s + 1 2833+6'5pn U2 ' 4 • 32s+1pn _ 2 1 3 3 3 + 6 . 5 p 2 n V I 2 - 3 s + V ^ 3 2 s + V -2733+6sp2" V2 -8 • 3 2 s + 1 21433+6spn V I ' 32s+1pn -2733+6sp2n V2 ' 4-3-^V2^ -8 • 3 2 5 + 1 21433+6spn Theorem 3.20 The elliptic curves E defined over Q, of conductor 2836p, and having at least one rational point of order 2, are the ones such that one of the following conditions is satisfied: 1. there exist integers I > 2 - b and n > 1 such that ^P™-1 is a square and E is Q-isomorphic to one of the elliptic curves: Chapter 3. Elliptic Curves with 2-torsion and conductor 2a3f3ps 88 a-2 04 A A l 2 . 3 « + 2 ( b - l ) p n _ 2 9 3 M + 6 ( b _ 1 ) p 2 n A 2 8 . 3 6 - 1 ^ 3 ^ 1 - 8 • 3 2 ( B _ 1 ) 2l53*"+6(6-l)pn A l ' 2 • 3 « + 2 ( f t - l ) p " _2932^+6(6-l) p2n A 2 ' 3 . 3 6 - 1 ^ 3 ^ 1 - 8 • 3 2 ( B _ 1 ) 2l53^+6(6-l) pn BI 4 . 3 6 - 1 ^ / 3 ^ 1 _ 2.32(«>-i) 293«"+6(6-l)pn B2 8 - 3 6 - y 3 f p ; ' - ) 8 • tf+^-Vp71 _ 2 1 5 3 2 « + 6 ( 6 - l ) p 2 n BI ' - 4 . 3 - v ^ i -2 -3 2 ( F E - 1 ) 293(?+6(6-l)pn B2' 3 . 3 6 - 1 ^ 3 ^ 8 . 3 < + 2 ( 6 - l ) p n _2l532t*+6(6-l)p2n 2. there exist integers £ > 2 — b and n > 1 such that 2 V is a square and E is Q-isomorphic to one of the elliptic curves: a2 a 4 A C l 4 - 3 - V ^ 2 . 3 £ + 2 ( 6 - l ) 2932/J+6(6-l)pn C2 8 • 36"1 y j 3 ^ 8 • 32l-h-1)pn 2l5 3<M-6(6-l)p2n cr -4-3»-V^F 2. 3«+2(6-l) 2 9 3 2 £ + 6 ( b - l ) p n C2' 8 . 3 6 - 1 ^ 3 ^ 8-3 2( 6-V 2 1 5 3 t " + 6 ( 6 - l ) p 2 n DI 4 . 3 " - 1 v / 3 ' + / " 2 • 3 2 ( 6 - V 2 93t"+6(6 - l )p2n D2 g . 3^+2(6-1) 2l532/:+6(6-l)p7i D I ' 4 - 3 b - i y 3 ' + p " 2 - 3 2 ( f c - 1 V 293f+6(6-l)p2rc. D2' 8 . 3 6 - i y 3 ' + P - g . 3^+2(6-1) 2l532("+6(b-1) n 0 / _ n 3. there exist integers £ > 2 — b and n > 1 such that —^— is a square and E is Q-isomorphic to one of the elliptic curves: Chapter 3. Elliptic Curves with 2-torsion and conductor 2a3l3ps 89 a.2 a 4 A E l 2 . 3<?+2(h-i) _ 2 9 3 2 ^ + 6 ( b - l ) p n E 2 - 8 • 3 2 ( b _ 1 ) p n 2 1 5 3 £ + 6 ( b - l ) p 2 n E l ' 2 . 3*+2(b-i) _2 93 2 £ + 6 ( b - 1 )p r l E 2 ' 8 . 3 6 - 1 ^ / 3 1 ^ 1 _ 8 . 3 2 ( 6 - l ) p n 2 l 5 3 « + 6 ( 6 - l ) p 2 n F I 4 . 3 b - 1 v / 3 ^ - 2 • &V>-Vpn 2 9 3 £ + 6 ( b - l ) p 2 n F 2 8 . 3 ^ + 2((>-i) _2l5 3 2€+6(b-l) n F I ' 4 . 3 " - 1 A / 3 ^ _ 2 . 3 2 ( 6 - l ) p n 2 9 3 « + 6 ( b - l ) p 2 n F 2 ' g . 3f+ 2 ( 6-l) _2 153 2^+6(b-l) pn 4. there exist integers £ > 2 — b and n > 1 such that —^— is a square and E is Q-isomorphic to one of the elliptic curves: a 2 a 4 A G I 4 . 3 6 - 1 ^ 2 ^ 2 • 3 2 ( ' , _ 1 ) p" _ 2 9 3 « + 6 ( 6-l ) p 2 n G 2 - 8 - 3 b - y ^ _ g . 3 « + 2 ( 6-l) 2 1532«+6(6-l) pn G I ' 4 . 3 6 - 1 ^ ^ 2 . 3 2 ( 6 - l ) p n _293« + 6(b-l ) p 2 n G 2 ' 8 . 3 6 - 1 ^ 2 2 ^ _ 8 . 3 « + 2 ( 6 - l ) 21532f:+6(b-l) pn H I 4 . 3 b - V ^ _ 2 . 3/+2(6-l) 2 9 3 2 f + 6(b-l) pn H 2 s-s"-1^"-3' 8 • 3 2 ( f c _ 1 ) p n _2l5 3 <?+6(6-l) p 2n H I ' 4 . 36-1 ^ £ ^ 1 _ 2 . 3 £ + 2(b-l) 29 3 2t3+6(b-l) p n H 2 ' 8 . 3 - V ^ g . 3 2(b-l) pn _ 2 1 5 3 « + 6(b-l ) p 2 n In the case that 6 = 2 , i.e. N = 2832p, we furthermore could have one of the following conditions satisfied: 5. there exists an integer n > 1 such that is a square and E is Q-isomorphic to one of the elliptic curves: Chapter 3. Elliptic Curves with 2-torsion and conductor 2 Q 3 / V 90 02 a 4 A 11 2 • 3 2 s + V 2 9 3 3+6. S p 2n 12 8 • 3 2 s + 1 2l5 33+6. S pn 11' _4.3"+VE^1 2 • 3 2 s + 1 p n 2 9 3 3 + G s p 2 n 12' 8 • 3 S + 1 ^ ^ ± 1 8 • 3 2 s + 1 2 1 5 3 3 + 6 ' s p n J l 4-3»+V^ 2 • 3 2 s + 1 2 9 3 3 + 6 s p n J2 g 3 S + 1 y P " + i 8 • 3 2 s + V 215 33+6.sp2n J l ' -4-3'+V^ 2 • 3 2 s + 1 2 9 3 3 + 6 - 5 p" J2' 8 • 3 2 s ' + 1 p n 2l523+6.Sp2n 6. f^ ert? exists an integer n > 1 swc/h f/zat ^ ^ r - ^ z's a square and E is Q-isomorphic to one of the elliptic curves: 0.2 a 4 A K I 4 - 3 ? + y ^ 2 • 32*'+V _ 2 9 3 3+6.s p 2n K2 - 8 • 32s+1 K I ' - 4 - 3 s + V ^ 2 • 3 2 s + 1 p " - 2 9 3 3 + G s p 2 n K2 ' - 8 • 3 2 s + 1 2 1 5 3 3 + G s p " L I 4 - 3 s + V ^ - 2 • 3 2 s + 1 2 9 3 3 + 6 s p " L2 - 8 - 3 - 5 + y ^ 8 • 3 2 s + 1 p n 2l5^3-f-6.s^2n L I ' -4-3'+V^ - 2 • 3 2 s + 1 2 9 3 3 + 6 s p n L2' 8 • 3 2 s + 1 p n _ 2 15 3 3+6, p 2n 3.3 C u r v e s o f C o n d u c t o r 2a3(3p2 As we mentioned in the introduction to this chapter the models presented in the following table are minimal except in the case when the conductor is not divisible by 4. In these cases (i.e. Theorems 3.21 and 3.22) the model is minimal except at 2, and a minimal model can be found using Corollary 2.2. We choose not to do this here. Chapter 3. Elliptic Curves with 2-torsion and conductor 2a3l3ps 91 Theorem 3.21 The elliptic curves E defined over Q, of conductor 3bp2, and having at least one rational point of order 2, are the ones such that one of the following conditions is satisfied: 1. there exist integers £ > 2 — b and n > 0 such that 263epn + 1 is a square and E is Q-isomorphic to one of the elliptic curves: (12 0,4 A A l e - 3 ' ' - V \ / 2 6 3 V l + l 24g/J+2(6-l)pn+2 2l2 32f:+6(b-l)p2n+6 A 2 - f - 2 - 3 b - V 2 6 3 y + l 3 2(b- l ) p 2 2l2 3 f+6(6-l)pn+6 where e G {±1} is the residue of3b lp modulo 4. 2. there exist integers £ > 2 — b and n > 0 such that 2n3e + p" is a square and E is Q-isomorphic to one of the elliptic curves: a2 a4 A BI e • 3b-V\/263<? + p n 2 4 3 £ + 2 0 - i y 2l2 32/?+6(b-l)pn+6 B2 - e - 2 - 3 b - 1 p ^ / 2 ( i 3 e + p n 32(b-l)pn+2 2l23<?+6(b-l)p2n+6 where e G {±1} is the residue of3b - J p modulo 4. 3. there exist integers £ > 2 - b and n > 0 such that 2G3e - pn is a square and E is Q-isomorphic to one of the elliptic curves: a-2 a 4 A C l e-3h-1p^/2G3e -pn 2 4 3 « + 2 ( b - l ) p 2 _ 2 l 2 3 2 £ + 6 ( b - l ) p n + 6 C2 -e-2-3h~^psf2G3i -pn _ 3 2(6- l ) p n+2 2 l 2 3 « + 6 ( 6 - I ) p 2 n + 6 where e € {±1} is the residue of3b 1p modulo 4. 4. there exist integers £> 2 - band n > 0 such that 26pn + 3e is a square and E is Q-isomorphic to one of the elliptic curves: 0.2 a 4 A DI e-3b-lpsj2Gpn +3* 24 32((,-l) j 9n+2 2 l 2 3 j ° + G ( 6 - l ) p 2 T i + 6 D2 - £ - 2 - 3 b -V \ / 2 V + 3^ 3 « + 2 ( b - l ) p 2 2 l 2 3 2 « ? + 6 ( f c - l ) p n + 6 where e £ {11} is the residue of3b lp modulo 4. Chapter 3. Elliptic Curves with 2-torsion and conductor 2a3(3ps 92 5. there exist integers £ > 2 — b and n > 0 such that 2 6 + 3epn is a square and E is Q-isomorphic to one of the elliptic curves: a2 a4 A E l 2 4 3 2 ( 6 - l ) p 2 2l23«+6(b-l) n+6 E2 - e - 2 - 3 6 _ 1 p \ / 2 6 + 3 V 3<?+2(t.-l)pn+2 2l232£+6(b-l) p2n+6 where e £ {±1} is the residue of3b lp modulo 4. 6. there exist integers £ > 2 — b and n > 0 such that 2 6 — 3ipn is a square and E is Q-isomorphic to one of the elliptic curves: «2 a4 A F l e - 3 b - V \ / 2 6 - 3 V l 2 4 3 2 ( 6 - l ) p 2 _ 2 l 2 3 < ° + 6 ( b - l ) p 7 1 + 6 F2 -t • 2- 3h~lp^26 - 3lpn _ 3 « + 2 (6 - l ) p n+ 2 2 1 2 3 2 £ + 6 ( b - l ) p 2 n + 6 where e £ {±1} is the residue of3b lp modulo 4. 7. there exist integers £ > 2 — b and n > 0 such that 3e — 26pn is a square and E is Q-isomorphic to one of the elliptic curves: a 2 a4 A G I t • 3 b ^ V \ / 3 ^ - 26pn ~2 4 3 2 ( b - 1 V+ 2 2l23C+6(b-l)p2n+6 G2 -e-2-3 b - 1 p\/3 ' ! -2V 3«+2(b-l) p2 _2l232f+6(b-l) pn+6 where e G { ± 1 } is the residue of3b lp modulo 4. 8. there exist integers £>2 — b and n > 0 such that pn - 263e is a square and E is Q-isomorphic to one of the elliptic curves: 0,2 a4 A H I e - 3 b - W P n - 263<» _ 2 4 3 « + 2 ( b - l ) p 2 2 1 2 3 2 £ + 6 ( b - l ) p n + 6 H2 -e -2-3 b " 1 p\ /p n - 263^ 32 (b - l ) p n+2 _2l23*?+6(b-l) p2n+6 where e £ {±1} is the residue of3b lp modulo 4. 9. there exist integers £ > 2 - b and t £ {0,1} such that is a square and E is Q-isomorphic to one of the elliptic curves: Chapter 3. Elliptic Curves with 2-torsion and conductor 2a3l3ps 93 0-2 (24 A 11 6 - 3 6 - y + y 2 6 3 ; + i 24 3^+2(6-l) p2t+l 21232*3+6(6-1)^3+64 12 £ . 2 . 3 6 - y + y 2 0 3 ; + i 32(6-1) 24+1 2l23*3+6(6-l)p3+6t where e € {±1} is the residue o /3 b l p t + 1 modulo 4. 10. there exist integers £ > 2 — b and t e {0,1} such that 2 6 3 p - 1 is a square and E is Q-isomorphic to one of the elliptic curves: a-2 a4 A J l e - 3 b - y + y 2 ° 3 ; - ' 2 43f+2(6- l ) p 2(+l _2l2 3 2*3+6(6- l)p3+6t J2 £ . 2 . 3 6 - y + y 2 « ' 3 « - i _32(i>-l)p2t+l 2123*3+6(b-l)p3+6t where e <E {±1} is ffte residue o /3 b 1 p t + 1 modulo 4. 22. fftere exz'st integers £ > 2 — b and t € {0,1} swc/z f/za£ 2-^- is a square and E is Q-isomorphic to one of the elliptic curves: a 2 04 A KI 2 4 3 2 ( 6 - l ) p 2 t + l 2l23*3+6(6- l)p3+6t K2 £ . 2 - 3 6 - y + l N / 2 8 + 3 ' 3 « + 2 ( 6 - l ) 2 t + l 21232*3+6(6-1)^3+64 where e & {±1} is the residue of3b y + 1 modulo 4. 12. there exist integers £>2 — b and £ G {0,1} such that 2—^ is a square and E is Q-isomorphic to one of the elliptic curves: a2 (24 A LI 2 4 3 2 ( 6 - i y t + i _ 2 123*?+6(6- l)p3+6i L2 £ - 2 . 3 b - y + y 2 6 ; 3 ' _ 3*3+2(6- i y t + i 2 1232*3+6(6- iy+6i where e 6 {±1} is the residue o /3 b 1pt+1 modulo 4. 13. there exist integers £>2 — b and t e {0,1} such that 3<?~26 is a square and E is Q-isomorphic to one of the elliptic curves: Chapter 3. Elliptic Curves with 2-torsion and conductor 2a3l3p3 94 « 2 0,4 A Ml £ - 3 6 - y + y 3 ' ; 2 8 _ 2 4 3 2 ( b - l ) p 2 t + l 2l23^+6(6 - l )p3+6t M2 e . 2 . 3 6 - y + y 3 ' - 2 6 3 « ? + 2 ( b - l ) 2t + l _ 2 l 2 3 2 « + 6 ( b - l ) p 3 + 6 « zu/jere e 6 {±1} is the residue of3b 1pt+1 modulo 4. In the case that 6 = 2, i.e. N = 2 • 32p2, we furthermore could have one of the following conditions satisfied: 14. there exist integers n > 0 and s £ {0,1} such that 2b~^p is a square and E is Q-isomorphic to one of the elliptic curves: 0,2 a.4 A NI e • 3- V x / ^ 2 43 2 s + 1 p 2 21233+6.5^71+6 N2 t • 2 • y+ipyJ*&i 3 2 s + l p n + 2 2 l233+6s p 2n+C where e e {±1} is the residue of3s+1p modulo 4. 15. there exist integers n > 0 and s e {0,1} such that 2 & gP" is a square and E is Q-isomorphic to one of the elliptic curves: 02 a4 A Ol 2 4 3 2 S + I p 2 _ 2 1 2 3 3 + 6 . S p n + 6 02 -e • 2 • 3 ^ P y J ^ _ 3 2 s + l n + 2 2 12334 -6Sp? i - f 6 where e G {±1} is the residue of3s+1p modulo 4. 16. there exist integers n > 0 and s e {0,1} such that p ' 3 2 is a square and E is Q-isomorphic to one of the elliptic curves: a2 a4 A PI -2432s+1p2 2 1 2 3 3 + 6 s p n + 6 P2 -e • 2 - 3-^pJ^ 3 2 s + l p n + 2 — 2 1 2 3 3 + 6 s p 2 n + 6 where e G {±1} is the residue of3s+1p modulo 4. Chapter 3. Elliptic Curves with 2-torsion and conductor 2cc3t3ps 95 Theorem 3.22 The elliptic curves E defined over Q, of conductor 2-3bp2, and having at least one rational point of order 2, are the ones such that one of the following conditions is satisfied: 1. there exist integers m>7,£>2-b and n > 0 such that 2rn3epn + 1 is a square and E is Q-isomorphic to one of the elliptic curves: a2 (I4 A A l e • 3b-1p^2m3epn + 1 2m-23e+2(b-l)pji+2 2 2 m 3 2 < ° + 6 ( 6 - l ) p 2 7 i + 6 A 2 -e • 2 • 3b-1py/2m3epn + 1 3 2 ( 6 - i y 2m+63e+6(b-l)pn+6 where e e {±1} is the residue of3b 1p modulo 4. 2. there exist integers m > 7, £ > 2 — b and n > 0 such that 2rn3e + pn is a square and E is Q-isomorphic to one of the elliptic curves: 0-2 a 4 A BI e • 3b-1py/2m3e +pn 2 m - 2 3 « + 2 ( h - l ) p 2 22m32t?+6(6-l)p7i+6 B2 -e-2-3b-1p^2m3e +pn 3 2 ( b - l ) p 7 i + 2 277i+63<?+6(b-l)p2n+6 where e G {±1} is the residue of3b Yp modulo 4. 3. there exist integers rn > 7, £ > 2 — b and n > 0 such that 2r"3e - pn is a square and E is Q-isomorphic to one of the elliptic curves: 02 a4 A C l e-3b~1p^2m3e -pn 2 m - 2 3 « + 2 ( 6 - l ) 2 __22 TO 3 2^+6 (b - l )p77+6 C2 -e- 2- 3b-1py/2m3e -pn _ 3 2 ( b - l ) p 7 i + 2 2 m+6 3 <?+6 ( b - l ) p 2n+6 where e G { ± 1 } is the residue of3b 1p modulo 4. 4. there exist integers m > 7, £ > 2 — b and n > 0 such that 2mpn + 3e is a square and E is Q-isomorphic to one of the elliptic curves: 02 a4 A D I e • 3b~1py/2mpn + 3 £ 2 m - 2 3 2 ( b - l ) p n + 2 2277i3<?+6(b-1 )p2n+G D2 -e-2-3b-1p^2mpn + 3e 3 £ + 2 ( b - l ) p 2 2 7 7 i + 6 3 2 t ? + 6 ( b - l ) p T i + 6 where e G { ± 1 } is the residue of3b 1p modulo 4. Chapter 3. Elliptic Curves with 2-torsion and conductor 2a3l3p6 96 5. there exist integers m > 7, £ > 2 — b and n > 0 such that 2rn + 3epn is a square and E is Q-isomorphic to one of the elliptic curves: a 2 a.4 A E l e • 3b-lp\/2m + 3 V 2m-232(b-l)p2 E2 -e • 2 • Zh~1pyj2m +Z*pn 3 « ? + 2 ( b - l ) p n + 2 2m+6g2e+6(b- l)p2n+6 where e G {±1} is the residue of3b lp modulo 4. 6. there exist integers m > 7, £ > 2 — b and n > 0 such that 2rn — 3epn is a square and E is Q-isomorphic to one of the elliptic curves: a 2 04 A FI e • 3h~1py/2m - 3 V 2 m - 2 3 2 ( 6 - l ) p 2 _22m^e+6(b-l) pn+6 F2 -t • 2 • 3 b - 1 p v / 2 m - 3 V _ 3 f 3 + 2 ( b - l ) p n + 2 2m+6 3 2(?+6 (b -1 )p2n+6 where e € {±1} is the residue of3b 1p modulo 4. 7. there exist integers m>7,£>2-b and n > 0 szzc/z tTzaf 3epn — 2m is a square and E is Q-isomorphic to one of the elliptic curves: a.2 0,4 A GI e • 3 f c - 1 p v / 3 V - 2 m _2m-2^2(b-l)p2 2 2 7 7 i 3 * ? + 6 ( 6 - l ) p n + 6 G2 - e • 2 • 3b~1py/3epn - 2m 3 ^ + 2 ( b - l ) ^ n + 2 _ 2 " i + 6 3 2 « + 6 ( b - l ) p 2 n + 6 zf/zere e e {±1} z's zTze residue of3b lp modulo 4. 8. t/zere exist integers m > 7, £ > 2 — b and n > 0 sizc/z that 3e — 2rnpn is a square and E is Q-isomorphic to one of the elliptic curves: a 2 a4 A HI e3 6 "Vv / 3 £ - 2mpn _ 2 " J - 2 3 2 ( b - l ) p n + 2 22m^e+6(b-l)p2n+6 H2 -e2 • tf-^py/tf - 2mpn 3«+2 (b- l )p2 _2m+632f?+6(6-l)pn+6 where e € {±1} z's f/ze residue of 3b 1p modulo 4. 9. ffoere exist integers m > 7, £ > 2 — b and n > 0 swc/z f/zaf p" - 2m3^ z's a square and E is Q-isomorphic to one of the elliptic curves: Chapter 3. Elliptic Curves with 2-torsion and conductor 2ct3l3p5 97 a-2 0.4 A 11 e • 3h~lpy/pn - 2m3e _2m-23e+2{b-l)p2 22m32e+6(b-l)pn+6 12 -e • 2 • 3b-lp^/pn - 2m3e 32{b-l)pn+2 _2m+63e+6(b-l)p2n+6 where e G {±1} is the residue of3b 1p modulo 4. 10. there exist integers m>7,£>2-b and t G {0,1} such that 2 T "^+ i js a square and E is Q-isomorphic to one of the elliptic curves: a 2 0,4 A JI e-3b-Y+1yJ2m3'+1 2m-23l.+2(b-\)p2t+l 2277132^+6(6-1 )^3+6* J2 e - 2 - 3 h - ' p t + 1 ^ 2 i n 3 e + l 3 2 ( 6 - 1 ) 2t+l 2 7 7 1 + 6 3 ^ + 6 ( 6 - l ) p 3 + 6 t where e G {±1} is the residue of 3b 1pt+1 modulo 4. 11. there exist integers m > 7, £ > 2 - b and t G {0,1} such that 2'"^~1 is a square and E is Q-isomorphic to one of the elliptic curves: a 2 04 A K I e 3 6 - y + y 2 " . 3 < - i 2 7 7 1 - 2 3 ^ + 2 ( 6 - l ) p 2 t + l _ 2 2 m 3 2 ) ? + 6 ( 6 - l ) p 3 + 6 i K2 £ 2 • s b - y + y s " ' 3 ; - 1 _ 3 2 ( 6 - l ) 2t+l 2 7 7 7 + 6 3 ^ + 6 ( 6 - l)p3+6t where e G {±1} is the residue of3b l p i + l modulo 4. 12. there exist integers m>7,£>2-b and t G {0,1} such that 2T"+3<! is a square and E is Q-isomorphic to one of the elliptic curves: a 2 0 4 A L I e - 3 f t - y + y 2 7 3 f 2 7 7 7 . - 2 3 2 ( 6 - 1 ) ^ 2 4 + 1 2277i3ts+6(6-l)p3+6t L2 _ e . 2 • 3 f c - y + y 2 = ± £ 3 ^ + 2 ( 6 - 1 ) 2t+l 2TO+632»?+6(6- 1)^3+6* where e e {±1} is the residue of3b 1pt+1 modulo 4. 13. there exist integers m > 7, £ > 2 — b and t e {0,1} such that 2 ' " ~ 3 £ is a square and E is Q-isomorphic to one of the elliptic curves: Chapter 3. Elliptic Curves with 2-torsion and conductor 2°3l3p6 98 0.2 0,4 A M l £ . 3 6 - y + y 2™^3£ 2 m -2 3 2(b - l )p2 t+ l _ 2 2 m 3 « ? + 6 ( b - l ) p 3 + 6 t M 2 e-2-Z»-V+1y/2m;3' _3*i+2(b-l)p2t+l 2 m + 6 3 2 » ? + 6 ( b - l ) p 3 + 6 t w/zere e G {±1} zs fTze residue ofSb l p t + l modulo 4. 24. there exist integers m > 7 , £ > 2 - b and t G {0,1} swdz that 3^~2T" z's a square and E is Q-isomorphic to one of the elliptic curves: a 2 a 4 A N I e 3 b - y + y 3 £ - _ 2 ^ - 2 " l _ 2 3 2 ( b _ 1 ' p 2 t + 1 22m 3*?+6(b- l)p3+6t N2 -e2 • 3 " - V+y 3 £ f = - 3«?+2(b-l) 2 t+l _ 2 m + 6 3 2 * + 6 ( ' , - 1 ) j r , 3 + 6 i where e G {±1} is the residue o/3b 1 p t + 1 modulo 4. 7M zTze case zTzaf 6 = 2, z'.e. N = 2 • 32p2, we furthermore could have one of the following conditions satisfied: 15. there exist integers m, > 7, n > 0 and s G {0,1} such that 2™ * p" z's a square and E is Q-isomorphic to one of the elliptic curves: 0.2 a 4 A O l 2 m - 2 3 2 3 + y 2 22777,^ 3 + 65^71 + 6 0 2 - e . 2 . 3 - + ^ ^ 32.s+l pn+2 2^+6^3+65^271+6 where e G {±1} is the residue of3s+1p modulo 4. 16. there exist integers m > 7, n > 0 anti s G {0,1} such that 2 ^p is a square and E is Q-isomorphic to one of the elliptic curves: a 2 a 4 A PI e . 3 * + 2 , „ - 2 3 2 s + l p 2 P2 -e . 2 • 3 - + J P v / 3 ^ _ 3 2s+l p n+2 2*71+6^3+65^271+6 where e G {±1} is the residue of3s+lp modulo 4. Chapter 3. Elliptic Curves with 2-torsion and conductor 2 oc3 l3p s 99 17. there exist integers m > 7, n > 0 and s G {0,1} such that '-—3— is a square and E is Q-isomorphic to one of the elliptic curves: 0.2 04 A Q l e • 3 - 5 + V \ / P " a 2'" _ 2 m - 2 3 2 s + l p 2 Q2 - e - 2 - ^ p s J ^ 32s+l p n+2 where e G {±1} is the residue of3s+lp modulo 4. 18. there exist integers m > 7 and s, t e {0,1} such that 2-^T" l s a square and E is Q-isomorphic to one of the elliptic curves: 02 04 A R l c-3 s+'p t+ 1^/ 2'"7i 1'" 2m-2^2s+lp2t+l 22m.^3-\-6Sp3-\-6t R2 e - 2 - 3 s + y + 1 x / 2 m 3 P " 32s+lp2t+l 2 +6 ^  3-f-6 s p 3 + 6 £ where e G {±1} is the residue of3s+1pt+1 modulo 4. 19. there exist integers rn > 7 and s,t e {0,1} such that 2 " 3 ~ 1 is a square and E is Q-isomorphic to one of the elliptic curves: a 2 04 A SI e . 3 , + y + y 2 ^ - 2 m - 2 3 2 s - + - l p 2 t + l — 2 2 m 3 3 + 6 s p 3 + 6 ' -S2 £ - 2 - 3 s + y + I N / 2 " , 3 P " _ 3 2 s + l p 2 t + l 2 m + 6 3 3 + 6 s p 3 + 6 i tf«ere e G {±1} is the residue of3s+lpt+1 modulo 4. Theorem 3.23 The elliptic curves E defined over Q, of conductor 223bp2, and hav-ing at least one rational point of order 2, are the ones such that one of the following conditions is satisfied: 1. there exist integers £ > 2 — b and n > 0 such that 4 • 3e + pn is a square, 3e = — 1 (mod 4) and E is Q-isomorphic to one of the elliptic curves: a2 04 A A l (.•3 h~ lp^A-3 l + p n 3e+2(b-i)p2 24g2£+6(6- l ) n+6 A 2 -e-2-3 b- 1py/A-3 f- +p n 3 2 ( b - l ) p n + 2 2 8 3 M - 6 ( b - l ) p 2 n + 6 Chapter 3. Elliptic Curves with 2-torsion and conductor 2a3@p5 100 where e G {±1} is the residue of3b lp modulo 4. 2. there exist integers £ > 2 — b and n > 0 such that 4 • 3e — pn is a square, 3e = — 1 (mod 4) and E is Q-isomorphic to one of the elliptic curves: a2 OL4 A B l e • 3 b-V\/ 4 -3e - P n 3 « + 2 ( f c - l ) p 2 _2432^+6(fe-1)pn+6 B2 -t-2-3h~xpy/4-3l -pn _ 3 2 ( 6 - l ) p n + 2 283^+6(b-l) p 2n+6 where e G {±1} is the residue of3b lp modulo 4. 3. there exist integers £ > 2 — b and n > 0 such that 4pn — 3e is a square, pn = -1 (mod 4), and E is Q-isomorphic to one of the elliptic curves: a-2 a4 A C I e • 3b~1px/4pn - 3e 32 (6-1)^+2 _ 2 4 3 ^ + 6 ( b - l ) p 2 n + 6 C2 - e • 2 • 3h-xp^4pn - 3l- _ 3<?+2(6-l) p2 2 8 3 2 ^ + 6 ( 6 - 1 ) ^ + 6 where e G {±1} z's the residue of3b lp modulo 4. 4. there exist integers £ > 2 — b and n > 0 such that pn — 4 • 3e is a square, 3e =. 1 (mod 4), and E is Q-isomorphic to one of the elliptic curves: a2 0,4 A DI £-3b-1py/pn-4-3e _ 3 ^ + 2 ( 6 - l ) p 2 2 4 3 2 ^ + 6 ( 6 - l ) p n + 6 D2 -e-2-3b-1p^pn -4-3e- 3 2 ( 6 - i y + 2 _ 2 8 3 f + 6 ( 6 - l ) p 2 n + 6 where e G {±1} z's the residue of3b lp modulo 4. 5. there exist integers £ > 2 - b and t G {0,1} such that 4 ' 3 p + 1 z's a square, 3ep = — 1 (mod 4), and E is Q-isomorphic to one of the elliptic curves: 0.2 0,4 A E l ( . 36-y+y 4-3^+1 3«+2 ( 6 - l)p2t+ l 2432«?+6 ( 6 - l)p3+6t E2 e • 2 • 3b- y +y 4 3 ; + i 3 2 ( 6 - i y t + i 2 8 3^+6 ( 6 - l) p3+6t where e G {±1} z's the residue of3b 1 p t + 1 modulo 4. Chapter 3. Elliptic Curves with 2-torsion and conductor 2a3l3ps 101 6. there exist integers £ > 2 — b and t <E {0,1} such that 4 3 p 1 is a square, 3ep = — 1 (mod A), and E is Q-isomorphic to one of the elliptic curves: a-2 d 4 A FI 3<"+2(b-l)p2t+l _2432f+6(b-l)p3+6t F2 _ 3 2(b- l ) p 2t+ l 2832«?+6(6-l)p3+6t where e 6 {±1} is the residue of3b l p t + 1 modulo 4. 7. t/zere exist integers £ > 2 - b and t e {0,1} swc/t f/zflf z's a square, p = — 1 (mod 4), and E is Q-isomorphic to one of the elliptic curves: «2 a 4 A GI £ . 3 6 - y + 1 ^ 4 + ^ 1 3 2(b- l ) p 2t+l 243/?+6(b-l)p3+6i G2 - e . 2 • S ^ V + V ^ 1 3 « + 2 ( b - l ) 2t+l 28 32«+6(b-l)p3+6t <x>/zere e € {±1} is tTze residue of3b 1 p i + 1 modulo 4. 8. there exist integers £ > 2 - b and t e {0,1} such that z's a square, p = 1 (mod A), and E is Q-isomorphic to one of the elliptic curves: a2 a 4 A HI _ 3 2(b- l ) p 2t+ l 243<'+6(b-l)p3+6t H2 3^+2(b-l)p2t+l _2832f+6(6-i)p3+6t where e e {±1} is f/ze residue of3b V t + 1 modulo A. In the case that 6 = 2, i.e. AT = 2 2 3 2 p 2 , we furthermore could have one of the following conditions satisfied: 9. there exist integers n > 0 and s 6 {0,1} such that 4 p " 3 "" 1 z's a square, pn = 1 (mod 4), and £ is Q-isomorphic to one of the elliptic curves: «2 a 4 A 11 32s+lpn+2 - 2 4 3 3 + 6 s p 2 n + 6 12 t-2-y+lp\lAp"zx - 3 2 s + 1 p 2 2833 + 6 8 ^ 7 1 + 6 Chapter 3. Elliptic Curves with 2-torsion and conductor 2a3l3p5 102 where e € {±1} is the residue of 3s lp modulo 4. 10. there exists an integers n > 0 and s G {0,1} such that p-^- is a square and E is Q-isomorphic to one of the elliptic curves: 0.2 (14 A Jl 3 2 s + y 2433+6 . 5^71+6 J2 e • 2 • 3s+1PxJpn+4 g2 .s+l pn+2 28g3+6 .Sp2n+6 where c. G {±1} is the residue of 3s lp modulo 4. Theorem 3.24 The elliptic curves E defined over Q, of conductor 2 3 3 6 p 2 / and hav-ing at least one rational point of order 2, are the ones such that one of the following conditions is satisfied: 1. there exist integers m G {4, 5}, £ > 2 - 6 and n > 0 such that 2m3epn + 1 is a square and E is Q-isomorphic to one of the elliptic curves: a2 a 4 A A l e • 3b~lpy/2m3ipn + 1 2m-23^+2(6-l)pn+2 22m .32£+6(f)- l) p 2n+6 A 2 - e • 2 • 3 6 _ 1 pv / 2 m 3V l + 1 3 2(fc-l )p2 2^+63^+6(6- l) pn+6 where e G {±1} is the residue of 3b 1p modulo 4. 2. there exist integers £ > 2 — b and n > 0 swc« tTzflf 4 • 3e + pn is a square, 3e = 1 (mod 4), and E is Q-isomorphic to one of the elliptic curves: 02 a.4 A B l -e-3b~1py/4-3e +pn 3fJ+ 2 ( 6-l ) p 2 24 32^+6(6 -l) pn+6 B2 e-2-3b-ipy/4-3e + pn 32(6-l)pn+2 283<"+6(6~l) p2Ti+6 where e G {±1} is the residue of3b 1p modulo 4. 3. there exist integers m G {4,5}, £ > 2 - b and n > 0 such that 2m3e + pn is a square and E is Q-isomorphic to one of the elliptic curves: a 2 0,4 A . CI e • 3 6 _ 1 pv / 2 m 3 € +pn 2m - 2 3 « + 2 ( 6-l ) p 2 22m32£+6(6-l)pn+6 C2 - e • 2 • 3b-1py/2m3e +pn 32(b-l) pn+2 2m+63<2+6(6-l)p2n+6 Chapter 3. Elliptic Curves with 2-torsion and conductor 2a3f3ps 103 where e G {±1} is the residue of3b 1p modulo 4. 4. there exist integers £ > 2 — b and n > 0 such that 4 • 3e — pn is a square, 3e = 1 (mod 4), and E is Q-isomorphic to one of the elliptic curves: G2 (I4 A DI -e • 3h-1ps/4 • 3e - p n 3 ^+2(6 -V - 2 4 3 2 ^ + 6 ^ _ 1 ^ p n + 6 D2 e ^ ^ ^ p s / A ^ 1 1 - p n _ 3 2(b- l ) p n+2 283t"+6(c>-l)p2n+6 where c G { ± 1 } is the residue of3b 1p modulo 4. 5. there exist integers m 6 {4, 5}, £ > 2 - b and n > 0 such that 2m3e - p" is a square and E is Q-isomorphic to one of the elliptic curves: «2 a4 A E l e-3h-ipy/2m3e - p n 277i-23<M-2(6-l)p2 —2 2 m 3 2 f + 6 ( b _ 1)pn+G E2 - c - 2 - 3 b - 1 p v / 2 m 3 ^ - p n _ 32(fc-l) n+2 2m+63<?+6(b-l) p2n+6 where e G {±1} is the residue of3b 1p modulo 4. 6. there exist integers m G {4,5}, £ > 2 - b and n > 0 such that 2mpn + 3e is a square and E is Q-isomorphic to one of the elliptic curves: 0.2 a.4 A F l e • 3b-Ypyj2™pn + 3e 2m-232(b~l)pn+2 22m.3<'+6(6-l)p27i+6 F2 -e • 2 • 3 b - V \ / 2 m p " + 3e 3C+2(6-l) p2 2m+632<!+6(b-l)p?i+6 where e G {±1} is the residue of3b 1p modulo 4. 7. there exist integers £ > 2 — b and n > 0 such that 4pn — 3e is a square, pn = 1 (mod 4), and E is Q-isomorphic to one of the elliptic curves: 0.2 a.4 A G I -e • 3b~1py/4pn - 3 £ 32(b-l) pn+2 _ 2 4 3 « " + 6 ( b - l ) p 2 n + 6 G2 £ • 2 • 3b~1psj4pn - 3* _3**+2(b-l)p2 2 8 3 2 £ + 6 ( 6 - l ) p n + 6 where e G {±1} is the residue of3b lp modulo 4. 8. there exist integers £ > 2 - b and n > 0 such that 4 + 3epn is a square and E is Q-isomorphic to one of the elliptic curves: Chapter 3. Elliptic Curves with 2-torsion and conductor 2a3l3ps 104 «2 0,4 A H I -e • 3b~lp^/4 + 3epn 3 2(J,-l) p 2 24 3 £+6(f . - l )pn+6 H2 e-2-3b-^p*J'4 + 3ipn 3£+2(b-l)pn+2 2832«?+6(b-l)p2n+6 where e € {±1} is the residue of3b 1p modulo 4. 9. there exist integers m, e {4,5}, £ > 2 — b and n > 0 such that 2 m + 3ep" is a square and E is Q-isomorphic to one of the elliptic curves: a 2 (Z4 A n e • 3 f t - 1 pv / 2 m + 3 V 2m-2 32(b-l)p2 22m3<»+6(b-l)pn+6 12 - e - 2 - 3 b - 1 p V ' 2 m + 3 V 3«+2(b-l)pn+2 2m+632«+6(b- 1 )p2n+6 where e G {±1} is the residue of 3b 1p modulo 4. 10. there exist integers m e {4, 5}, £ > 2 — b and n > 0 such that 2 m — 3^pn fs a square and E is Q-isomorphic to one of the elliptic curves: a2 04 A JI e • 3h-ipv/2m - 3epn 2 m - 2 3 2 ( b - l ) p 2 _22m3<f+6(b-l)pTi+6 J2 - e - 2 - 3 b - 1 p \ / 2 m - 3 V 1 _ 3 £+2(b- l )pn+2 2m+632 l?+6(b-l)p2n+6 where e £ {±1} is the residue of3b 1p modulo 4. 11. there exist integers m e {4, 5}, £ > 2 - b and n > 0 such that 3epn - 2m is a square and E is Q-isomorphic to one of the elliptic curves: a-2 0 4 A K I e • 3b-1p^3epn - 2 m _ 2 " i - 2 3 2 ( b - l ) p 2 2 2 7 7 1 3 ^ + 6 ( 6 - l ) p ? i + 6 K2 - e - 2 - 3 f t - 1 p v / 3 V - 2 m 3£+2 (b- l )pn+2 _ 2 7n+ 6 3 2 i " + 6 ( b - l ) p 2 n + 6 where e e {±1} is the residue of3b 1p modulo 4. 12. there exist integers m e {4,5}, £ > 2 - 6 and n>0 such that 3e - 2mpn is a square and E is Q-isomorphic to one of the elliptic curves: «2 04 A L I e • 3b~1p^3e - 2mpn _2«i--232(b-l)p7i+2 22m3»?+6(b-l)p27i+6 L2 -e • 2 • 3b~lps/3i - 2 mp" 3 £+2(6-l)p2 _2T O + 6 3 2 ^+ 6 ( t > - ; 1 )p™+6 Chapter 3. Elliptic Curves with 2-torsion and conductor 2a3!3p6 105 where e e { ± 1 } is the residue of3b 1p modulo 4. 23. there exist integers £ > 2 — b and n > 1 swcn that pn — 4 - 3^ is a square, 3e = — 1 (mod 4), and 2? is Q-isomorphic to one of the elliptic curves: « 2 a 4 A M l - e • 3b-1ps/pn -4-3e „ 3 ^ + 2 ( f e - l ) p 2 2 4 3 2 ^ + 6 ( b - l ) p n + 6 M 2 t-2- 3b-1py/pn -4-31 3 2 ( 6 - l ) p n + 2 _ 2 8 3 ^ + 6 ( 6 - l ) p 2 n + 6 where e £ { ± 1 } is the residue of3b 1p modulo 4. 14. there exist integers rn e {4,5}, I > 2 — b and n > 0 such that pn — 2m3e is a square and E is Q-isomorphic to one of the elliptic curves: « 2 aA A N I t • 3b~lPy/pn - 2m3e _ 2 m - 2 3 ^ + 2 ( 6 - l ) p 2 22m£2e+6(b-l)pn+6 N2 t-2- 3b-lp^pn - 2m3e 3 2 ( 6 - l ) p ? i + 2 _2m+6^e+6{b-l)p2n+6 where e G { ± 1 } is the residue of3b 1p modulo 4. 15. there exist integers £ > 2 — b and t e {0,1} such that 4 ' 3 ^ + 1 is a square, 3ep = 1 (mod 4) and E is Q-isomorphic to one of the elliptic curves: a-2 a 4 A O l £ -3"-y + V 4 ' 3 p + 1 3 f + 2 ( b - l ) p 2 t + l 2 4 3 2 f + 6 ( b - l ) p 3 + 6 t 0 2 £-2.3"-y+1v/^±i 3 2 ( b - l ) p 2 t + l 2 83*!+6(b- l)p3+64 where e £ { ± 1 } is fne residue of3b 1 p t + 1 modulo 4. 26. fn<?r<? exist* integers rn £ {4,5}, £ > 2 — b and t € {0,1} swcn f/W 2 , " 3 < + 1 is a square and E is Q-isomorphic to one of the elliptic curves: a-2 a 4 A PI £ - 3 6 - y + y 2 m 3 ; + i 2m-23<?+2(6-l) 2 t+l 2 2 m 3 2 f + 6 ( b - l ) p 3 + 6 t P2 - e . 2 . 3 " - y + y ^ f t i 3 2 ( 6 - l ) p 2 t + l 2?n+6 3t !+6(b - l ) p 3+6i where e G {±1} is the residue of3b 1pt+1 modulo 4. Chapter 3. Elliptic Curves with 2-torsion and conductor 2a3@ps 106 17. there exist integers £ > 2 - b and t € {0,1} such that 4 3 p 1 is a square, 3ep = 1 (mod 4) and E is Q-isomorphic to one of the elliptic curves: 0,4 A Q l 3 ^ + 2 ( 6 - l ) p 2 t + l _2432<»+6 (b- l )p3+6t Q2 _ 3 2 ( 6 - l ) 2 i + l 2 8 3 ^ + 6 ( 6 - 1 ) 3 + 6 < where e £ {±1} is the residue o/3 6 l p t + 1 modulo 4. 18. there exist integers m <E {4, 5}, £ > 2 - b and t e {0,1} swcn that 2 " ' 3 J ~ 1 is A square and E is Q-isomorphic to one of the elliptic curves: a2 a4 A R l c.^-y+y2"'3;-1 2 m - 2 3 « + 2 ( b - l ) p 2 t + l _22m32 t !+6 (b - l )p3+6t R2 e . 2 - 3 b - y + 1 ) / 2 ' " 3 ' - 1 _ 3 2 ( 6 - l ) p 2 t + l 2m+63f?+6 (b - l )p3+6t where e € {±1} is the residue of3b 1 p t + 1 modulo 4. 19. there exist integers £ > 2 — b and t e {0,1} swc« that is a square, p = 1 (mod 4) and E is Q-isomorphic to one of the elliptic curves: a2 a4 A SI - e - 3 b - y + y ^ 32(1—1) 21+1 243*!+6(b-l)p3+6t S2 e . 2 . 3 " - y+y^ 3 « + 2 ( b - l ) p 2 t + l 2 8 3 2 « + 6 ( b - l ) p 3 + 6 t where e 6 {±1} is the residue o/3 b 1 p t + 1 modulo 4. 20. there exist integers m 6 {4,5}, £ > 2 — b and t€ {0,1} SUCH f/wf 2^~- is a square and E is Q-isomorphic to one of the elliptic curves: a2 a4 A TI £-3b-y+y2";+3C 2 m - 2 3 2 ( b - l ) p 2 t + l 22Tn3<?+6(b- 1 )p3+6t T2 e -2 -3 b - y+y 2 7 3 ' 3 « + 2 ( b - l ) p 2 t + l 2 7 7 1 + 6 3 2 ^ + 6 ( 6 - 1 ) ^ 3 + 6 * where e € {±1} is t«e residue of3b 1 p t + 1 modulo 4. Chapter 3. Elliptic Curves with 2-torsion and conductor 2a3@p6 107 21. there exist integers m G {4,5}, £ > 2 - b and t G {0,1} such that 2 m p 3 is a square and E is Q-isomorphic to one of the elliptic curves: 0 4 A U l 2rn -232 (b - l )p2 t+ l _22m3»f+6(6-l)p3+6t U2 - e • 2 • 3 « - y + V _ 3 « + 2 ( b - l ) i D 2 t + l 2 m + 6 3 2 £ + 6 ( 6 - l ) p 3 + 6 t where t G {±1} fs the residue o/3fe- y + 1 modulo 4. 22. fnere exisf integers £ > 2 - b and t 6 {0,1} such that is a square, p = - 1 (mod 4), and E is Q-isomorphic to one of the elliptic curves: 0 4 A V I - c - 3 b - y + y ^ _32(b-l)p2(+l 2432«+6(b-l) 3+6t V2 e • 2 • 3"-y+y 2 ^ 3^+2(6-1) 2t+l _ 2 8 3 ( !+0 {b - l )p3+6 t where e G {±1} fs the residue of3b 1pt+1 modulo 4. 23. inert? exfsf integers m G {4,5}, £ > 2 — b and t G {0,1} swcra that 3 t ~ 2 "* fs fl square and E is Q-isomorphic to one of the elliptic curves: a 2 0 4 A W I ^ - y - V ^ - 2m _ 2 3 2 ( 6 _ 1 ) p 2 t + 1 22m 31^+6(6- 1 )p3+6t W2 e 2 . 3 " - y + 1 V / 3 ' - p 2 ' " 3<?+2(6-l)p2t + l _ 2 ' » + 6 3 2 < ? + 6 ( b - l ) p 3 + 6 t where e G {±1} fs fne residue o/3 b y + 1 modulo 4. Jn f«e case f/wf 6 = 2, f.e. = 2 3 3 2 p 2 , we furthermore could have one of the following conditions satisfied: 24. there exist integers m G {4,5}, n > 0, and s G {0,1} swc/z that 2 3 P fs A square and E is Q-isomorphic to one of the elliptic curves: 02 04 A X I e . 3*+ 2 m - 2 3 2 , + l p 2 X2 e-2-3s+lVsJ2-^ 32s+lpn+2 2?7r+6^34-6Sp2n+6 Chapter 3. Elliptic Curves with 2-torsion and conductor 2a3l3p5 108 where e 6 {±1} is the residue o / 3 s + 1 p modulo 4. 25. there exist integers rn G {4, 5}, n > 0, and s € {0,1} such that 2'"~p" is a square and E is Q-isomorphic to one of the elliptic curves: a2 a 4 A YI 2 m - 2 3 2 . 5 + l p 2 Y2 _ e . 2 • 3 ^ p y / ^ - _ 3 2 s +l p n + 2 2m+6 ^ 3+6s^2n+6 where e 6 {±1} is the residue of3s+1p modulo 4. 26. there exist integers n > 1 and s € {0,1} such that 4p,'3~1 is a square, pn = — 1 (mod 4) and E is Q-isomorphic to one of the elliptic curves: a 2 a 4 A ZI e . 3 * + ^ ^ 3 2 s + l p T i + 2 _ 2 4 3 3 + 6 s p 2 n + 6 Z2 e • 2 • 3 8 + 1 P y ^ l = i - 3 2 s + V 2833+6sp?x+6 where e G {±1} is the residue of3s+1p modulo 4. 27. there exist integers n > 1 and s G {0,1} swch that ^Ap4- is a square and E is Q-isomorphic to one of the elliptic curves: a2 a.4 A ZI _ 3 2 s + i y 2433+65^71+6 Z2 e • 2 • 3s+1p^Jp"34 325+lpTl+2 — 2 8 3 3 + 6 s j > 2 n + 6 where e G {±1} is the residue o / 3 s + 1 p modulo 4. 28. there exist integers m G {4, 5}, n > 1, and s G {0,1} such that p ~ 2 is a square and E is Q-isomorphic to one of the elliptic curves: a 2 a 4 A AA1 C • 30+lpy/*^. _ 2 m - 2 3 2 S + l p 2 22m^3+6spTi+6 AA2 _C.2.3'+V^ 32.5+1^71+2 2m+^33+6.Sp2n+6 where e G {±1} is the residue of3s+1p modulo 4. Chapter 3. Elliptic Curves with 2-torsion and conductor 2a3l3ps 109 29. there exist integers m e {5}, s e {0,1} and t G {0,1} such that is a square and E is Q-isomorphic to one of the elliptic curves: a-2 0 4 A BBl 2m-2g2s+lp2t+l 22m33+6Sp3+6t BB2 32s+lp2t+l 2^71+6 Q3+6S p3+6t where e G {±1} is the residue of3s+1pt+1 modulo A. 30. there exist integers m e {4}, s e {0,1} and t € {0,1} SMC/I frtflf 2 - ^ - is a square and E is Q-isomorphic to one of the elliptic curves: a2 a 4 A CC1 2 m - 2 3 2 s + l p 2 t + l _ 22m23+6Sp3+6t CC2 _ 3 2 s + l p 2 t + l 2"i+6^3+6.Sp3+6£ wrtere e G {±1} is fne residue of3s+1pl+1 modulo 4. Theorem 3.25 T«e elliptic curves E defined over Q, of conductor 243bp2, and hav-ing at least one rational point of order 2, are the ones such that one of the following conditions is satisfied: 1. there exist integers m > A, £ > 2 — b, and n > 0 such that 2m3epn + 1 is a square and E is Q-isomorphic to one of the elliptic curves: a2 a.4 A A l -e • 3 b - 1 p v / 2 m 3 V + 1 2m-23e+2(b-l)pn+2 2 2 m 3 2 f + 6 ( b - l ) p 2 7 i + 6 A2 e-2-3b-1py/2m3t-pn + 1 3 2 ( f c - i y 2 m + 6 3 t ' + 6 ( b - l ) p n + 6 where t G {±1} is the residue of3b lp modulo-A. 2. there exist integers £ > 2 - b and n > 0 such that A • 3e + pn is a square and E is Q-isomorphic to one of the elliptic curves: a2 o 4 A BI e-3b-lp^/A-3e+pn 3 « + 2 ( b - l ) p 2 2432f?+6(b-l)pn+6 B2 -e-2-3b-lp^A-3* +pn 3 2(6-l) p n+2 283<?+6(6-l)p2n+6 Chapter 3. Elliptic Curves with 2-torsion and conductor 2a3l3p5 110 where e G {±1} is the residue of3e+b lp modulo 4. 3. there exist integers m > 4, £ > 2 — b, and n > 0 such that 2m3e + pn is a square and E is Q-isomorphic to one of the elliptic curves: o 2 a4 A C l -e • 3 b - 1 pv / 2 m 3^ +pn 2m-23e+2(b-l)p2 2 2 m 3 2 £ + 6 ( b - l ) p n + 6 C2 t-2-3b-1p^/2m3e +pn 3 2 ( 6 - l ) p n + 2 2tn+63t?+6(t>- l)p2n+6 where e € {±1} is the residue of3b lp modulo 4. 4. there exist integers £ > 2 — b and n > 0 such that 4 • 3e — pn is a square and E is Q-isomorphic to one of the elliptic curves: 0 2 a.4 A DI e-3b~1py/4-3i -pn 3e+2(b-i)p2 _ 2 4 3 2 t ° + 6 ( b - l ) p T i + 6 D2 -e-2-3b-1p^4-3e ~pn - 3 2 ( b _ 1 ) p " + 2 2 8 3 « + 6 ( b - l ) p 2 n + 6 where e € {±1} is the residue of 3e+b lp modulo 4. 5. there exist integers m > 4, £ > 2 — b, and n > 0 such that 2rn3e — pn is a square and E is Q-isomorphic to one of the elliptic curves: o 2 a4 A E l -e •-3h~1py/2m3e -pn 2m-23e+2(b-\)p2 _22m32£+6(b-l)pn+6 E2 e • 2 • 3b-1p^2m3e - pn _32(b-l)pn+2 2m+63e+6{b-l)p2n+6 where c e {±1} is the residue of 3b xp modulo 4. 6. there exist integers m > 4, £ > 2 — b, and n > 0 such that 2mpn + 3e is a square and E is Q-isomorphic to one of the elliptic curves: o 2 04 A' F l -t • 3b-1p^/2mpn + 3e 2m-232(6- l)pn+2 22m3<"+6(b-l)p2n+6 F2 e • 2 • 3 b _ 1 p \ / 2 m P n + 3e 3 £+2 (b - l ) p 2 2m+632i'+6(6-l)pn+6 where e € {±1} is the residue of3b 1p modulo 4. 7. there exist integers £ > 2 — b and n > 0 such that 4pn — 3e is a square and E is Q-isomorphic to one of the elliptic curves: Chapter 3. Elliptic Curves with 2-torsion and conductor 2 Q 3 / V 5 UI a2 0,4 A GI e • 3b~1p^4pn - & 3 2 ( 6 - l ) p n + 2 _ 2 4 3 < " + 6 ( b - l ) p 2 n + 6 G2 -e • 2 • 3 b - 1 p ^ 4 p n - 3e _3«M-2(t.-l) 2 where e G {±1} is the residue of3b 1 p n + 1 modulo 4. S. fTzere exz'sf integers I > 2 — £> and n > 0 such that 4 + 3epn is a square and E is Q-isomorphic to one of the elliptic curves: a-2 0,4 A HI e - 3 6 - V \ / 4 + 3 V 3 2 ( f > - i y H2 -e -2 -3 ' ' - 1 p\ / 4 + 3 V 3 < ! + 2 ( b - l ) p n + 2 2832t ' +6 (b - l ) p 2n+6 where e G {±1} is the residue of3b xp modulo 4. 9. there exist integers m > 4, I > 2 — b, and n > 0 such that 2 m + 3epn is a square and E is Q-isomorphic to one of the elliptic curves: a2 a4 A 11 - £ - 3 h ~ 1 p y / 2 m + 3epn 2 m - 2 3 2 ( 6 - l ) p 2 22?n 3 ^ + 6 ( 6 - l ) p n + 6 12 e - 2 - 3 h - 1 p v / 2 m + 3 V 3 < : + 2 ( b - i ) p n + 2 2 i r a+6g2> :+6 (b - l )p2n+6 where e G {±1} is the residue of3b 1 p modulo 4. 10. there exist integers m > 4, I > 2 — b, and n > 0 such that 2m - 3epn is a square and E is Q-isomorphic to one of the elliptic curves: a 2 a4 A Jl -t • 3b-lpy/2m - 3epn 2 m - 2 3 2 ( b - i y _22m3<?+6(6-l) n+6 J2 e-2-3h-1p^2m - 3 ^ _ 3 ( ? + 2 ( b - l ) p n + 2 2m+ 6 3 2 * ? + 6 ( 6 - l ) p 2 n + 6 where e G {±1} is the residue of3b 1p modulo 4. 11. there exist integers m > 4, I > 2 — b, and n > 0 such that 3epn — 2rn is a square and E is Q-isomorphic to one of the elliptic curves: a 2 0 4 A KI -e • 3b-lpy/3epn - 2m _ 2 « J - 2 3 2 ( b - l ) p 2 22m^e+6(b-l) pn+6 K2 e • 2 • 3 b - 1 p s / 3 l p n - 2m 3 « + 2 ( b - l ) p n + 2 _ 2 m + 6 3 2 » ? + 6 ( b - l ) p 2 n + 6 Chapter 3. Elliptic Curves with 2-torsion and conductor 2ct3f3p5 112 where e € {±1} is the residue o/3 b lp modulo 4. 12. there exist integers m > 4, £ > 2 — b, and n > 0 such that 3e — 2mpn is a square and E is Q-isomorphic to one of the elliptic curves: a 2 a4 A LI —2" l- 23 2( b- 1)p n+ 2 22?Ti3('+6(l)-l)p2n+6 L2 e - 2 - 3 f c - W 3 £ - 2 m p " 3 £ + 2 ( 6 - l ) p 2 _2m+6^2e+6{b-l)pn+6 where e e {it 1} z's the residue o/3 b 1p modulo 4. 13. there exist integers £ > 2 — b and n > 0 such that pn - 4 • 3e is a square and E is Q-isomorphic to one of the elliptic curves: a-2 a.4 A M l t • 3h-lpyjpn - 4 • 3£ _ 3f!+2(fc-l) p2 2432f!+6(6-l)pn+6 M2 -t-2- 3h~xpyjpn -4-3t 32(b-l)pn+2 _283<?+6(6-l)p2n+6 where e e {±1} z's the residue of3e+bp modulo 4. 14. there exist integers m > 4, £ > 2 — b, and n > 0 such that pn — 2rn3e is a square and E is Q-isomorphic to one of the elliptic curves: a2 a4 A NI -e • 3 b _ 1 p \ /p n - 2m3* _ 2m-2 3<?+2(b-l) p2 22m32<?+6(b-l)pn+6 N2 t-2- 3 f t- ]p\/p" - 2m3* 32(ft-l)pn+2 _ 2 m + 6 3 « + 6 ( b - 1)^ 271+6 where e e {±1} is the residue of 3b 1p modulo 4. 15. there exist integers £ > 2 — b and t E {0,1} such that 4 3 * + 1 z's a square and E is Q-isomorphic to one of the elliptic curves: a 2 04 A O l £ . 3 6 - y + y 4.3^+1 2m 3<?+2(b-l) p2t+l 24 32*!+6(b-iy+6t 02 e . 2 • 3 6 - V + y ^ ± l 3 2 ( b - l ) p 2 t + l 2 8 3 2 « ? + 6 ( b - i y + 6 t where e e {±1} is the residue of3e+b 1pt modulo 4. Chapter 3. Elliptic Curves with 2-torsion and conductor 2"3/V5 113 16. there exist integers m > 4, £ > 2 — b and t G {0,1} such that 2 m 3 J + i is a square and E is Q-isomorphic to one of the elliptic curves: a-2 0-4 A PI 2m3e+2(b-l)p2t+l 2 2 7 n c j 2 £ + 6 ( b - l ) p 3 + 6 t P2 32(b-l) J 92(+l 2m+63<?+6(b- l)p3+6t where e G {±1} is the residue of3b 1 p t + 1 modulo 4. 17. there exist integers £ > 2 — b and t £ {0,1} such that 4'3p"1 is a square and E is Q-isomorphic to one of the elliptic curves: a2 a.4 A Q l e - s o - y + y 4 3 ; - 1 2m 3 i+2(b-l) 21+1 _ 24 32^+6(b-l) p3+6t Q 2 e • 2 . 3 & - y + y 4 - 3 ; - 1 _ 3 2(b- l ) p 2t+ l 283<"+6(b-l)p3+6t where e G {±1} is the residue of3e+b 1pt modulo 4. 18. there exist integers m > 4, £ > 2 - b and t G {0,1} such that 2m3*~1 is a square and E is Q-isomorphic to one of the elliptic curves: a2 a.4 A Rl ^ - y + y 2 " ' 3 ; - 1 2 m 3 < » + 2 ( b - ] ) p 2 / . + l _22moj2<f+6(b-l)p3+6t R2 £ . 2 - 3 " - y + y 2 ' " 3 ; - 1 _ 3 2(b- l ) p 2t+ l 2?n+63<f+6(b-l)p3+6t where e G {±1} is the residue of3b 1 p t + 1 modulo 4. 19. there exist integers £ > 2 - b and t G {0,1} such that js a square and E is Q-isomorphic to one of the elliptic curves: a2 04 A SI e . 3 b - y + y 4 ^ i 3 2 ( b - l ) p 2 £ + l 24 3 f + 6(6 - l)p3+ 6 t S2 -e • 2 • 3 ^ - y + y ^ f i 3 «?+2(b- l ) 2t+l 2 8 3 2f+6(b-l)p3+6t w«ere e G {±1} zs ifte residue of3b lpt modulo 4. Chapter 3. Elliptic Curves with 2-torsion and conductor 2c"3l3ps 114 20. there exist integers m > 4, £ > 2 - b and t G {0,1} such that 2m+3 is a square and E is Q-isomorphic to one of the elliptic curves: « 2 <24 A TI 2 m - 2 g 2 ( 6 - l ) p 2 t + l 2 2 m 3<?+6(6 - l )p3+6 t T2 £ - 2 - 3 b - y + y 2 y 3 < + 2 ( 6 - l ) p 2 i + l 2 m + 6 3 2 ( ? + 6 ( b - l ) p 3 + 6 « where e G {±1} is the residue of3b 1 p i + 1 modulo 4. 22. f/zere exist integers m > 4, £ > 2 - b and t G {0,1} such that 2 " p 3 < ! fs square and E is Q-isomorphic to one of the elliptic curves: 0-2 a 4 A U l _ £ . 3 6 - y + y 2 ^ 3 i 2 m - 2 3 2 ( 6 - l ) p 2 t + l _ 2 2 m 3 « + 6 ( 6 - l ) p 3 + 6 ( U2 e - 2 - 3 6 - y + y 2 m ; 3 ' _3«?+2(6-l) 2t+l 2m + 6 3 2 £ + 6 ( b - l ) p 3 + 6 t where a £ {±1} is the residue of3b y + 1 modulo 4. 22. there exist integers £ > 2 - b and t e {0,1} such that is a square and E is Q-isomorphic to one of the elliptic curves: a-2 a 4 A VI e - 3 " - y + i x / 3 ^ _ 3 2 ( b - l ) p 2 f + l 24 3 «+6 (b - l )p3+6t V2 - e - 2 . 3 b - y + y y 3«+2(b-l) 2t+l _ 2 8 3 2 f + 6 ( b - l ) p 3 + 6 £ where e G {±1} is the residue of3bpi modido 4. 23. there exist integers m > 4, £ > 2 - b and t G {0,1} such that 3^~2"' is a square and E is Q-isomorphic to one of the elliptic curves: a2 a 4 A WI _ e . 3 6 - 1 ^ + 1 ^ 3 5 ^ _2" i - 2 3 2 (b - l ) 2t + l 22m3<?+6(6-l)p3+6t W2 £ • 2 • 3 6 - y + y 3 ' ; 2 m 3 «+2 (b- l ) 2t+l _2m+632<?+6(b-l) p 3+6i Chapter 3. Elliptic Curves with 2-torsion and conductor 2a3l3ps 115 where e G {±1} is the residue of3b 1pt+1 modulo 4. In the case that b — 2, i.e. N — 2432p2, we furthermore could have one of the following conditions satisfied: 24. there exist integers n > 0 and s e {0,1} such that 4p7'~1 is a square and E is Q-isomorphic to one of the elliptic curves: a 2 a 4 A XI e . 3-+ 3 2 s + l p n + 2 _2433+6«p2n+6 X2 £ - 2 . 3 s + V V / 4 T 1 - 32 s + y 2833+6.5^71+6 w«ere e £ {±1} is the residue of3spn+1 modulo 4. 25. there exist integers n > 1 and s € {0,1} such that t^l+A js a square and E is Q-isomorphic to one of the elliptic curves: a 2 a 4 A YI 32 !+V2 2433+6.5^,77+6 Y2 e • 2 • 3 s + 1 P x f p " + 4 3 2 s + l p 7 7 + 2 2833+6.5^277+6 u;/zere c G {±1} is the residue of 3sp modulo 4. 26. there exist integers m > 4, n > 1 and s € {0,1} such that 2'"+P" is a square and E is Q-isomorphic to one of the elliptic curves: a 2 a 4 A ZI 2m~232s+1p2 2 2 m ^ 3 + 6 S p n + 6 Z2 e • 2 - 3s+1PyJ2'"+pn 3 2 s + l p 7 i + 2 2m+6^3+6Sp2n+6 ztf/iere e G { i l } zs the residue of3sp modulo 4. 27. there exist integers m > 4, n > 1 and s G {0,1} swc/z f/zaf 2 g P zs A square and E is Q-isomorphic to one of the elliptic curves: a 2 a 4 A AA1 e . 3-+ 2 m - 2 3 2 s + V _22m<j>3+6.$pTi+6 AA2 - £ . 2 . 3 ' + ^ ^ _ 3 2 S + l p n + 2 2m+623+65p2n+6 Chapter 3. Elliptic Curves with 2-torsion and conductor 2a3l3ps 116 where e e {±1} is the residue of3sp modulo 4. 28. there exist integer n > 1 and s G {0,1} such that is a square and E is Q-isomorphic to one of the elliptic curves: « 2 (I4 A BBl - 3 2 s + V 2 4 3 3 + 6 . 5 p n + 6 BB2 £ • 2 • Z'+1pyJ>£f± g 2 . s + l p n + 2 — 2 8 3 3 + 6 , 5 j 9 2 , l + 6 where e G {±1} zs tTze residue of3sp modulo 4. 29. fizere exzsf integers m > 4, n > 1 arzd s G {0,1} sz^ c/z f/zat1 p " g 2 " zs a square and E is Q-isomorphic to one of the elliptic curves: a 4 A CC1 _ 2 m - 2 3 2 » + l p 2 2 2 m ^ 3+6 s pn+6 CC2 32.S+I n + 2 _ 2^+633+6s^ 2 n - ( -6 z^ /zere e G {±1} zs z7ze residue of3sp modulo 4. 30. f/zere exz'sf integers m > 4 and s, £ G {0,1} swc/z rTzrtf 2 - ^ ~ zs a square and E is Q-isomorphic to one of the elliptic curves: a-2 a 4 A DDI 2 7 7 i . - 2 3 2 . s + l p 2 t + l 2 2 m 33+65^3+64 DD2 -£ • 2 • 3"+V+y £g±! 3 2 . s + l p 2 t + l 2 'm+633+65 p 3+6f . where e G {±1} is the residue of3spi+l modulo 4. 31. there exist integers m > 4 and s, t G {0.1} such that 2,"i~1 is a square and E is Q-isomorphic to one of the elliptic curves: a-2 a 4 A EE1 ^ • 3 s + v + y ^ 2m-232 .5+1^24+1 _ 2 2 m - 3 3 + 6 S p 3 + 6 t EE2 _ e . 2 • 3 « + y + y 2 = = i _ 3 2 s + l p 2 t + l 2 m + 6 3 3 + 6 s p 3 + 6 t where e G {±1} zs the residue of3spt+1 modulo 4. Chapter 3. Elliptic Curves with 2-torsion and conductor 2a3l3ps 117 T h e o r e m 3.26 The elliptic curves E defined over Q, of conductor 253bp2, and hav-ing at least one rational point of order 2, are the ones such that one of the following conditions is satisfied: 1. there exist integers £ > 2 — b and n > 0 such that 3epn + 1 is a square and E is Q-isomorphic to one of the elliptic curves: a2 a4 A A l 2 • 3 f t" V v / S V + 1 3 < " + 2 ( 6 - l ) p n + 2 2632£+6(6-l) p27i+6 A2 - 4 • 3b" V \ / 3 V + 1 4 • 3 2 ( f c -V 2 l 2 3 < ? + 6 ( 6 - l ) p 7 i + 6 A l ' -2 • 3b~1p^3ipn + 1 3 ^ + 2 ( 6 - 1 ) ^ 7 1 + 2 2 6 3 2 ^ + 6 ( 6 - 1 ) ^ 2 7 7 + 6 A2' 4 • 3 6-V\/3y + 1 4 . 3 2 ( b - i y 2l2g<!+6(6- l) p n+6 2. there exist integers £ > 1 and n > 0 such that 3e + pn is a square and E is Q-isomorphic to one of the elliptic curves: (a) £ is even; a 2 a4 A B l 2 -3 f e - 1 pv / 3*+P n 3 2 ( 6 - l ) p n + 2 263<!+6(6-l)p2n+6 B2 -4-3h-lp^/3e +pn 4 . 3 £ + 2 ( 6 - l ) p 2 21232^+6(6-l)p7i+6 B l ' -2-3h~lpy/3l +pn 3 2 ( 6 - l ) p n + 2 263<?+6(&-l)p2n+6 B2' 4 -3 6 " 1 pv / 3 f +pn 4 • 3 ' + 2 ( b - V 2 1 2 3 2 ^ + 6 ( 6 - 1 ) ^ 7 7 + 6 (b) £ is odd; a-2 a4 A C I 2-3h-ipy/3i +pn 3f.+ 2(b-l)p2 2 6 3 2 ^ + 6 ( 6 - i y + 6 C2 -4 -3 f e - 1 p \ / 3 £ +p' 1 4 • 32(h-1)-pn+2 2 l 2 3 < » + 6 ( 6 - l ) p 2 n + 6 C I ' -2-3h-1p^3e +pn 3e+2(b-i)p2 2 6 3 2 ^ + 6 ( 6 - 1 ) ^ 7 1 + 6 C2' 4 • 3h-1ps/3e +pn 4 • 3 2 ( b - l ) p n + 2 2 1 2 3 < " + 6 ( 6 - l ) p 2 7 i + 6 3. there exist integers £ > 1 and n > 0 such that 3 — pn is a square and E is Q-isomorphic to one of the elliptic curves: (a) £ is even; a2 a4 A D I 2-3h~lpsj3l -pn _ 3 2 ( 6 - l ) p 7 7 + 2 263<?+6(6-l) p2n+6 D2 - 4 - 3 f c - 1 p \ / 3 ^ - p n 4 . 3 « + 2 ( 6 - l ) p 2 _ 212 32 ,5+6(6-1 )pn+6 D I ' - 2 - 3 b - 1 p v / 3 * -pn _ 3 2 ( f t - l ) p n + 2 2 6 3 ^ + 6 ( 6 - l ) p 2 n + 6 D2' 4 - 3 6 - 1 p v / 3 ^ ~Pn 4 . 3 « + 2 ( b - l ) p 2 _ 2 l 2 3 2 « ? + 6 ( 6 - l ) p n + 6 Chapter 3. Elliptic Curves with 2-torsion and conductor 2a3@p5 118 (b) £ is odd; a 2 0,4 A E l 2 - 3 6 - V 3 ( - P " 3^+2(6-l)p2 _26^e+6(b-i)pn+6 E2 - 4 - 3b-lp^/3l ~Pn _ 4 . 32(h-l)pn+2 2123f+6(6-l) p2n+6 E l ' -2 • 3 h -V\/3 f -Pn 3«+2(b-l)p2 _2632^+6(b-l)pn+6 E2' 4 • 3 b-V\/3 £ ~Pn _ 4 . 3 2 (6 - l ) p n+2 2l23«+6(6-l)p2n+6 4. fhere exz'sf integers £ > 2 — b and n > 0 swc/t f/iaf p n — 3 f is a square and E is Q-isomorphic to one of the elliptic curves: (a) £ is even; ai a4 A FI 2 • 3b~1p^pn - 3l _3e+2(b-i)p2 2 6 3 2 £ + 6 ( 6 - l ) n+6 F2 - 4 - 3 b " V \ / p n - 3 ' ! 4 • 3 2 ( f c - i ) p n + 2 - 2 1 2 3 £ + 6 ( b _ 1 ^ p 2 n + 6 FI' - 2 • 3 b " 1 p v / P n - 3^ 2 6 3 2 f + 6 ( t — l ) n + 6 F2' 4 • 3b^p^pn - 3£ 4 . 32(f.-l) pn+2 _2 l23«+6 ( (> - l )p2n+6 (b) £ is odd; 0-2 0 4 A GI 2 • 3h~1py/pn - 3f 3 2 ( 6 - 1 ) ^ + 2 _ 2 6 3 « + 6 ( b - l ) p 2 n + 6 G2 - 4 - 3 b ^ 1 p v / P " - 3 f - 4 . 3 ^ + 2 ( 6 - l ) p 2 2 1 2 3 2 ^ + 6 ( 6 - l ) p n + 6 GI ' -2 • 3b"V\/pn - 3* 3 2 ( b - l ) p n + 2 2 6 3 ^ + 6 ( 6 - l ) p 2 n + 6 G2' 4 • 3b~Wpn - 3* - 4 . 3 ^ + 2 ( 6 - 1 ) ^ 2 2 1 2 3 2 ( ? + 6 ( 6 - l ) p 7 l + 6 5. fhere exz'sf integers £ > 1 and t € {0,1} swc/t f/iaf 3-± 1- is a square and E is Q-isomorphic to one of the elliptic curves: (a) p = 1 (mod 4); a 2 0 4 A HI 2 . 3 b - y + y ^ i 3«+2(b-l) 2t+l 2632*:+6(b-l)p3+6t H2 - A - 3 b - l p t + ^ ^ 4 • 3 2 ( b " 1 )p 2 4 + 1 2 1 2 3 « + 6 ( 6 - l ) p 3 + 6 ( HI ' - 2 - 3 b - y + i v / 3 ^ i 3*+2(6-l) 2*+l 2 632tM -6 (6- l ) p 3+6t H2' 4 - 3 6 - y+y 3 ^ i 4 . 3 2 ( 6 - l ) p 2 t + l 2 1 2 3 « + 6 ( 6 - l ) p 3 + 6 t (b) p = - 1 (mod 4); Chapter 3. Elliptic Curves with 2-torsion and conductor 2a3l3ps 119 0-2 a 4 A 11 2 - 3 b - y + i x / 2 £ ± i 3 2 ( 6 - l ) p 2 t + l 263^+6(6-1)^3+64 12 _ 4 . 36-y+y 3 f ± l 4 . 3 ^ + 2 ( 6 - 1 ) 2 t + l 2 l 2 3 2 * : + 6 ( 6 - l ) p 3 + 6 4 11' - 2 - 3 h - y + y ^ 3 2 ( b - l ) p 2 t + l 2 6 3 t " + 6 ( 6 - l ) p 3 + 6 4 1 2 ' 4 - 3 f e - V + 1 v / 3 ^ i 4 . 3 ^ + 2 ( 6 - 1 ) 2 4 + 1 2 1 2 3 2 ^ + 6 ( 6 - 1 ) ^ 3 + 6 4 6. there exist integers £ > 1 and £ € {0,1} f/wf - — - is a square and E is Q-isomorphic to one of the elliptic curves: (a) p = 1 (mod 4); 0 2 a 4 A J I 2 . 3b - y + i v / 3 ^ i _ 3 2 ( 6 - 1 ) ^ 2 4 + 1 263^+6(6-1)^3+64 J 2 - 4 . 3 f e - y + y ^ 4 . 3 ^ + 2 ( 6 - 1 ) 24+1 _ 2 1 2 3 2 ^ + 6 ( 6 - l ) p 3 + 6 4 J I ' - 2 . 3 b - y + V ^ f i __32 (6 - l )p24+l 263<?+6(6-l)p3+64 J 2 ' 4-3 f e -y+y 3 ^ i 4 . 3<!+2(6-l)p24+l _ 2 l 2 3 2 £ + 6 ( 6 - l ) p 3 + 6 4 (b) p = - 1 (mod 4); a 2 a 4 A K I 2 - 3 b - y + y 3 ^ 3 1 + 2 ( 6 - 1 ) ^ 2 4 + 1 _2632<?+6(6-l)p3+64 K 2 ^•s^-y+y 3^ _ 4 . 3 2 ( 6 - 1 ) ^ 2 4 + 1 2123^+6(6 -1 ) ^ 3 + 6 4 K I ' - 2 . 3 6 - y + y ^ 3 £ + 2 ( 6 - l ) p 2 4 + l _2632t"+6(6-l)p3+64 K 2 ' 4 . 3 h - y + y y - 4 • 3 2 ( b - l ) p 2 4 + l 2 123 ( '+6 (6 - l )p3+64 7. there exist integers £> 2 — b and n > 0 SMC/I that 8 • 3 p™ + 1 is a square and E is Q-isomorphic to one of the elliptic curves: a2 a 4 A L I S ^ P N / 8 • 3 V + 1 2 . 3 £ + 2 ( 6 - l ) n + 2 2632^+6(6-l)p2n+6 L 2 - 2 - 3 b - 1 p v / 8 - 3 V + 1 3 2 ( 6 - l ) p 2 293«+6(6-l)pn+6 L I ' S ^ P ^ / S • 3 V +1 2 . 3«+2(6-l)pn+2 2632^+6(6-l)p2n+6 L 2 ' 2 • 3b-lp^8-3epn + 1 3 2 ( 6 - i y 293^+6(6- l)pn+6 8. there exist integers £ > 2 — b and n > 0 such that 8 • 3e + pn is a square and E is Q-isomorphic to one of the elliptic curves: Chapter 3. Elliptic Curves with 2-torsion and conductor 2a3l3ps 120 a2 0,4 A M l 3b~lpyj8-3i +pn 2 . 3 / + 2 ( 6 - i y 2 6 3 2 M - 6 ( b - l ) p n + 6 M 2 -2 • s ^ y / 8 • 3 * + Pn 32(b-l)pn+2 2 93^+6 (b - l ) p 2n+6 M l ' 2 . 3 « + 2 ( f c - l ) p 2 2 6 3 2 ^ + 6 ( b - l ) p n + 6 M 2 ' 2 - 3 6 - 1 p v / 8 - 3 * + P n 3 2 ( b - l ) p n + 2 293<!+6(b-l)p2n+6 9. there exist integers £ > 2 — b and n > 0 such that 8 • 3 ^ — pn is a square and E is Q-isomorphic to one of the elliptic curves: a2 0,4 A N I 3 b - W 8 - 3 ^ -Pn 2 • 3 « + 2 ( b - D p 2 _ 2 6 3 2 f ! + 6 ( 6 - l ) p n + 6 N 2 - 2 - 3 6 - V \ / 8 - 3 « - p n _ 3 2 ( 6 - l ) p n + 2 29 3 f + 6 ( b - l ) p 2 n + 6 N I ' 2 • 3 < + 2 ( 6 - l ) p 2 _ 26 3 2 * + 6 C > - l ) p n + 6 N 2 ' 2 - 3 b " 1 p \ / 8 - 3 < -pn _ 3 2 ( b - l ) p n + 2 2 9 3 « + 6 ( b - l ) p 2 n + 6 10. there exist integers £ > 2 — b and n > 0 such that 8pn + 3e is a square and E is Q-isomorphic to one of the elliptic curves: a2 a.4 A O l 3 b " 1 p \ / 8 p n + 3* 2. 3 2 ( b - l ) p n + 2 263*!+6(b-l)p2n+6 0 2 - 2 - 3 b * 1 p \ / 8 p n + 3 f 3f+2(b - l )^ 2 9 3 2 £ + 6 ( b - l ) p 7 i + G O l ' - 3 b " V \ / 8 p n + 3e 2 • 32(b-1)pn+2 263f+6(b-l) p2n+6 or 2 • 3 b ~ V \ / 8 p n + 3* 3^+2(b-l)p^ 2932<:+6(b-l)pn+6 11. there exist integers £ > 2 — b and n > 0 such that 3 pn — Sis a square and E is Q-isomorphic to one of the elliptic curves: a2 04 A P I 3 b " 1 pv / 3V - 8 - 2 - 3 2 ( b - V 263<?+6(6-l)p77,+6 P 2 - 2 -3b-lpy/3lpn - 8 3*?+2(b-l) pn+2 _ 2 932(, +6(b-l) p 2n+6 P I ' -3b~1pyj3lpn - 8 - 2 • 3 2 < b ~ V 2 6 3 « ? + 6 ( b - l ) p n + 6 P 2 ' 2 • S ^ W S V 1 - 8 3 « + 2 ( b - l ) p n + 2 _ 2 9 3 2 « + 6 ( b - l ) p 2 » 2 + 6 12. there exist integers £ > 1 and n > 0 SMC/Z that 3( — 8pn is a square and E is Q-isomorphic to one of the elliptic curves: Chapter 3. Elliptic Curves with 2-torsion and conductor 2cc3l3p6 121 a-2 a 4 A Q l 3h~'ipy/3l - 8pn -2 • 3 2 ( f c - l ) p n + 2 2 6 3 < ' + 6 ( 6 - l ) p 2n + 6 Q2 -2 • 3h~lpy/3e- - 8pn 3^+2(6-l) p2 _ 2 9 3 2 ^ + 6 ( 6 - l ) p n + 6 Q l ' -t^py/V - 8pn -2-32(b-1)p"+2 2 6 3 € + 6 ( 6 - l ) p 2 n + 6 Q2' 2 • 3b-1py/3e - 8pn 3 £ + 2 ( 6 - l ) p 2 - 2 9 3 M + 6 ( f c _ 1 V 1 + 6 13. there exist integers £ > 2 - b and n > 0 such that pn - 8 • 3e is a square and E is Q-isomorphic to one of the elliptic curves: a2 0,4 A R l 3h-xpx/pn -8-31 -2 • 3^+2(ft-Dp2 26 3 2 f + 6 ( b - l ) p n + 6 R2 -2-3b~lp^/pn -8-3* 3 2 ( b - l ) p n + 2 _ 2 9 3 ^ + 6 ( 6 - l ) p 2 n + 6 R l ' S^py/p" -8-3e _23e+2(b-i)p2 2 6 3 2 £ + 6 ( 6 - l ) n + 6 R2' 2 • 3 6 _ 1pvV -8-3i 3 2 ( b - l ) p n + 2 _ 2 9 3 £ + 6 ( b - l ) p 2 T i + 6 14. there exist integers £ > 2 - b and t e {0,1} such that 8 3 p + 1 is a square and E is Q-isomorphic to one of the elliptic curves: a 2 0 4 A SI 2 . 3 ^ + 2 ( 6 - 1 ) 2t+l 2 6 3 2 < M - 6 ( b-l ) p 3 + 6 ( S2 2 - 3 b - V + 1 v / 8 ' 3 f + 1 3 2 ( 6 - 1 ) 2 t+l 293£ + 6 ( 6-l ) p 3 + 6 t SI' _ 3 6 - y + y 8 ^ t i 2 . 3 ^ + 2 ( b - l ) p 2 t + l 2 6 3 2 ^ + 6 ( 6 - 1 ) ^ 3 + 6 * S2' 2 - 3 b - V + 1 V / 8 3 p + 1 3 2 ( 6 - l ) p 2 t + l 29 3 ( , + 6 ( 6-l ) p 3 + 6 ( 15. there exist integers £ > 2 - b and t € {0,1} such that 8 , 3 1 is a square and E is Q-isomorphic to one of the elliptic curves: a 2 a.4 A TI 3 6 - y + y 8.3^-1 2 . 3 ^ + 2 ( b - l ) p 2 t + l _ 2 6 3 2 « + 6 ( 6 - l ) p 3 + 6 t T2 2 . 3 6 - I t + l / 8 - 3 « - l ^ V P _ 3 2 ( 6 - l ) p 2 t + l 2 9 3 « + 6 ( 6 - l ) p 3 + 6 t TI' 3 b - y + y 8.3^-1 2 . 3 * ? + 2 ( b - l ) p 2 t + l — 2632£+6(b_1)p3+6' T2' 2 - 3 b - y + i v / 8 ' 3 f - 1 _ 3 2 ( b - l ) p 2 t + l 2 9 3 ^ + 6 ( 6 - l ) p 3 + 6 t 16. there exist integers £ > 2 - b and t e {0,1} such that 8-^3- is a square and E is Q-isomorphic to one of the elliptic curves: Chapter 3. Elliptic Curves with 2-torsion and conductor 2a3^p5 122 a2 A UI 3 b -y+y^ 2 . 3 2 ( 6 - l ) p 2 t + l 2 6 3 € + 6 ( b - l ) p 3 + 6 t U2 -2-3b-1pt+1y/^fL 3<?+2 ( fc- l )p2t+l 2 9 3 2 « + 6 ( b - l ) p 3 + 6 t UI ' 2 • 32(b-Dp2t+l 2 6 3 ^ + 6 ( 6 - l ) p 3 + 6 i U2' 2 . 3 6 - y + i v / 5 ± 3 i 3 € + 2 ( 6 - l ) p 2 t + l 2932 * ;+6(6 - l )p3+6t 27. there exist integers £ > 2 — b and t € {0,1} swch that ^-T- 3 - is a square and E is Q-isomorphic to one of the elliptic curves: a 2 . 0,4 A VI 2 • 3 2 ( b - D p 2 i + l _ 2 6 3 € + 6 ( 6 - l ) p 3 + 6 i V2 -2 -3 h -y+y^ _ 3 « + 2 ( f c - l ) 2 t+l 2 9 3 2 ^ + 6 ( 6 - 1 ) ^ 3 + 6 * V I ' 2 . 3 2 ( 6 - l ) p 2 ( + l _ 2 6 3 ^ + 6 ( h - ! ) p 3 + 6 t V2' 2-3 b -y+y^ _ 3 £ + 2 ( 6 - l ) p 2 t + l 2 9 3 2 £ + 6 ( 6 - l ) p 3 + 6 ( . 18. there exist integers £ > 2 — b and t G {0,1} such that 3-~ is a square and E is Q-isomorphic to one of the elliptic curves: a-2 a4 A WI s " - y + V ^ -2 • 3 2 ( 6 - l ) p 2 t + l 2 63«+6 (6 - l )p3+6t W2 - 2 - 3 b - y + y ^ 3^+2(6-1) 2 t+l - 2 9 3 2 £ + 6 ( b _ i y + 6 ( WI' - 3 b - v + V ^ - 2 - 32 ( b - i y t + i 263^+6(6-1)^3+61 W2' 2 . 3 6 - y + y 3 ^ 8 3 « + 2 ( 6 - l ) p 2 £ + l _29 3 2 ^+6 (6 - l )p3+6i In the case that 6 = 2, i.e. N = 2 5 3 2 p 2 , we furthermore could have one of the following conditions satisfied: 19. there exist integers n > 0 and s 6 {0,1} such that fs a square and E is Q-isomorphic to one of the elliptic curves: Chapter 3. Elliptic Curves with 2-torsion and conductor 2a3l3ps 123 a2 a4 A XI 2 • 3 s + 1 p ^ ^ - 3 2 s+y 2633+65^71+6 X2 _4. y"Py/e±i 4 . 32s+lpn+2 21233+65^271+6 X I ' 2 • 3»+lpy/lZ±± 3 2 s + y 2633+65^77+6 X2 ' 4 . 32s+\pn+2 21233+65^277+6 20. there exist integers n > 0 and s e {0,1} such that -^g-^  zs a square and E is Q-isomorphic to one of the elliptic curves: a-2 a4 A YI 325+1^71+2 2*^3+ 6s jyZn+6 Y2 - 4 • 32*+V 2l233+6s^n+6 YI ' 32s+l P 77+2 _ 2633 + 6 5 ^ 2 7 1 +6 Y2 ' 4 . 3 ^ ^ - 4 • 3 2 s+V 2 l 2 3 3 + 6 S p n + 6 21. there exist integers s,t <E {0,1} such that E is Q-isomorphic to one of the elliptic curves: (a) p = 1 (mod 4): a-2 a4 A ZI 0 32s + l p 2 t + l 2633+6Sp3 + 6£ Z2 0 _ 4 . 3 2 5 + l p 2 t + l 2l233+6s^3 + 6/. -(b) p = - 1 (mod 4): 0.2 a4 A A A 1 0 _ 3 2 5 + l p 2 t + l 2633 + 65^3+6* A A 2 0 4 . 3 2 s + l p 2 t + l 2l233+6SpT?.+6 22. E is Q-isomorphic to one of the elliptic curves: a2 a4 A BBl 0 _ 3 2 ( 6 - l ) p 2 2 6 3 6 ( 6 - l ) p 6 BB2 0 4 . 3 2 ( 6 - l ) p 2 _ 2 1 2 3 6 ( b - l ) p 6 23. there exists an integer t e {0,1} such that E is Q-isomorphic to one of the elliptic curves: Chapter 3. Elliptic Curves with 2-torsion and conductor 2°3l3p5 124 (a) p = 1 (mod 4): a2 0,4 A CC1 0 _32(b-l) 2t+l 2636(f)-l)p3+6t CC2 0 4 . 32(6-l) p2t+l _ 21236(b-l)p3+6i (b) p= - 1 (mod 4): 0,2 0 4 A D D I 0 3 2 ( 6 - l ) p 2 t + l _ 2 6 3 6 ( f e - l)p3+ 6 t DD2 0 _4 . 3 2 ( 6 - 1 ) ^ + 1 2 1 2 3 6 ( 6 - 1)^3+ 6 * 24. fhere exist integers n > 1 and s £ {0,1} swch that ^- j— is a square and E is Q-isomorphic to one of the elliptic curves: 0,2 04 A EE1 2 • 3 2 s + y 2633+6s pn+6 EE2 2 - 3 s + ' P y / ^ 32.s+lpn+2 2933+6sp2n+6 EE1' 2 • 32s+1p2 2633+65^71+6 EE2' 2 • 3 s + 1 P y J ^ 3 2 « + l p n + 2 2933+6sp2?i+6 25. there exist integers n > 1 and s e {0,1} such that 8 .£ is a square and E is Q-isomorphic to one of the elliptic curves: 02 04 A FF1 2 • 32s+lp2 _263 3+6spn+6 FF2 -2 • 3s+1Ps/^=f- _32s+lpn+2 2933+6Sp2n+6 FF1' - 3 ' + 1 P ^ 2 • 32s+1p2 _ 2 6 3 3+6spn+6 FF2' 2-3s+1pyJ^=f- _32s+l pn+2 2933+6.Sp2n+6 26. there exists an integer n > 1 and s 6 {0,1} such that -^g— is a square and E is Q-isomorphic to one of the elliptic curves: Chapter 3. Elliptic Curves with 2-torsion and conductor 2a3l3p5 125 a 2 0,4 A G G 1 -2 • 3 2 s + V 26g3+6Spn+6 G G 2 2 • y+1py/z^ 32s+lpn+2 — 2 93 3+ 6 sp 2 n+ 6 G G 1 ' - 2 • 3 2 s + y 2633+65^71+6 G G 2 ' 2 • 3 S + V y 7 ^ 32s+lpn+2 — 2 93 3+ 6 sp 2 n+ 6 Theorem 3.27 The elliptic curves E defined over Q, of conductor 2e3bp2, and hav-ing at least one rational point of order 2, are the ones such that one of the following conditions is satisfied: 1. there exist integers £> 2 — b and n > 0 such that 3epn + 1 is a square and E is Q-isomorphic to one of the elliptic curves: a-2 a 4 A A l 2 • 3 b - 'p \/ 3 V + 1 32(b-iy 263f+6(fe-l)pn+6 A2 -4 • 3b-1p^3epn + 1 4 . 3f+2(b-l)pn+2 2 l 2 3 2 f + 6 ( 6 - l ) p 2 n + 6 A l ' - 2 • 3b-1py/3ipn + 1 3 2 ( 6 - l ) p 2 2 6 3 ^ + 6 ( 6 - 1 ) ^ 7 7 + 6 A2' 4-3 b - 1 p\/3V + l 4 . 3e+2(b-l)pn+2 2 1 2 3 2 ^ + 6 ( 6 - l)p2n+6 2. there exist integers £ > 1 and n > 0 such that 3e + pTl is a square and E is Q-isomorphic to one of the elliptic curves: (a) £ is even; a 2 a 4 A BI 2-3b-lpx/3e +pn 3f+2(6-l)p2 2 6 3 2 ^ + 6 ( 6 - 1 ) ^ 7 7 + 6 B2 -4-3 f e - 1 pv / 3 £ +P n 4 . 3 2 (6 - l ) p n+2 2 l 2 3 « + 6 ( 6 - l ) p 2 n + 6 BI' -2-3 f e-V\/3 f +Pn 3 2 (6 - l ) p 2 2 6 3 2 ^ + 6 ( 6 - 1)^77+6 B2' 4 - 3 b - W 3 * + j D n 4 . 3^+2(6- l)p77+2 2l23<?+6(6-1)^277+6 (b) £ is odd; a2 0,4 A C l 2 • 3b-1p^/3l+pn 3 2 ( 6 - 1 ) ^ 7 7 + 2 263«+6(6-l)p2n+6 C2 -4-3b~lp^3l +pn 4 . 3 < + 2 ( b - l ) p 2 2 1 2 3 2 ^ + 6 ( 6 - 1 ) ^ 7 7 + 6 C l ' -2-3 6- 1pv /3*+Pn 32(6-l)pn+2 2 6 3 ^ + 6 ( 6 - 1 ) ^ 2 7 7 + 6 C2' A-3b~lp^3t +pn 4 . 3 ^ + 2 ( 6 - l ) p 2 21232^+6 (6 -1 ) ^ , 7 7 +6 Chapter 3. Elliptic Curves with 2-torsion and conductor 2a3l3p5 126 3. there exist integers £ > 1 and n > 0 such that 3 - pn is a square and E is Q-isomorphic to one of the elliptic curves: (a) £ is even; 02 0,4 A D I 2-3b-1py/3e-pn 3e+2(b-i)p2 - 2 6 3 2 f + 6 ( 6 - 1 ) p n + 6 D2 -4-3b-1py/3e -pn - 4 . 32(b-l)p«+2 2123f+6(b-l)p2n+6 D I ' -2 • 3b~1py/3e - pn 3«+2(6-l)p2 _2632/:+6(6-l)pn+6 D2' 4 • 3b-1py/3e-pn - 4 . 32(fe-l)pn+2 2 123^+6(6- l ) p 2n+6 (b) £ is odd; a-2 0,4 A E l 2 • 3b~lps/3l - pn _32(b-l)p"+2 2 6 3 £ + 6 ( 6 - l ) p 2 n + 6 E2 - 4 •3b'lpy/3i -pn 4 . 3^+2(fc-l)p2 _21232#:+6(b-l)p7i+6 E l ' - 2 - 3 h ~ 1 p \ / 3 < - p n _32(fc-l)pn+2 2 6 3 t ?+6 (6 - l ) p 2 n +6 E2' 4-3b-lpy/3l -pn 4 . 3^ +2(b-l)p2 _21232^+6(b-l)pn+6 4. there exist integers £ > 2 — b and n > 0 such that pn - 3 is a square and E is Q-isomorphic to one of the elliptic curves: (a) £ is even; a2 04 A FI 2 • 3b-V\7pn - 3* 32(b-l)pn+2 _ 2 6 3 < ' + 6 ( 6 - l ) p 2 n + 6 F2 -4-36-1pv /P n - 3 f _ 4 . 3*:+2(b-i)p2 2 1 2 3 2 f + 6 ( 6 - l ) p J l + 6 F T - 2 • 3b~lpy/pn - 3* 3 2 ( 6 - 1 ) n+2 _ 2 6 3 * : + 6 ( 6 - l ) p 2 n + 6 F2' 4 • 3b-1p^/pn - 3e - 4 . 3^+2(b-l)p2 21232£+6 (6-l) p n+6 (b) £ is odd; 0,2 04 A GI 2 • 3b-lps/pn - 3! _ 3 * + 2 ( 6 - l ) p 2 2632£+6(b-l)pn+6 G2 - 4 - 3 b - 1 p v / P n - 3 f 4 • 32( fc-1)p™+2 _ 2 1 2 3 > : + 6 ( 6 - l ) p 2 r j + 6 G I ' -2-3 b"V\/p n-3* _3*:+2(b-i)p2 2 6 3 2f+6 (6- l ) p n+6 G2' 4 • 3b~1pJ'pn - 3e 4 • 3 2( f c- 1)p n+ 2 _ 2 1 2 3 £ + 6 ( 6 - l ) p 2 n + 6 5. there exist integers £ > 1 ana* £ G {0,1} such that 3-y^ is a square and E is Q-isomorphic to one of the elliptic curves: (a) p = 1 (mod 4); Chapter 3. Elliptic Curves with 2-torsion and conductor 2a3f3p5 127 a2 a 4 A HI 2-36-y+1y/3^  3 2 ( 6 - l ) p 2 t + l 2 6 3 « + 6 ( f t - l ) p 3 + 6 i H2 _ 4 . 36-y+y 3f±l 4 . 3 € + 2 ( b - l ) p 2 t + l 2 1 2 3 2 £ + 6 ( 6 - l ) p 3 + 6 t H I ' ~2 • 3b-y+y 2f±i 3 2 ( 6 - l ) p 2 t + l 2 6 3 £ + 6 ( b - l ) p 3 + 6 t H2' 4-3b-y+1v/2f+l 4 . 3 < + 2 ( b - l ) p 2 t + l 2 1 2 3 2 £ + 6 ( b - l ) p 3 + 6 t (W p = - 1 (mod 4); a2 a 4 A 11 2-3h-lpt+1xJ^f- 3<+2( t - l )p2i+l 2 6 3 2«+6(6-iy+6t 12 _4.3fc-y+y3f+_L 4 . 3 2 ( 6 - l ) p 2 t + l 212 3*5 (6-l)p3 t 11' -2-3b-y+y 3^i 3 £ + 2 ( b - l ) p 2 £ + l 2 6 3 2 M - 6 ( b - l ) p 3 + 6 i 12' 4.3"-y+y3^i 4 • 32(b-l)j,2t + l 2 1 2 3 ^ + 6 ( 6 - l ) p 3 + 6 t 6. f/rere exzsf integers £ > 1 and i G {0,1} swc/z that '^—^ is a square and E is Q-isomorphic to one of the elliptic curves: (a) p = 1 (mod 4); «2 a 4 A JI 2-3"-y+y^ 3<+2(b-iy(+i _ 2 6 3 2f+6(6-iy+6i J2 -4-3''-y+ 1 v / 5^ i _ 4 . 3 ( - l ) p 2 t + l 2 12 3 ^+6(b - l )p3+6 t JI' 3 * + 2 ( b - l ) 2t+l _ 2 6 3 2«+6(b - iy+6t J2' 4-3"-y+y^i _4 . 3 2 ( 6 - l ) p 2 t + l 2 1 2 3 « + 6 ( b - l ) p 3 + 6 i (b) p=-l (mod 4); a2 a 4 A KI 2 . 3 b - y + y 3 ^ i _ 3 2 ( b - l ) p 2 t + l 2 6 3 ^ + 6 ( b - l ) p 3 + 6 t K2 _ 4 . 3 b - y + y 3 ^ i 4 . 3 <+2(b- l ) 2t + l _ 2 12 3 2«+6(b - iy+6t K I ' ^•^-y+y3^ _32(b-iyt + l 26 3f+6(b-iy+6t K2' 4 . 3b-y+y 3^1 4 . 3 f + 2 ( b - l ) p 2 t + l _ 2 1 2 3 2 € + 6 ( 6 - l ) 3+6t 7. there exist integers m > 3, £ > 2 — b and n > 0 such that 2m3 pn + 1 is a square and E is Q-isomorphic to one of the elliptic curves: Chapter 3. Elliptic Curves with 2-torsion and conductor 2a3l3p6 128 a2 0 4 A LI 2 • 3 b - V \ / 2 m 3 y i + 1 2»ng*+2(6-l)pn+2 22m+632e+6(b- l ) p 2 n + 6 L2 - 4 • 3 b - V v / 2 m 3 V + 1 4 - 3 2 ( b - i y 2m+123e+6(b-l)pn+6 LI ' -2 • 3 f c - 1 p \ / 2 m 3 V + 1 2 m 3 « + 2 ( H ) n + 2 2 2 m + 6 3 2 ^ + 6 ( t ) - l ) p 2 n + 6 L2' 4 • 3 b - 1 p v / 2 m 3 y + 1 4 . 3 2 ( b - i y 2771+12 C j j ? + 6 ( b - l ) p n + 6 8. there exist integers m > 2, £ > 2 — b and n > 0 such that 2rn3e + pn is a square and E is Q-isomorphic to one of the elliptic curves: 0.2 0 4 A M l 2 • 3b-1Py/2m3i +pn 2m3^+2(b-l)p2 22m+632i+6{b- 1 )^71+6 M 2 -4-3 b _ 1 pv / 2 m 3 £ +p n 4 • 32(fc-1)pn+2 2 7 7 1 + 1 2 3 ^ + 6 ( 6 - 1 ) 2 n + 6 M l ' -2 • 3b-lPy/2m3e +pn 2 7 7 i 3 ( ' + 2 ( b - l ) p 2 2 2m+632<£+6(b-l)pn+6 M 2 ' 4 • 3b-1p^/2m3e +pn 4 • 32(''-1)pn+2 2 7 7 i + 1 2 3 < ? + 6 ( 6 - l ) p 2 7 ( + 6 9. there exist integers m > 2, £ > 2 — b and n > 0 such that 2 m 3 - p" is a square and E is Q-isomorphic to one of the elliptic curves: a 2 0 4 A NI 2 • 3b~1py/2m3e - p n 2 7 7 i 3 « + 2 ( 6 - l ) p 2 _22m+632e+6(b-l) pn+6 N 2 - 4 • 3b-1py/2m3e - p n _ 4 . 3 2 ( 6 - l ) p n + 2 2 1 7 1 + 1 2 3 ^ + 6 ( 6 - 1 ) ^ 2 7 1 + 6 N I ' ~2-3b-1p^2m3i -pn 2 7 7 i 3 < + 2 ( 6 - l ) p 2 _ 2 2 m + 6 3 2 « + 6 ( 6 - l ) p 7 i + 6 N 2 ' 4 • 3b-1py/2m3e - p n _ 4 . 3 2 ( 6 - l ) p 7 i + 2 2 7 7 i + 1 2 3 « + 6 ( 6 - l ) p 2 n + 6 10. there exist integers m > 3, £ > 2 — b and n > 0 such that 2mp" + 3 is a square and E is Q-isomorphic to one of the elliptic curves: a-2 o4 A 0 1 2 • 3 b - 1 p v / 2 m J » n + 3e 2771.32(6-1)^71+2 2 2 m + 6 3 < M - 6 ( b - l ) p 2 7 i + 6 0 2 -4-3b-lpy/2mpn + 3e 4 . 3«+2(<>-i)p2 2 7 7 i + 1 2 3 2 ( ? + 6 ( 6 - l ) p n + 6 o r - 2 - 3 b " 1 p \ / 2 T n P n + 3 ^ 277132 (6 -1)^71+2 2 2 m + 6 3 £ + 6 ( b - l ) p 2 7 i + 6 0 2 ' 4 - 3 b - 1 p V ' 2 m P n + 3 ^ 4 . 3^+ 2(b-l) p 2 2771+1232^+6 (6 -1)^71+6 11. there exist integers £ > 2 - b and n > 0 such that Apn — 3e is a square and E is Q-isomorphic to one of the elliptic curves: Chapter 3. Elliptic Curves with 2-torsion and conductor 2°3l3p5 129 a2 a.4 A PI 2 • 3b~lp\j4pn - 3* 4 • 32((>-iy»+2 _ 2 l 0 3 f + 6 ( h - l ) p 2 n + 6 P2 - 4 - 3 b - 1 p v / 4 P n ~ 3 € _ 4 . 3 « + 2 ( 6 - l ) p 2 2 1 4 3 2 M - 6 ( b - i y + 6 PI ' - 2 • 3 l - 1 p v / ¥ - 3 f 4 • 3 2( f a- 1)p n+ 2 — 2 1 0 3£ + 6 ( b _ 1 )p 2 n + 6 P2' 4 • 3 b - x p \ / 4 p ™ - 3e -4 . 3^+2(i>-l)p2 214 3 2 ^ + 6 ( 6 - l ) p n + 6 12. there exist integers m > 2, £ > 2 — b and n > 0 such that 2rn 4- 3 pn is a square and E is Q-isomorphic to one of the elliptic curves: a 2 a 4 A Q l 2 • 3f t - V \ / 2 m + 3epn 2 m 3 2 ( 6 - l ) p 2 22?n+63*-+G(b- 1 )pn+6 Q2 - 4 - 3 b - 1 p v / 2 m + 3 V 4 . 3 « + 2 ( b - l ) p n + 2 2 m + 1 2 3 2 « + 6 ( b - l ) p 2 ? j + 6 Q l ' - 2 - 3 b " 1 p v / 2 m + 3 V 2m 3 2 ( 6 - i y 22m ,+63<?+6 (b - iy i+G Q2' 4 • 3 b - 1 p \ / 2 m + 3 V 4 . 3 ^ + 2 ( b - l ) p n + 2 2 m + 1 2 3 2 / : + 6 ( b - l ) p 2 n + 6 13. there exist integers m > 2, £ > 2 — b and n > 0 such that 2rn — 3 pn is a square and E is Q-isomorphic to one of the elliptic curves: a 2 a.4 A R l 2 • 3 b - 1 p \ / 2 m - 3 fp n 2 m 3 2 ( b - l ) p 2 _ 2 2 m .+6 3^+6 (b-1 )pn+6 R2 -4 • 3b-1py/2m - 3epn _ 4 . 3 £ + 2 ( b - l ) n+2 2 m + 1 2 3 2 f + 6 ( b - l ) p 2 n + 6 R l ' - 2 • 3h~1p-s/2m - 3epn 2 m 3 2 ( b - i y _ 2 2 ? n+63^+6 ( b - l ) p n+6 R2' 4 • 3 b - 1 p v / 2 m - 3epn _ 4 . 3 « + 2 ( b - i y + 2 2r7i.+ 1 2 3 2 £ + 6 ( b - l ) p 2 n + 6 14. there exist integers m > 2, £ > 2 — b and n > 0 such that 3tp" — 2m is a square and E is Q-isomorphic to one of the elliptic curves: a 2 a4 A SI 2 • 3h~'lpy/3ipn - 2™ _2m 32(6-l)p2 2 2 m + 6 3 * ? + 6 ( b - l ) p n + 6 S2 -4 • 3 b - 1 p \ / 3 V - 2 m 4 . 3 > _ + 2 ( b - i y + 2 _ 2 m + 1 2 3 2 * ? + 6 ( b - l ) p 2 ? i + 6 SI' - 2 - 3 b - 1 p v / 3 V - 2 m _ 2 m 3 2 ( b - i y 2 2 m + 6 3 ^ + 6 ( b - l ) p T i + 6 S2' 4 • 3 b _ 1 p v / 3 V - 2 m 4 . 3<+2(4-iy+2 _ 2 " i + 1 2 3 2 f + 6 ( b - l ) p 2 n + 6 15. f/zere exzsf integers m > 2, £ > 2 — b and n > 0 S M C A Z f/wf 3 £ - 2 m p " is a square and E is Q-isomorphic to one of the elliptic curves: Chapter 3. Elliptic Curves with 2-torsion and conductor 2 a 3 0 p s 130 0-2 a 4 A TI 2 • 3h~1py/3e - 2 m p n _2m32(b-l)p-n+2 2 27Ti+63i?+6(6-l)p2n+6 T2 -4 • 3 h - 1 p y / 3 e - 2 m p n 4 . 3i+2(b-\)p2 _ 2 m+1232 (?+6 (6- l )pn+6 TI' -2-3^ 1 pV / 3^ - 2mpn _2")32(b-l)pn+2 22m+63<?+6(6-l)p2n+6 T2' 4 • 3h-lp%/3t - 2 m p n 4 . 3 <?+2(b-l) p 2 _2m+1232i?+6(6-l)pn+6 16. there exist integers m > 2, £ > 2 — b and n > 0 such that pn — 2m3 is a square and E is Q-isomorphic to one of the elliptic curves: «2 a 4 A U l 2 • 3 b - l p ^ p n - 2m3* _ 2 m 3 ^ + 2 ( 6 - l ) p 2 22m.+632t+6{b-l) pii+6 U2 -4 • 3 b ~ V v V - 2m3e 4 • 32(b-1)p"+2 _2m+123<?+6(6-l)p2n+6 U l ' -2 • 3b~1ps/pn - 2™3f _2m.3*!+2(b-l)p2 22771+632^+6(6- l )pn+6 U2' 4 • 3b-1py/pn - 2m3e 4 • 3 2( h- 1)p n+ 2 _2m+l 23^+ 6( b- 1)p 2 n+ 6 17. there exist integers m > 2, £ > 2 - b and t G {0,1} such that 2 ' " 3 + 1 fs a square and E is Q-isomorphic to one of the elliptic curves: a-2 a 4 A VI 2.3 f t-y+y 2"' 3; + i 2 m . 3 ^ + 2 ( 6 - l ) p 2 t + l 22777.+632 l?+6(b-l)p3+6( V2 4 • 3 6 - v t + V 2 " ' 3 f + i 4 • 3 2C>-i) p 2 *+i 2777+123<?+6(b-l)p3+6t VI' 2 - 3 b - y + y 2 m 3 j + 1 2 7 7 7 . 3 f + 2 ( b - l ) p 2 t + l 22777+632£+6(6- l)p3+6t V2' 4 - 3 b - y + y 2 m 3 f + 1 4 • 3 2 ( b - l ) p 2 * + l 2777,+ 123<?+6(b-l)p3+6f 18. there exist integers rn > 2, £ > 2 - b and t £ {0,1} such that 2 m 3 J 1 is a square and E is Q-isomorphic to one of the elliptic curves: a.2 a 4 A WI 2-3b-y+y2"'3;-1 2m3«+2(b-l)p2f+l _22m+632£+6(6-l)p3+6t W2 4-3b-y+y2'"3'-1 _4 . 3 2 ( b - l ) p 2 t + l 27T7.+123J?+6(b-l)p3+6t WI' 2 . 3 b - y + 1 V / 2 " T 1 27773('+ 2(b-l)p2t+l _22m+632<?+6(b-l)p3+6t W2' 4-3b-y+y2"'3'-1 -4 • 3 2(<>-iy+i 2m+ 1 23^+ 6( b- 1)p 3+ 6 t 19. there exist integers m > 2, £ > 2 - b and t 6 {0,1} such that 2m+3 is a square and E is Q-isomorphic to one of the elliptic curves: Chapter 3. Elliptic Curves with 2-torsion and conductor 2a3l3ps 131 0.2 0 4 A X I 2 . 3 6 - y + y 2™+3< 2 m 3 2 ( b - l ) p 2 t + l 2 2 m + 6 3 £ + 6 ( b - l ) p 3 + 6 ( X2 4 - 3 b -y+ i y 2 ' " + 3 ' 4 . 3 ^ + 2 ( 6 - 1 ) 2 t + l 2 m + 1 2 3 2 ( ? + 6 ( b - l ) p 3 + 6 t X I ' 2-3 b-y +V 27 3 ' 2 m 3 2 ( 6 - l ) p 2 t + l 2 2 m + 6 3 l + 6 ( ( b - l ) p 3 + 6 t X2' 4-36-y+ix/2'"+3f 4 . 3<+2( fc - l )p2 t+l 2 m + 1 2 3 2 f + 6 ( 6 - l ) p 3 + 6 « 20. there exist integers m > 2, £ > 2 - b and t e {0,1} such that 2"* 3 is a square and E is Q-isomorphic to one of the elliptic curves: a 2 a4 A YI 2 . 3 f t - y + y ^ 2 m 3 2 ( f a - l ) 2 t + l _ 2 2m+63<M -6 (6 - i y + 6 t Y2 4 - 3 6 - y + y 2 m p - 3 ' _ 4 . 3 < ! + 2 ( 6 - l ) p 2 i + l 2771+1232^+6(6-1)^3+61 Y I ' 2 . 3 6 - y + y 2 - - 3 ' 2 m 3 2 ( 6 - l ) p 2 t + l _2 2 m+ 63^+ 6(' ,- 1)p 3+ 6* Y2 ' 4 . 3 6 - y + y ^ _ 4 . 3 * + 2 ( H y [ + l 277i+1232i?+6(6-l)p3+6t 21. there exist integers m > 2, £ > 2 — b and t € {0,1} such that 3 2™ is a square and E is Q-isomorphic to one of the elliptic curves: « 2 a4 A Z I 2 • 3b-y+1 y 3 ' - / " _ _ 2 " i -32 (6 - i y t + i 2277i+63<"+6(6 - l ) p 3+6( Z2 4.3"-y+V3f/" 4 . 3 f + 2 ( 6 - l ) p 2 t + l _ 2 7 n + l 2 3 2 ^ + 6 ( 6 - l ) p 3 + 6 t Z I ' _ 2 . 3 b - y + ' y t e _ 2 7 7 7 3 2 ( 6 - 1 ) 2i+l 2 2 m + 6 3 < ? + 6 ( 6 - i y + 6 t Z2 ' 4 . 3 6 - y + y ^ 4 . 3 < ? + 2 ( 6 - i y * + l _ 2 m + 1 2 3 2 < ? + 6 ( 6 - l ) p 3 + 6 t In the case that 6 = 2, i.e. N = 2 6 3 2 p 2 , we furthermorexould have one of the following conditions satisfied: 22. there exist integers n > 1 and s e {0,1} such that is a square and E is Q-isomofphic to one of the elliptic curves: Chapter 3. Elliptic Curves with 2-torsion and conductor 2a3l3ps 132 a-2 04 A A A 1 2 • 3s+1pyfp"3+1 32s+lpn+2 26^3-f-6sp2n-f 6 A A 2 4 • 3 2 - s + y 21233+65^71+6 A A 1 ' 2 • 3s+1pyf^l±l 32,+ lpn+2 2633+65^271+6 A A 2 ' 4 . 3 2 S + l p 2 21233+65^11+6 23. there exist integers n > 1 and s 6 {0,1} such that p~3-^ is a square and E is Q-isomorphic to one of the elliptic curves: a2 04 A BB1 2 • 3s+1pyj£ff± _ 3 2 S + l p 2 26 33+6s pn+6 BB2 _ 4 . 3'+ 4 . 32s+lp?i+2 — 2 1 2 3 3 + 6 - s p 2 "+ 6 BB1' 2 • 3s+lpyj^±l _ 3 2 S + I p 2 26 33+6s pn+6 BB2' 4 . s ' + ^ y ^ 4 . 32.s+lpn+2 — 2 1 2 3 3 + 6 s p 2 n + 6 24. there exist integers s,t e {0,1} such that E is Q-isomorphic to one of the elliptic curves: (a) p = 1 (mod 4): a2 04 A CC1 0 _32s+lp2t+l 26 33+6s p3+6t CC2 0 4 • 3 2 s + y t + i _2l 23 3+6.s p3+6t (b) p = -1 (mod 4): a 2 04 A D D I 0 32s+lp2/.+ l _26 33 +6. 5 p 3+C( DD2 0 _4 . 32s+l p2i+l 2l2 33+6s p3+6t 25. E is Q-isomorphic to one of the elliptic curves: a2 a4 A EE1 0 3 2(6 - l ) p 2 _ 2 6 3 6 ( b - l ) p 6 EE2 0 - 4 • 3 2 < 6 - V 2 12 3 6 (b- l ) p 6 26. f/iere exz'sfs an integer t e {0,1} swc/z fTiaf E 1 zs Q-isomorphic to one of the elliptic curves: Chapter 3. Elliptic Curves with 2-torsion and conductor 2a3)3ps 133 (a) p = 1 (mod 4): a2 0 4 A FF1 0 3 2 ( 6 - l ) p 2 t + l _ 2 6 3 6 ( 6 - l ) p 3 + 6 t FF2 0 _ 4 . 3 2 ( b - l ) p 2 t + l 2 l 2 3 6 ( b - l ) p 3 + 6 t (b) p = - 1 (mod 4): 0-2 0,4 A GG1 0 _ 3 2 ( 6 - l ) p 2 t + l 2 6 3 6 ( 6 - l ) p 3 + 6 t G G 2 0 4 • 3 2 ( b - l ) p 2 t + l _ 2 1 2 3 6 ( 6 - l ) p 3 + 6 t 27. f/zere exzsf integers n > 0 and s e {0,1} SMCTZ (AM.* 4 p g 1 is a square and E is Q-isomorphic to one of the elliptic curves: a2 a 4 A H H 1 2 • y+ipy[&^± 4 • 32s+l)pn+2 _ 21 0 3 3 + 6 s p 2 n + 6 H H 2 4-3' 5+ 1pV / 4 p" 3" 1 - 4 • 32 s + V 2 l 4 3 3 + 6 S p n + 6 H H 1 ' - 2 - 3 ^ ^ 4 • 32«+l)p"+2 —2 1 033+ 6 sp 2 n+ 6 H H 2 ' 4 - 3 s + 1 p v / 4 P 3 ^ 1 - 4 • 32 s + 1 V 2 1 43 3+ 6 sp"+ 6 28. there exist integers m > 2, n > 0 and s e {0,1} swc/z rTzflt 1 > n + 2 " is a square and E is Q-isomorphic to one of the elliptic curves: 02 a.4 A III 2 • 3s+1PxJ^p- 2 m 3 2 . s + y 22m+633+6s^n- f 6 112 4 . 3 2 5 + 1 ^ 7 7 + 2 2 7 7 1 + 1 2 3 3 + 6 5 ^ 2 7 1 + 6 III' 2 - 3 s + V V / 2 l L F i 2 M 3 2 s + 1 p 2 2 2 7 7 1 + 6 3 3 + 6 5 ^ 7 1 + 6 112' 4 • 3 ^ ^ 4 . 3 2 5 + 1 ^ 7 7 + 2 2 m + 1 2 3 3 + 6 s ^ 2 n + 6 29. there exist integers m > 2, n > 0 and s e {0,1} szzc/z that 2 " 3 P ' zs fl square and E is Q-isomorphic to one of the elliptic curves: Chapter 3. Elliptic Curves with 2-torsion and conductor 2cc3l3p5 134 a-2 a4 A JJ1 2 . y + i p y / ^ £ L 2 m 3 2 s + V 22m-\-6^3-\-6sp7i-\-G JJ2 - 4 - 3 ^ 2 = = ^ _ 4 . 3 2 5 + 1 ^ + 2 2m+12 ^3+6Sp2n-ir6 j jr 2 m 3 2 s + V 2 2 7 7 1 + 6 3 3 + 6 . 5 ^ 7 1 + 6 JJ2' 4 . 3 * + ^ ^ — 4 • 3 2 « + l p n + 2 2m+l2^3+6Sp2n-\-6 30. there exist integers m > 2, n > 0 and s € {0,1} such that p " 3 2 ' ' fs a square and E is Q-isomorphic to one of the elliptic curves: 02 a4 A K K 1 2 . 3 ^ ^ _ 2 m 3 2 S + l p 2 22m+6^3+6 .Sp7i+6 K K 2 - 4 . 3 ' + ^ ^ 4 . 3 2 5 + 1 ^ + 2 _2m+1223+6 .Sp27i+6 K K 1 ' " 2 • S-'+Vv 7 2^ _ 2 r n . 3 2 S + l p 2 2 2 7 7 1 + 6 3 3 + 6 . 9 ^ 7 1 + 6 K K 2 ' 4 • 3* + V y 7 ^ 4 . 32.5+1^72+2 _ 2m+1233+6 .s^2n+6 31. there exist integers m > 2 and s,t G {0,1} such that 2 ™ + 1 fs a square and E is Q-isomorphic to one of the elliptic curves: « 2 a4 A LL1 2 . 3 . + v + y 2 = ± i 2 m 3 2 s + l p 2 t + l 2 2 m + 6 3 3 + 6 S p 3 + 6 t LL2 _4 . 3 . s + y + y r^i 4 . 3 2 s + l p 2 t + l 2 m + 1 2 3 3 + 6 . S p 3 + 6 i L L 1 ' - 2 . 3 '+y+y™ 2 m 3 2 s + V t + 1 2 2 7 7 i . + 6 3 3 + 6 . S p 3 + 6 t L L 2 ' 4.3,+y+ym 4 . 3 2 s + l p 2 t + l 2 m + 1 2 3 3 + 6 . S p 3 + 6 t 32. there exist integers m > 2 and s,t <E {0,1} such that 2 ' 3 p 1 fs A square and E is Q-isomorphic to one of the elliptic curves: a2 a4 A M M 1 2. 3,+y+y2^i 2 m 3 2 s + l p 2 t + l _ 2 2 m + 6 3 3 + 6 S p 3 + 6 t M M 2 - 4 - 3 ' ^ V + V 2 ^ _ 4 . 3 2 s + l p 2 ( + l 2 m + 1 2 33+6.ip3+6t M M 1 ' - 2 . 3 ^ y + y ^ i 2 m 3 2 s + l p 2 t + l _2 777+63 +6.Sp3+6t M M 2 ' 4 - 3 ^ V + V ^ i _ 4 . 3 2 s + l p 2 i + l 2 m + 1 2 3 3 + 6 S p 3 + 6 4 Chapter 3. Elliptic Curves with 2-torsion and conductor 2a3,3ps 135 Theorem 3.28 The elliptic curves E defined over Q, of conductor 273bp2, and hav-ing at least one rational point of order 2, are the ones such that one of the following conditions is satisfied: 1. there exist integers £ > 2 — b and n > 0 such that 2 • 3e + pn is a square and E is Q-isomorphic to one of the elliptic curves: 0,2 a4 A A l 2-3h-lpyj2-3e +pn 2 • 3^+2 (b- i) p 2 2 8 3 2 £ + 6 ( b - l ) p n + 6 A2 ~4-3 6"Vv /2-3*+p n 4 . 3 2 ( 6 - 1 ) ^ + 2 2l3 3<?+6(b-l)p2n+6 A l ' -2 • 3b~lpy/2 • 3e + pn 2 • 3 f + 2( b" 1)p 2 2 832t !+6(b-l )pn+6 A2' 4-3 b" 1pv /2-3 f +pn 4 . 3 2 ( b - l ) p n + 2 2l33>:+6(6-l)p2n+6 B l 2-3 6 -V\ /2-3^ +pn 3 2 ( b - l ) p n + 2 2 7 3 ^ + 6 ( b - l ) p 2 n + 6 B2 -A-3b-lp^j2-3l +pn 8 • 3<+2<6-V 2l432^+6(b-l) pn+6 B l ' -2-3b-lp^/2-3e +pn 3 2 ( b - l ) p n + 2 2 73<?+6(b-l) p 2n+6 B2' 4 - 3 b - 1 p \ / 2 - 3 € +Pn 8 • 3 '+ 2 ( b - i y 2l4 3 2J!+6(b-l) p n+6 2. there exist integers £ > 2 — b and n > 0 such that 2 • 3 — pn is a square and E is Q-isomorphic to one of the elliptic curves: «2 a.4 A CI 2 • 3 b" 1p\/2 • 3e -pn 2 . 3 ^ + 2 ( 6 - l ) p 2 ( _2832£+6(6-l)pn+6 C2 - 4 - 3 b - 1 p V / 2 - 3 ^ - p ™ _ 4 . 32(b-l)pn+2 2l33*?+6(b-l)p2n+6 c r ~2-3b-}py/2-3e -pn 2 . 3 « + 2 ( 6 - l ) p 2 - 2 8 3 M + 6 ( b _ 1 ) p ™ + 6 C2' 4-3 b - 1 pv / 2- 3^-p™ -4 . 32(b-l)pn+2 2133f!+6(b-l)p2n+6 DI 2 -3 b " 1 pV / 2-3^ -p r l _ 3 2 ( b - l ) p n + 2 273f+6(b-l)p2n+6 D2 -4 • 3b-1py/2 • 3e - pn 8 • 3*+2(b"V _21432«+6(b-l)pn+6 DI' - 2 - 3 b - V \ / 2 - 3 * - p " _32(b-l)pn+2 2 7 3 « + 6 ( 6 - l ) 2 n + 6 D2' 4 - 3 b " 1 p V / 2 - 3 ^ - p n 8 • 3^+2(b~1)p2 _2l432t:+6(b-l)pn+6 3. there exist integers £ > 2 — b and n > 0 such that 2pn + 3 is a square and E is Q-isomorphic to one of the elliptic curves: Chapter 3. Elliptic Curves with 2-torsion and conductor 2c"3l3p5 136 a 2 CI4 A E l 2-3b-1ps/2pn + 3e 2 • 32(b~l)pn+2 283<'+6(b-l)p2n+6 E2 -4 -3 b - 1 pv / 2p n + 3^  4 . 3*+2 (b- l ) p 2 2 13 3 2f+6(6- l) p n+6 E l ' -2-3h-1p^/2pn + 3e 2 . 3 2 ( 6 - l ) p n + 2 283t5+6(6-l)p2n+6 E2' 4 • 3 b- 1p v/2p" + 3e 4 . 3 « + 2 ( b - l ) p 2 2 1 3 3 2 « + 6 ( b ~ l ) p n + 6 FI 2-3 f e- 1pv /2p n + 3 f 3 £ + 2 ( 6 - l ) p 2 2732<5+6(6-l)pn+6 F2 -4-3b-1py/2pn + 3e 8 • 3 2 ( b - 1 )p n + 2 2143<i+6(b-l)p27i+6 FI' - 2 • 3b-1py/2pn + 3e 3 « + 2 ( b - l ) p 2 2732<?+6(b-l)pn+6 F2' 4- 3 b - 1 pv / 2p n + 3^  8 • 3 2( b" 1)p"+ 2 2143<!+6(b-l)p2n+6 4. there exist integers £ > 2 — b and n > 0 such that 2pn — 3e is a square and E is Q-isomorphic to one of the elliptic curves: a2 0,4 A GI 2 • 3 b ~ W 2 p n - 3« 2 . 3 2 ( 6 - l ) p n + 2 _ 2 8 3 « + 6 ( b - l ) p 2 n + 6 G2 -4 • 3b~1p^2pn - 3e _4 . 3^+2 (b- l) p 2 21332(5+6(b-l)pn+6 GI ' - 2 - 3 b " 1 p \ / 2 p n - 3 f 2 • 32(b-Dp«+2 _ 2 8 3 f5+6(b-l) p 2n+6 G2' 4- 3b-lps/2pn - 3e _4 . 3 f+2(b-l)p' 2 21332<f+6(b-l)pn+6 HI 2-3b~1ps/2pn -3e _ 3 / 5+ 2 (b - l ) p 2 2 7 3 2^+6(b-l) p n+6 H2 -4 • 3>'-1psj2pn - 3* 8 . 3 2 ( b - l ) p n + 2 _ 214 3(?+6(6-l) p2n+6 H I ' -2 • 3 b- 1pv /2p" - 3* _ 3 « + 2 ( h - l ) p 2 2 7 3 2f+6(6- l ) p n+6 H2' 4-3b-1psV2pn -3e 8 . 3 2 ( b - l ) p n + 2 _ 2 14 3 f!+6(b~l) p 2n+6 5. there exist integers £ > 2 — b and n > 0 such that 2 + 3epn is a square and E is Q-isomorphic to one of the elliptic curves: a 2 a4 A 11 2-3 b~ 1pv /2 + 3 V 2 . 3 2 ( t - l ) p 2 283*M-6(b-l)pn+6 12 -4 • 3b~lp^2 + 3 V 4. 3 <+2 ( d - i y +2 21332<?+6(b~l) p2n+6 11' - 2 - 3 b - 1 p v / 2 + 3 V 2 • 32(b-V 283f?+6(b-l)pn+6 12' 4 • 3 b - 1 pv / 2 + 3 V 4 . 3 <H-2(b-l) p n+2 213 32^+6(b-l) p2n+6 J l 2-3 f t- 1pv /2 + 3 V 3^ +2(b-l)pn+2 2 7 3 2(?+6(b- l) p 2n+6 J2 - 4 - 3 b _ 1 p A / 2 + 3 V 8 • 3 2 ^ p 2 2143^+6(b-l)pn+6 J l ' - 2 -3 b - 1 pv / 2 + 3 V 3 M - 2 ( 6 - l ) p n + 2 2 7 3 2f+6 (b - l ) p 2n+6 J2' 4-3 b - 1 pv / 2 + 3 V 8 • 3 2 ^ p 2 2 1 4 3 £ + 6 ( 6 - l ) p n + 6 Chapter 3. Elliptic Curves with 2-torsion and conductor 2a3l3p5 137 6. there exist integers £ > 2 - b and n > 0 such that 3epn -2 is a square and E is Q-isomorphic to one of the elliptic curves: 0-2 <Z4 A KI 2 • 3b-1p^/3ipn - 2 „ 2 . 3 2 ( 6 - i y 283f+6(b-l)pn+6 K2 -4-3b~1py/3epn -2 4 . 3«+2(6-l)pn+2 _21332<?+6(b-l)p2n+6 K I ' -2 • 3 b - V \ / 3 V - 2 _2.32(b-l)p2 283^+6(b-l)pn+6 K2' 4 - 3 b " V \ / 3 V - 2 4 . 3£+2(6~l)pn+2 _21332<?+6(b-l)p2Ti+6 LI 2 - 3 b - V \ / 3 V - 2 3<?+2(b- l)pn+2 - 27 3 2 ^ + 6 ( b _ 1 ) p 2 T l + 6 L2 -4 • 3b"lps/3epn - 2 - 8 - 3 2 ^ V 2143£+6(b-l)pn+6 LI ' -2 • 3b~lps/3ipn - 2 3<M-2(b-l)pn+2 _ 2 7 3 2 f + 6 ( 6 - l ) p 2 n + 6 L2' 4 • 3b-1p^3ipn - 2 _8.32(b-l)p2 2143«+6(6-l)pn+6 7. there exist integers £ > 2 - b and n > 0 such that 3e - 2pn is a square and E is Q-isomorphic to one of the elliptic curves: 0.2 a.4 A M l 2-3b~1p^3e -2pn -2 • 3 2 ( 6 - 1 ) ^ + 2 2 8 3 ^ + 6 ( 6 - l ) p 2 n + 6 M2 -4-3 b "V\/3^-2p" 4 . 3«+2(b-l)p2 _ 21332 <»+6(b-l) p7i+6 M l ' - 2 • 3b~1py/3e - 2pn -2 • 32(b-l)pn+2 283^+6(b-l)p2n+6 M2' 4 • 3b-^py/3i - 2pn 4 . 3<!+2(6-l) p 2 _2l332«+6(b-l)pn+6 NI 2 • 3 b " W 3 * - 2pn 3«+2(b-l)p2 _2732^+6(b-l )pn+6 N2 - 4 - 3 b - 1 p \ / 3 ' ! - 2 p n -8 • 3 2 ( 6 - l ) p " + 2 2l4 3«+6(b-l) p 2n+6 N I ' -2-3 b -V\ /3 ' ? -2p n 3«+2(b-l)p2 _ 2732*5+6(6-l)pn+6 N2' 4 • 3b"V\/3£ - 2pn _8.32(b-l)pn+2 214 3(!+6(b-l) p 2n+6 8. there exist integers £ > 2 - b and n > 0 such that pn — 2 • 3 is a square and E is Q-isomorphic to one of the elliptic curves: Chapter 3. Elliptic Curves with 2-torsion and conductor 2°3l3ps 138 «2 CI4 A O l 2 - 3 b - 1 p \ / p n - 2 - 3 * -2 • 3^+2(b-D p2 2832t?+6(b-l) pn+6 02 - 4 • 3b-V\/pn - 2 • 3e 4 . 3 2(c.-l) pn+2 _2l33«+6(b-l)p2n+6 or -2 • 3 6 - 1 p V / P " - 2 - 3 ' -2 • S ^ 2 ^ " 1 ^ 2 2832^+6(b-l)pn+6 02' 4 • 3 b ~ W V - 2 - 3 ' 4 • 32(ft-i)p«+2 _2l33^+6(b-l) p 2n+6 PI 2 • S^VvV - 2 - 3 * 3 2(b-l) p n+2 _ 2 7 3 « + 6 (b - l ) p 2 n + 6 P2 - 4 • 3b~1py/pn -2-3e - 8 • 3<+2(6- l)p2 21432<;+6(b-l)pn+6 pr -2 • 3h-lp^pn - 2 • 3e 32(6-1)^71+2 _ 2 7 3 ^ + 6 ( 6 - l ) p 2 n + 6 P2' 4 • 3 b " V v V 1 ~2-3i _8.3^+2(b-iy 2l4 32£+6(b-l)pn+6 9. there exist integers £ > 2 - b and t e {0,1} such that 2 - 3 + 1 is a square and E is Q-isomorphic to one of the elliptic curves: a-2 a4 A Q l 2 . 3 h - V + 1 v / 2 ^ ± i 2 . 3^+2(6-1) 2t+l 2832^+6(6-1)^3+64 Q2 - 4 . 3 b - y + y 2 ^ i 4 • 32(6 - l)p2t+l 2133^+6(6-1) 3 + 6 t Q l ' 2 . 3 b -y+ 1 ^ 2 . 3 ^ + 1 2 . 3<?+2(b-l) p2t+l 2 8 3 2 / 5 + 6 ( 6 - l ) p 3 + 6 t Q2' 4-3b-y+y2'3'+1 4 . 3 2 (b - l ) p 2 t +l 2 1 3 3 « + 6 ( 6 - l ) p 3 + 6 t R l 2-3b-y+V2'3;+1 3 2 ( 6 - l ) p 2 t + l 273 (5+6(6- l ) p 3+6t R2 4-3b-y+y2-3;+i g . 3 * + 2 (b - l ) p 2 t +l 2 1432i?+6(6- l) 3 + 6 t Rl' 2 • 3b-y+y 2_3i±i 32(6-1) 2t+l 273(5+6(6- l ) p 3 + 6 t R2' 4-3 b-y+y 2^ti 3 . 3 € + 2 ( 6 - l ) p 2 t + l 2l432^+6(6- l) p 3+6t 10. there exist integers £ > 2 - b and t e {0,1} such that 2'3p 1 is a square and E is Q-isomorphic to one of the elliptic curves: Chapter 3. Elliptic Curves with 2-torsion and conductor 2a3/3p5 139 a2 a 4 A SI 2-3h~1pt+1J2-3'-1 1 \J p 2 . 3<!+2(b-l) 2t+l _2832f5+6(b-l)p3+6t S2 4 - 3 6 - y + 1 v / 2 ' 3 C - 1 _ 4 . 3 2 ( f r - l ) p 2 t + l 2 l 3 3 « + 6 ( 6 - l ) p 3 + 6 t SI' 2-3 b-y+y 2- 3;- 1 2 . 3 ^ + 2 ( 6 - 1 ) 2t+l _ 2 8 3 2 £ + 6 ( 6 - l ) p 3 + 6 t S2' 4 - 3 b - y + i v / 2 - 3 ' - 1 _ 4 . 3 2 ( 6 - l ) p 2 t + l 2 1 3 3 « ! + 6 ( 6 - l ) p 3 + 6 t TI 2 - 3 b - y + i x / 2 ' 3 ; - 1 _ 3 2 ( b - l ) p 2 l + l 273*5+6(6-1)^3+64 T2 4-3 b-y+y 2- 3;- 1 8 . 3^+2(6 -1)^24+1 _2l432t5+6(6-l) p3+6t TI' 2 - 3 b - y + i v / 2 - 3 ' - 1 _ 3 2 ( 6 - l ) p 2 i + l 2 7 3 « + 6 ( 6 - l ) p 3 + 6 ( T2' 4 - 3 b ~ y + i v / 2 3 ' - 1 g . 3*5+2(6- l) p24+l _ 2 1 4 3 2 ( 5 + 6 ( 6 - l ) p 3 + 6 t 11. there exist integers £ > 2 — b and t G {0,1} such that 2sy- is a square and E is Q-isomorphic to one of the elliptic curves: a 2 a 4 A U l 2 - 3 b - y + y ^ 2 . 3 2 ( 6 - l ) p 2 t + l 283»?+6(6-l) p 3+6t U2 - 4 - 3b - y + y 2 ± ^ 4 . 3<5+2(b-l ) p 2t+l 2 1 3 3 2 * 5 + 6 ( 6 - l ) p 3 + 6 t U l ' - 2 . 3 b - y + y ^ 2 . 3 2 ( 6 - l ) p 2 t + l 2 8 3 f 5 + 6 ( b - l ) p 3 + 6 t U2' 4 . 3 6 - y + y ^ 4 . 3 « + 2 ( 6 - l ) p 2 t + l 2l3 3 2i5+6(b-l) p 3+6t VI 2 . 3 6 - y + y 2 + ^ 3*5+2(6-1 ) p 2 t + l 2 732*5+6 (b - l ) p 3+6t V2 _ 4 . 3b - y + y 2 ± ^ g. 3 2 ( 6 - 1 )p2t+l 2 l 4 3 « + 6 ( b - l ) p 3 + 6 t V I ' - 2 . 3b - y + y 2 + ^ 3 < 5 + 2 ( 6 - l ) p 2 t + l 27 3 2*5+6(6 - l ) p 3+6t V2' 4 - 3 b - y + y ^ 8 . 3 2 ( b - l ) p 2 t + l 2l4 3 *5+6(b-l) p 3+6t 22. there exist integers £ > 2 — b and t 6 {0,1} such that is a square and E is Q-isomorphic to one of the elliptic curves: Chapter 3. Elliptic Curves with 2-torsion and conductor 2a3l3ps 140 a.2 a 4 A W I 2 - 3 6 - y + y ^ -2-3 2 < f , - i y t + i 283*?+6(6-iy+6t W2 - 4 . 3 b - y + y ^ 4 . 3<'+2(f,-l)p2(+l _2l332£+6(t>-l)p3+6t w r - 2 . 3b - v + y ^ - 2 - 32(6-iyt+i 2 8 3 « ! + 6 ( 6 - i y + 6 t W2' 4 . 3^+2(6-1) 2t+l _2l3 32)?+6(b-l)p3+6t XI 2 - 3 b - y + i v / 3 ^ 3*!+2(b-3)p2t + l - 2 7 3 2 £ + 6 ( b _ i y + 6 i X2 - 4 . 3 b - y + y 2 f ^ _ 8 . 3 2 ( 6 - i y t + l 2 1 4 3 « + 6 ( 6 - i y + 6 t X I ' - 2 . 3 b - y + y ^ 3*M-2(b-iyt+l _27 32<+6(b-iy+6t X2' 4 - 3 b - y + y 3 ^ _ 8 . 32(6-iyt+i 2143*!+6(b-iy+6t In the case that 6 = 2, i.e. N — 2732p2, we furthermore could have one of the following conditions satisfied: 13. there exist integers n > 0 and s 6 {0,1} such that 2 p 3 + 1 is a square and E is Q-isomorphic to one of the elliptic curves: a 2 a 4 A Y I 2-3 - s +V v / 2 j i r i 2 - 32 s + y + 2 2 833+6Sp2n+6 Y2 4-3 s+v v/ 2 p; + l 4-3 2 s + 1 p 2 2 1 3 3 3 + 6 5 ^ 7 1 + 6 Y I ' 2-3 s + 1 p V / 2 P 3 + 1 2 - 3 2 s + 1 p n + 2 2 8 3 3 + 6 ^ 2 7 7 + 6 Y2' 4-3 s +V V / 2 j l f ± i 4 - 32 s + > 2 2 1 3 3 3 + 6 . ^ 7 1 + 6 Z I 2 • 3 s + 1 p V / 2 P 3 + 1 3 2 a + y 2 7 3 3 + 6 s p n + 6 Z2 4 . 3 - 5 + V V / 2 P 3 + 1 8 • 32 s + y + 2 2l433+6 .Sp2n+6 Z I ' 2.3s+1pV /2 j^±i 3 2 s + 1 p 2 2733+6s p 7 i+6 Z2' 4 • 3 * + ^ ^ 8 • 3 2 - 5 + y + 2 2 l 4 3 3 + 6 s p 2 n + 6 14. there exist integers n > 0 and s £ {0,1} such that is a square and E is Q-isomorphic to one of the elliptic curves: Chapter 3. Elliptic Curves with 2-torsion and conductor 2a3f3ps 141 a2 04 A A A 1 2 • 3s+1pyJpT,+2 2 • 3 2 s+y 2833+6s p n+6 A A 2 - 4 • 3^py/^ 4 . 32s+lpn+2 2l3 3 3+6Sp2n+6 A A 1 ' -2-3'^py/^ 2 • 32s+1p2 2 8 3 3 + 6 5 ^ 7 1 + 6 A A 2 ' 4 • 3 ' + 1 P y J ^ 4 . 32.s+l pn+2 2 1 3 3 3 + 6 . S p 2 n + 6 BB1 2 • 3s+lpyl^±l 32.s+lpn+2 2 7 3 3 + 6 S p 2 n + 6 B B 2 - 4 - 3 ^ p ^ 8 • 3 2 s+y 2 l 4 3 3 + 6 s p 7 i + 6 B B 1 ' 2 • 3s+lpyj^ 32s+l p n+2 2733+6Sp2n+6 B B 2 ' 4 • 3 ^ ^ g . 3 2 , + l p 2 2 1 4 3 3 + 6 5 ^ 7 1 + 6 15. there exist integers n > 0 and s 6 {0,1} such that -^g-2- is a square and E is Q-isomorphic to one of the elliptic curves: a2 0,4 A C C 1 2 . 3 ' + ^ ^ - 2 - 3 2 s + V 2 8 3 3 + 6 5 ^ 7 7 + 6 C C 2 -4.3'+V^ 4 . 32.S+I n + 2 — 2 1 3 3 3 + C s p 2 n + 6 C O ' - 2 • 3 2 s + V 2833+6.5^71+6 C C 2 ' 4 . 3 ^ ^ 4 . g 2 s + l p n + 2 _2 l333+6.Sp2n+6 DDI 2 . 3 ' + ^ ^ 3 2 , + l p n + 2 — 2 7 3 3 + 6 s p 2 n + 6 D D 2 -4 • 3 * + i p > / £ £ 2 -8 • 3 2 " + 1 p 2 21433+6.5^71+6 DDI' - 2 " 3 ' + ^ ^ 3 2 s + l p 7 , . + 2 _2733+6 . s p 2n+6 D D 2 ' 4 . 3-^py/^ -8 • 3 2 s + V 2l433+6 .Sp7i+6 Theorem 3.29 The elliptic curves E defined over Q, of conductor 283bp2, and hav-ing at least one rational point of order 2, are the ones such that one of the following conditions is satisfied: 1. there exist integers I > 2 — b and n > 0 such that 3tp2'~1 is a square and E is Q-isomorphic to one of the elliptic curves: Chapter 3. Elliptic Curves with 2-torsion and conductor 2a3/3ps 142 ai 0 4 A A l 2 . 3^+2 (6- l ) p n+2 _ 2 9 3 2 « + 6 ( 6 - l ) p 2 n + 6 A2 _ 8 . 3 2 ( 6 - l ) p 2 2 1 5 3 £ + 6 ( 6 - l ) p n + 6 A l ' -4 • 3»~ V25^=1 2 . 3^+2 (6- l ) p n+2 _ 2 9 3 2 » ! + 6 ( 6 - l ) p 2 n + 6 A2' -8 • 32<6" V 2 l 5 3 £ + 6 ( 6 - l ) p n + 6 B l 4-3"-V v/ 3V'- 1 -2 • 32(fc-V 293(?+6(b-l)pn+6 B2 8 • s^py/3'";-1 g . 3^+2(6-1)^71+2 _ 2 i 5 3 2 ^ + 6 ( b - i ) p 2 n + 6 B l ' - 4 . 3 ' - ^ ^ _ 2 . 3 2 ( 6 - l ) p 2 293€+C(b-l)pn+6 B2' 8 . 3 6 - ^ 3 ^ g . 3«+2(fe-l) p 77+2 _ 21532f5+6(6-l) p2n+6 2. there exist integers £ > 2 — b and n > 0 such that 3 + p zs a square and E is Q-isomorphic to one of the elliptic curves: a 2 0 4 A CI 2 • 3^+2C'-i)p2 2932<+6(6-l) n+G C2 8 • 3b"V^/3f+p" 8 • 3 2 ( b - ! ) p ™ + 2 215 3^+6(6-1) 2n+6 c r " 4 • 3 b " 2 • 3<+2(6-Dp2 2 9 3 2 « + 6 ( 6 - l ) p n + 6 C2' 8 • 32(6-l)pn+2 2 l53£+6(6- l ) p 2n+6 DI 4 - 3 - V v ^ 2 • 3 2C>-i)p"+2 293^+6(6-1) 2n+6 D2 8 • 3 b - 1 p A / 3 '+ p " 8 • 3 « + V 2l532(5+6(b-l) p7i+6 DI' - 4 - 3 ^ ^ 2 • 3 2C'-i) ?,n+ 2 293<5+6(6-l)p2n+6 D2' 8 • 3h-1pyJ3'+/' 8 • 3*+V 2l532f+6(6-l) n+6 O f _ T] 3. f/tere exzsf integers £ > 2 — b and n > 0 swc/z tTzzzf — j 2 - is a square and E is Q-isomorphic to one of the elliptic curves: Chapter 3. Elliptic Curves with 2-torsion and conductor 2 a 3 / 3 p 5 143 a-2 (24 A E l 2 . 3 < + 2 ( f c - i y _2932<?+6(6-l)p7i+6 E2 -8 • 32(b-!)p«+2 2l53<;+6(b-l)p2n+6 E l ' - 4 . 3 ^ ^ 2 - 3 ' + 2 ( t - i y _2932^+6(fe-1)pn+6 E2' 8 - 3 " - ^ ^ -8 • 3 2 ( h - i ) p « + 2 2l5 3£+6(b-l) p2n+6 F l 4 • 3 b" V v 7 ^ -2 • 32(b-l)p«+2 293^+6(b-l)p2n+6 F2 - 8 - 3 - V v 7 ^ 8 • 3e+2V>-Vp2 _2l532<?+6(b-l)pn+6 F l ' - 4 . 3 - V y 7 ^ -2-32( f c-1V+2 293<J+6(b-l)p2n+6 F2' 8 - 3 b " V x / ^ 8 • 3 e + 2 ^ p 2 _2l532*?+6(b-l)pn+6 •n _o£ 4. there exist integers I > 2 — 6 and n > 0 such that p—2— is a square and E is Q-isomorphic to one of the elliptic curves: a-2 0 4 A GI 4 . 3 ^ / ^ 2.32(b-l)pn+2 _293«+6(b-l)p2n+6 G2 _ 8 . 3 b - v 7 ^ _ 8 .3£ + 2(b- l) p 2 2^2,2t+fi(b-\)pn+& GI ' _ 4 . 3 b - V V / ^ 2.32(b-l)pn+2 _293e+&(b-l) p2n+6 G2' 8-3fc-1pV/'E:T2£ _ 8 . 3 £+2 (b - i y 2l532/i+6(b-l)p7a+6 HI 4-3 b -V V / p " ; 3 f -2 • 3<+2<6-V- 2932f+6(b-l)pn+6 H2 - s ^ - V y 7 ^ 8 • 32C>-iy+2 _ 2153< ,+6(b-l)p2n+6 H I ' -4 • 3>" V v 7 ^ - 2 • 3* + 2 ( 6 -V 2932/5+6(b-l)pn+6 H2' S - ^ P V 7 2 ^ 8 • 32(fe-V+2 _2l53*?+6(b-l)p2n+6 5. i7zere exist integers £ > 2 — b and t e {0,1} such that z s a square and E is Q-isomorphic to one of the elliptic curves: Chapter 3. Elliptic Curves with 2-torsion and conductor 2Q3 /3p'5 144 a2 0 4 A 11 2 . 3<5+2(b~l)p2t+l 2932f+6(b-l) p3+6t 12 8 • 32 ( f e - i y t + i 2l5 3 f+6 (b- l ) p 3+6( 11' -4-3'-V+V3gi 2 . 3 f?+2 (b- l ) p 2i+l 2932«+6(b-l) p 3+6t 12' 8 . 3 b - y + l v / 3 i t l 8 • 3 2 ^ p 2 t + 1 2 153f :+6(6- l)p3+6t J l 4 - 3 b - y + y 2 i ± i 2 . 3 2 ( b - l ) p 2 t + l 293«5+6(b-l) p3+6t J2 _ 8 . 3b - y + y 3 i t i g . 3 f + 2 ( 6 - l ) p 2 t + l 2l532(?+6(b-l) p3+6t J l ' - ^ - y + y ^ 2 . 3 2 ( b - l ) p 2 t + l 293<+6(b-l) p3+6t J2' s - ^ - y + y 3 ^ g . 3 C + 2 ( b - l ) p 2 t + l 2l532(5+6(b-l) p3+6t 6. there exist integers £ > 2 — b and t e {0,1} such that is a square and E is Q-isomorphic to one of the elliptic curves: a2 04 A KI 4 - 3 b - y + V 3 ^ 1 2 . 3*5+2(b-l)p2t + l _2932f5+6(b-l)p3+64 K2 - s - ^ - y + y ^ i - 8 • 3 2 ( t , - 1 ) p 2 « + i 2l53*5+6(b-l)p3+6t K I ' ^ . ^ - y + y 3 ^ 2 . 3*5+2(6-l) p2t+l _2932(5+6(b-l) p3+6( K2' 8 . 3 b - y + y 3 ^ i - 8 • 3 2 ( f c - V t + 1 2l53^+6C>-l)p3+6t LI 4 - 3 b - y + V ^ - 2 - 32 ( b - i y + i 293^+6(6-l) p 3+6t L2 _ 8 . 3 b -y+y 2 ^ 1 g . 3^+2(6- l ) p 2 (+l _ 2 i 5 3 2 ( ! + 6 ( ( > - i ) p 3 + 6 t LI ' - 4 - 3 b - y + V ^ - 2 • 3 2 ( f - i y t + i 2 93«+6 (b - l ) p 3+6( L2' 8.3b-y+V^i 3 . 3 £ + 2 ( b - l ) p 2 ( + l _2 l532«+6 (b- l ) p 3+6t In the case that b — 2, i.e. N = 2 8 3 2 p 2 , we furthermore could have one of the following conditions satisfied: 7. there exists an integer n > 0 such that p g 1" 1 is a square and E is Q-isomorphic to one of the elliptic curves: Chapter 3. Elliptic Curves with 2-torsion and conductor 2a3@ps 145 « 2 a.4 A M l 4 . 3«+ 2 • 3 2 s + y + 2 2 9 3 3 + 6 S p 2 n + 6 M 2 8 • 3 s + 1 p \ / ^ 8-3 2 s + 1 p 2 2 l 5 3 3 + 6 s p n + 6 M l ' - 4 • S ' + V y ^ 2 • 32 s + 1 p n + 2 2933+6Sp2n+6 M 2 ' 8 • 3s+1pxJpn+1 8 . 3 2 , + l p 2 2 1 5 3 3 + 6 S p T i + 6 NI 4 • 3s+1PyJp"+1 2 • 3 2 s + 1 p 2 2 9g3+6 .5p77+6 N 2 8 • 3s+1pyJpn+* 8 • 3 2 s + 1 p n + 2 2l5|T{3+6.5p2n+6 N I ' 2-3 2 s + 1 p 2 2 9 3 3 + 6 S p n + 6 N 2 ' 8 • 3s+lpxJp,'+i 8 - 3 2 s + 1 p n + 2 2l5g3+6.Sp2n+6 8. there exists an integer n > 0 swcfr ffrfl/1 -^g— is a square and E is Q-isomorphic to one of the elliptic curves: a-2 (I4 A O l 2 • 3 2 s + 1 p n + 2 —2 93 3 + 6 sp 2 n + 6 02 -8 • 32s+V 215 33+65^77+6 O l ' - 4 - 3 ' + ^ ^ 2 • 32 s+V"+ 2 —2 93 3 + 6 : sp 2 n + 6 02' 8 - 3 ^ ^ -8 -3 2 s + 1 p 2 2l533- i-6s p n+6 PI 4 - 3 ' + V V / £ ^ - 2 - 3 2 s + 1 p 2 2933+6 . s p 7i+6 P2 _ 8 . 3 - + I P N / ^ 8 • 3 2 s + y + 2 —2 1 5 3 3 + 6 s p 2 n + 6 PI ' - 4 - 3 ^ ^ -2 • 3 2 s + 1p 2 2933+6.5p77+6 P2' 8 •y+'Py/*^ 8 • 3 2 s + 1 p n + 2 _21533+6-sp2n+6 9. there exist integers s,t € {0,1} such that E is Q-isomorphic to one of the • elliptic curves: « 2 0.4 A Q i 0 2 • 3 2 s + 1 p 2 t + 1 —2 9 3 3 + 6 s p 3 + 6 t Q2 0 - 8 • 3 2 ; + 1 p 2 t + 1 2l533+6sp3+6£ R l 0 - 2 • 3 2 ; + 1 p 2 t + 1 2933+65^3+6* R2 0 8 • 3 2 s + 1 p 2 4 + 1 —2 1 5 3 3 + 6 s p 3 + 6 t Chapter 3. Elliptic Curves with 2-torsion and conductor 2cc3f3ps 146 3.4 Proofs of 2 a 3 / J p and 2"3V We sketch a constructive proof that the curves listed in the tables of Sections 3.2 and 3.3 are all the curves up to Q-isomorphism of the stated conductor. Most of the work has already been done in Appendix A ; there we have a classification of curves (up to Q-isomorphism, and containing a point of order 2) with conductor of the form 2a3l3p6. A l l that needs to be done now is to find the exact conductor, i.e. the values o f 0 < a < 8 , l < / 3 < 2 and 1 < 5 < 2. These wi l l depend on the values of m,£,n and the congruence class of d modulo 4 in the defining Diophantine equation. To compute the conductors of each of the curves in the tables of Appendix A we make extensive use of the tables in Chapter 2. Rather than get bogged down in all the details of computing the conductors of the curves we wi l l give a general overview of how the computations can be done. This should be enough to give the reader the flavor of the proof and allow us to save some trees in the process. In what follows we refer to the elliptic curve y2 = x3 + ax2 + bx by its coefficients a and b. We split the curves appearing in Lemma A . l into three classes: let A.I.I be the class consisting of curves numbered 1 through 9, A. l . I I be the class of curves numbered 10 through 18, and A . l . I l l be the class of curves numbered 19 through 27. It is straightforward to check that the 2-valuations of a, b and A for curves in each of the three classes are as follows. A.1.I A. l . I I A . l . I l l v2{a) n = r i + l (if m > 1) > n + 1 (if m = 0) n + 2 v2(b) m + 2r x - 2 2n 2n + 1 v2(A) 2m + 6ri 2rj + 1 6n +9 Chapter 3. Elliptic Curves with 2-torsion and conductor 2a3l3p5 147 It follows directly from Theorem 2.1 that for curves in A.I . I v2(N) = { A if r i = 0, m = 2, a = l (mod 4), b = 1 (mod 4) 2 if 7*1 = 0, m = 2, a = 1 (mod A), b = -1 (mod 4) 3 i f r x = 0,m = 2, a = - 1 (mod 4), b = 1 (mod 4) 4 if 7*1 = 0, m = 2, a EE - 1 (mod 4), 6 EE - 1 (mod 4) 5 if 7*1 = 0, 777 = 3, 4 if 7*1 = 0, m > 4,. a = - 1 (mod 4) 3 if T*I = 0, TO = 4, 5, a EE 1 (mod 4) 0 if 7*i = 0, rn = 6, a EE 1 (mod 4) 1 if 7*1 = 0, m > 7, a EE 1 (mod 4) 7 if 7*1 = 1, 777 = 1, 6 if 7*1 = 1, 777 > 2, 7 if 7*1 = 2 ,777=1, Chapter 3. Elliptic Curves with 2-torsion and conductor 2a3/3ps 148 and for the curves in A. l . I I we have v2(N) = '6 if r\ = 0, rn = 0, 6 = 1 (mod 4) 5 if r\ = 0, m = 0, 6 = -1 (mod 4) 7 if ri = 0, m = 1, 4 if r i = 0, m = 2, a - 6 = 13 (mod 16), 3 if rj = 0, m = 2, a — 6 = 5 (mod 16), 4 if rj = 0, m = 2, a - 6 = l (mod 16), 2 if r\ = 0, m = 2, a - 6 - = 9 (mod 16), 5 / if ri = 0, m = 3, \ 4 if ri = 0, m > 4, a/2 = 1 (mod 4) 3 if ri = 0, m = 4,5, a/2 = - 1 (mod 4) 0 if rj = 0, m = 6, a/2 = - 1 (mod 4) 1 if r i = 0,777 > 7, a/2 = - 1 (mod 4) 5 if ri = 1, 777, = 0, 6/4 = 1 (mod 4) 6 if r\ = l , m = 0, 6/4 = - 1 (mod 4) 6 if n = 1, m > 1, 6 /4=1 (mod 4) if 7 -i = 2, m >1 . 6/4 = - 1 (mod 4). As for the curves in A . l . I l l we simply have V2(N) = 8. The values of v-i(N) can be directly'computed from Theorem 2.3. We find that if E is one of the curves in Lemma A . l and D E the corresponding Dio-phantine equation which p satisfies, then, if 3 appears as a coefficient of d 2 in DE then v3(N) = 2, otherwise {0 if r 2 = 0 and I = 0 1 if ra = 0 and I ^ 0 2 i f r 2 = l . Similarly, we find that if p appears as a coefficient of d? in D E then vp(N) = 2, Chapter 3. Elliptic Curves with 2-torsion and conductor 2 a3 / 3p 5 149 otherwise VP(N) = < 1 if r 3 if r 3 if r 3 0 and n = 0 0 and n ^ 0 1. Recall, we also made the convention in Appendix A that, in the Diophantine equations listed in Lemma A . l , £, m, or n can only be zero if they appear on the right-hand side of the equation. This just avoids redundancy in our list of curves. This is all the information needed to distribute the curves in A . l across the appropriate theorems in Sections 3.2 and 3.3. Notice that, by taking £ = 0, we get curves of conductor 2ap2. This is how the curves in the theorems of Section 3.1 were originally found, though the proof we gave there did not reflect this. Similar considerations can be applied to the curves in Lemma A.2. This completes the proof of the theorems in Sections 3.2 and 3.3. Chapter 4 Diophantine Lemmata In this chapter we prepare all the Diophantine lemmata we w i l l need in sub-sequent chapters. 4.1 Useful Results In this section we collect together, for the convenience of the reader, all the re-sults that we wi l l need to solve the Diophantine equations in the next section. The first result we wi l l need is Catalan's Conjecture, which is now a theo-rem of Mihailescu [52]. In this work we wi l l refer to it as "Catalan's Theorem", or simply as "Catalan". Theorem 4.1 (Mihailescu) The only solutionlo the diophantine equation xn - ym = 1 in positive integers x,y,m,n with n,m > lis given by 3 2 — 2 3 = 1. Some results of Cohn and Ljunggren that we wi l l make use of are the following. Theorem 4.2 (Cohn [19]) Let k be odd. The only solutions to x2 + 2k = yn in positive integers x, y and n > 3 are k x y n 6a + 1 5 • 2 3 ° 3- 22a 3~ 4 a + 5 7 - 2 2 q 3 - 2 a 4 10a+ 5 l l - 2 5 a + 3 3 - 2 2 a + 1 5 150 Chapter 4. Diophantine Lemmata 151 with a > 0. Theorem 4.3 (Ljunggren [48]) The equation x4 — 3y2 = 1 has no solution in pos-itive integers. The main results that we wi l l use in attacking the diophantine equations arising in the table of the previous chapter are the results of Bennett, Skinner, Vatsal and Yazdani. Here we restate the relevant parts of their results. Theorem 4.4 (Bennett, Skinner [5]) If n > 4 and C € {1, 2,3, 6} then the equa-tion has no solutions in nonzero pairwise coprime integers (x, y, z) with, say, x > y unless C = 2 and (n, x, y, z) G {(5, 3, - 1 , ±11), (4,1, - 1 , ± 1 ) } . Theorem 4.5 (Bennett, Vatsal, Yazdani [6]) If C G {1,2,3}, n > 5 is prime and a, (3 are nonnegative integers, then the diophantine equation or, possibly, (a, C) = (1, 2). Theorem 4.6 (Bennett [3]) Suppose that a < b are positive integers with ab = 2a3/3 for nonegative integers a, (3. If n > 3 is an integer, then the only solutions in positive integers x and y to the diophantine equation ax7' — by71 = ± 1 are given by The proofs of the above theorem rely heavily on results on ternary dio-phantine equations coming from the theory of Galois representations and modular forms. In a few cases of the proofs of the results in the next sec-tion we wi l l need to make use of this theory, so we briefly outline the main idea as it applies here. See [5] for the general details. xn + yn = Cz2 xn + 3ayn = C^z3 has no solutions in coprime integers (x, y, z) with \xy\ > 1, unless ( | x | , | y | , a ,n , |Cz 3 | ) = (2 , l , l I 7,125) Chapter 4. Diophantine Lemmata 152 Consider the equation 3exq + 2myq = d2 with £, m, q fixed integers. We want to show this equation has no solutions for x,y, d, with \xy\ > 1, in the case that q > 7 is prime and m > 6. Suppose that (x, y, d) = (a, b, c) is a solution in this case. Without loss of generality we may assume c E E 1 (mod 4). Then, as in [5], we associate to this solution the elliptic curve E a A c • Y2 + XY = X3 + (—^-X2 + 2m~2bqX, whose discriminant is A = 22m-'23e(ab2)" and conductor is N(EaAc) = 2a3l[p, p\ab E where a € {—1,0}. We then associate to E a A c a Galois representation pq a'b'c which is irreducible (see [5] corollary 3.1 , this is where \xy\ > 1 and q > 7 is required). The representation pq ",b,c arises from a cuspidal newform of weight 2 and level N = 2 Q + 1 3 (see [5] Lemma 3.3). This is where we reach a contradiction, since there are no newforms at level 3 or 6. So how does this help us with the diophantine equations we wi l l be con-sidering in the next section? Well, the above result applies to the equations d2 = 3 V + 2m and d2 = 3e + 2mpn to show that there are no solutions with both m > 6 and n having a prime divisor > 7. Then we'll use other methods to deal with the other cases of n and m. 4.2 Diophantine lemmata In the following p denotes a prime > 5. The following results generalize those of Ivorra [37] (and of Hadano [34]). In particular, Ivorra's work concerns the case when £ = 0, thus in our proofs we can refer to Ivorra's work and assume £ > 1. For an integer n, let Pm;„(n) denote its smallest prime factor and Pmax(n) denote its largest prime factor. This notation wi l l be used throughout the rest of this section. Chapter 4. Diophantine Lemmata 153 Lemma 4.7 1. The solutions to d2 - 2rn3epn = 1 with TO, £, n > 0 and d > 1 satisfy one of the following (i) n = Oand (d, m, £) e {(2,0,1), (3,3,0), (5, 3,1), (7,4,1), (17, 5, 2)}, (ii) n = 1 and (a) p = 5and (d,m,£) € {(9,4,0), (161,6,4)}, (b) p = 3e ±2 with£> 1 and (d, m) = (p + 1,0), (c) p = 3-2=± with £ odd and (d, rn) = (4p + 1, 3), (d) p = ^rp- with £ even and (d, m) = (4p — 1,3), (e) p= with £ odd and (d,m) = {8p - 1,4), (f) p = 2m-23e ± 1 with TO > 3 and d = 2p + 1, (g) p = 2m-2 ± 1 with m>hand (d, £) = (2p +1,0), (h) p = 2 ' " ~ ' + 1 wff/i TO > 5, > 1 and d = 2m~l +1, wrtere 3* | m - 2. (iii) n = 2and (p,m,£,d) e {(5,0, 3, 26), (5,5,1,49), (7, 6,1,97), (11,3,5,485),(17,7,2,577)}. 2. 77ze solutions to d2 - 2rn3epn = - 1 zfif/i m,£,n> 0 and d > 1 ynwsi1 /zaue ^ = 0 and satisfy one of the following (i) p > 5 and (ci, m, n) = (1,1,0), (n) p = 13and (d,m,n) e {(5,1,1), (239,1,4)}, (iii) p^l3,p= 1 (mod 4) and (m,n) € {(0,1), (1,1), (1, 2)}. Proof . 1) It follows from Corollary 1.4 of [3] that n e {0,1,2}. By considering the equation modulo 8 either rn = 0 or TO > 3, since 3 and 5 are quadratic non-residues modulo 8. If rn = 0 then the equation can be written ( d + l ) ( d - l ) = 3 V . It follows d + 1 and d — 1 are coprime and so d + l = 3 £ f d - 1 = 3* or < d - l = p n l d + l = p " . Chapter 4. Diophantine Lemmata 154 We wi l l consider such cases many times throughout the proofs in this section so to save space we wi l l collapse them into one single pair of equations: (d±l = 3e \d+l=pn. Subtracting the two equations gives ± 2 = 3 ^ - pn. If n = 0 then £ = 1 and d = 2; if n = 1 then p = 3e T 2. Finally, if n = 2 then p2 = 3e =F 2, but modulo 3 implies the sign must be negative, and modulo 4 implies £ is odd. For the rest of the proof we assume m > 3 and consider the cases n = 0, 1, and 2 separately. n = 0: Again, we could write the equation as d2 — 1 = 2m3e and factor the left-hand side to obtain the solutions in an elementary way, however, a direct application of Corollary 1.4 of [3] suffices to show the only solutions are (d,m,e) = {(3,3,0), (5,3,1), (7,4,1), (17,5,2)}. n — 1 or 2: The equation can be written as (d+ l ) ( d - 1) = 2 m 3 V where gcd (d + 1, d — 1) = 2, so one of the following three cases must hold: ( d ± l = 2 - 3 V \d±l = 2-pn \d±l = 2-3e < A „ o r \ (4-1) [dTl = 2m-1 {d+l=3e \dTl=pn Case 1 of (4.1): Subtracting the equations and dividing through by 2 gives 3epn - 2m~2 = ± 1 . (4.2) Suppose 3epn — 2m~2 = — 1, then consideration modulo 3 implies m is even. Writing m — 2 = 2k the equation can be written as 3epn = (2k + l)(2 f c - 1), thus 2k ± 1 = 3e 2kTl=pn Chapter 4. Diophantine Lemmata 155 The second equation has no solutions by Catalan's theorem. Thus (4.2) is 3epn — 2m~~2 = 1. Clearly m > 5 and consideration modulo 3 implies m odd. If n = 1 then we are in case (ii)(g) or (ii)(h) of the lemma. In this case 22(m-2) = 1 ( m o d 3e^ a n d S Q 3^-1 | m _ 2 ( s e 6 / f o r instance, [BeOO] Lemma 3.1). On the other hand, if n = 2 then £ is even (look modulo 3). The equation can then be written (3 £/ 2p) 2 - 2m~2 = 1 which has no solutions with m > 5 by Catalan's theorem. Case 2 of (4.1): Subtracting the equations and dividing through by 2 gives pn - 2m~23e = ± 1 . If n = 1 then p = 2 m ~ 2 3* ± 1 which is case (ii)(f) of the lemma. If n = 2 then p 2 = 2 m _ 2 3 * ± 1. A simple inspection modulo 4 reveals that the negative sign cannot occur, therefore p2 = 2™-23e + l . Moving the 1 to the left-hand side and factoring, or simply applying corollary 1.4 of [Be:2004], reveals the only solutions are (m,e,p) € {(5,1,5), (6,1,7), (7, 2,17)}. Case 3 of (4.1): Subtracting the equations and dividing through by 2 gives 3 < _ 2 T O - V = ± 1 . n — 1: If rn, = 3 then we are in (ii)(c) or (ii)(d) of the lemma. If m = 4 then we are in case (ii)(e) of the lemma. Now suppose m > 5, so the equation is 3* -2 T " - 2 p = ± l . (4.3) The right-hand side must be positive and £ must be even by considering the equation modulo 8. Letting £ = 2k we may write (3fc + l ) ( 3 f c - 1) = 2 m " 2 p . Since gcd (3fc + 1,3fc — 1) = 2 we are in one of the two cases: 3 f c ± l = 2 f 3 f c ± l = 2p or < 3 f c =F 1 = 2m-3p [3k =F 1 = 2m~3. Chapter 4. Diophantine Lemmata 156 The first case has no solutions, so we must be in the second case. Subtracting the two equations and dividing through by 2 gives p — 2 m _ 4 = ± 1 , and so p = 2rn-4 ± 1 with m > 6. Going back to (4.3) (recall the right-hand side must be positive) where we now know m > 6 we get 3^  = 1 (mod 16). Thus 4 | £. Finally, taking (4.3) modulo 5 results in 2m~2p = 0 (mod 5), hence p = 5. Thus the only solution with m > 5 is (p, n, m, £, d) = (5,1,6,4,161). n = 2: In this case the equation is 2 m - y = 3 < ± i . If m is even then the equation is (2(-m~2^2p)2 — 3e = ± 1 which has no solutions by Catalan's theorem. Therefore m is odd and the equation can be written as 2x2 - 3 £ = ± 1 , where x = 2^n~3^2p. Clearly there are no solutions, with x of the desired form, when £ = 0, so assume £ > 1. The left-hand side must be negative by considering the equation modulo 3: 2x2 - 3 e = - 1 . Certainly the only solutions with £ < 2 are (x,£) € {(1,1), (2,2)}. A s for £> 3, Nagell [54] has shown the only solution is (x, £) — (11, 5). Of these three solutions only one has x of the desired form, namely (x,£) = (11,5). Thus 2 m _ 2 p 2 = 3* ± 1 has only the solution (p, rn, £) = (11,3,5). This completes the proof (1). 2) Considering the equation d 2 - 2m3-'p n = -1 modulo 3 implies £ = 0. Therefore, the lemma follows from Lemma 3 in [37]. • Lemma 4.8 1. The solutions to d2 - 2rnpn = 3e with £, n > 0, rn > 2, and d > 1 satisfy one of the following: Chapter 4. Diophantine Lemmata 157 (i) n = 0cind (d, m,£) = (3,3,0), (5,4, 2), (ii) n = 1 and (a) p = ^ J i i with £/2 even and (d, m) = {3e>2 + 2,3), (b) P = 2f^±l wifft £/2 odd and (d, m) = {3il2 + 2,4), (c) p = 2m~2 ± 3ll2 with m>3andd = 2PT 3i'2, (m) n = 2, and (p, rn,£, d) = (5, 6,4,41). (iv) n = 3and (p, m, d) = (5,9, 2, 253). 2. 77-.<? solutions to d2 + 3e = 2rnpn with rn > 2, £ > 1 and n > 0 satisfy one of the following: (i) n — 0, m = 2 and £ = 1, (ii) n = 2,p = 3-^p-/ £ odd, m = 2, and d = 2p - 1, (m) ?i odd, n = 1 or Pmm(n) > 7, p = 1 (mod 3), m — 2 and £ odd. 3. The solutions to d2 + 2mpn = 3' rx>jf/i m > 2, <? > 1 and n > 0 satisfy one of the following: (i) n = 0and(m,£,d) E {(3,2,1), (5,4, 7)}, (ii) n = 1 and (a) p = 5 and (m, £, d) = (6, 8, 79), (b) p = 3 ^ p ± , £/2 odd, m = 3,andd = ±(4p - 3^ 2 ) , (c) p = 3 £ / 2 - 2 m - 2 , m>3,£ even, and d = ±(2p - 3ll2), (iii) n = 2 and (a) p = 5 and (m, £, d) = (3,6,23), (b) p = 7 and (m, d) = (7,8,17)., (c) p2 = 21^=1, £ / 2 odd, m = 3,andd = ±(4p2 - 3e'2), Proof . 1) The case when £ = 0 is done in [37], so we may assume £ > 1 henceforth. We break up the proof into a series of statements or "assertions". Chapter 4. Diophantine Lemmata 158 Assertion 1: If m > 6 and Pmax(n) > 7 then there are no solutions. Write the equation as 3*(± l ) g + 2 r V = d2 where q = Pmax(n) > 7 and y = pnlq. This is a particular form of the equation 3 V + 2my" = d2. Let Eaj,fi be the elliptic curve associated to a solution (x, y, d) = (a, b, c) of this equation as described in case (v) of [5]. It follows from Lemma 3.3 in [5] that Ea^c has conductor I 3» if m = 6 N(EaAc) = \ F [6p if m > 7. and corresponds to a cuspidal newform of weight 2, and level {3 if rn = 6 6 if m > 7. This gives a contradiction since there are no cuspidal newforms of weight 2 at these levels. Assertion 2: If m > 2 then SL is even and m ^ 2. This follows by inspection of the equation modulo 4 and 8. Assertion 3: If m > 2 and P m a x ( n ) > 5 then £ = 2. Furthermore, m > 8. By assertion 2 it follows that £ is even so we may write £ = 2k and factor the equation as (d + 3')(d-3*) = 2 > n Either one of the following two cases holds: jd±3k = 2 fd±3k = 2pn [dT 3 fc = 2 m " V \dT 3 fc = 2™- 1 . Suppose we are in the first case. Subtracting the two equations and dividing through by 2 gives 2m- V ± 3 f c = 1. Chapter 4. Diophantine Lemmata 159 Clearly there are no solutions when the sign is positive, and when the sign is negative there are no solutions by Theorem 1.2 of [3]. N o w suppose we are in the second case. Subtracting the two equations and dividing through by 2 gives x t + 3 f c ( l ) i = 2 m - 2 ( - l ) 3 , where t = Pma.x{n) and x = ±pn/t. By Theorem 1.5 in [6] it follows that k = 1, i.e. £ = 2. Additionally, one may easily check by hand that there are no solutions for m < 7. This proves assertion 3. Assertion 4: If m > 2 and P m a x (n) > 7 then there are no solutions. This is a direct consequence of assertions 1 and 3. Assertion 5: If m > 2 and Pmax(n) = 5 then there are no solutions. Recall £ = 2 by assertion 3 so the equation can be written as d 3 — 2™ x 5 = 9 where x = pn/b. Letting 0 < a < 4 be the residue of m modulo 5 we may write (2 2 m d) 2 = ( 2 ^ x ) 5 + 2 4"9. If a > 3 we may scale through by 2 ~ 1 0 b where 4a = 106 + r, and 0 < r < 9 is even, to get 2 2 ™ d \ 2 f2m^x\5 25b l I 22b 2r9. The point is that solutions to our original equation correspond to rational points of a particular form on a genus 2 hyperelliptic curve: Y2 = X5 + 22s9, 0 < s < 4. We show in Chapter 5 (see Theorem 5.1) that the rational points on these curves all have X £ {—2, 0,4}. Thus, there are no solutions to our original equation. This proves assertion 5. Assertion 6: If m > 2 and Pma,x(n) = 3 then the only solution is (p, n, ra, £) = (5,3,9,2). In this case the equation can be written as d 2 = 2 m x 3 + 3^  where x = pn/3. Let 0 < a < 2 be the residue of m modulo 3 and 0 < 6 < 5 be the residue of £ modulo 6. Recall £ is even, so 6 must be even also. Making the change of variables v _ 2("'+ 2«)/ 4x A n _ 2Qd X ~ 3 (*-6)/3 a n d D ~ Chapter 4. Diophantine Lemmata 160 the equation becomes D2 = X3 + 22aSb where 2a, b € {0,2,4}. We are only interested in solutions where X and D are of the form above, in particular (X, D) is to be a {3, oo}-integral solution. The table in appendix B lists all the 5-integral points on these elliptic curves. The only solution to this equation of the desired form occurs when (a, b) — (0, 2) and it is (X, D) = (40,253). This pulls back to the solution (p, n, m, I, d) =(5,3,9, 2, 253) of the original equation. This proves assertion 6. Assertion 7: If m > 2 and n = 2a where a > 1 then p = 5 and (m, I, d) = (6,4,41). By considering the equation modulo 3 it follows that m is even (2 | n means p" = 1 (mod 3)). Write m — 2s and n — 2t so equation becomes {d + 2spt){d-2spt) = 3e. It follows that d + 2sp* = 3e d-2spt = 1, and subtracting these equations gives 2 s + y = 3' - 1 . By assertion 2 we know I is even so we may write I — 2k and factor the right-hand side: 2S+V = (3fc + l)(3 f c - 1). It follows that 3 f e + l = 2V . f 3 f c ± l = 2p i or < 3 f c - l = 2 [ 3 f c T l = 2 s . The first case only has the solution (k,s,t) = (1,2,0) which implies n = 0, a contradiction. In the second case, it follows from Catalan and the sec-ond equation that (s, A;) € {(1,1), (2,1), (3,2)}, hence p = 5 and (rn,£,d) = (6,4,41). This proves assertion 7. Chapter 4. Diophantine Lemmata 161 Assertion 8: If m > 2 and n = 1, then the only solutions are the ones stated in the lemma. Since £ is even (assertion 2) the equation can be written as (d + 3 f c ) (d-3 f c ) = 2mp, where £ = 2k. It follows that fd±3k = 2p (d-3k = 2 \ c i T 3 f c = 2 m - 1 \ c i + 3 fe = 2 m - V -Eliminating d in the first case gives ± 3 f c = p - 2rn~2. Thus p = 27"~2 ± 3 f c. In the second case eliminating d gives — 3k = 1 — 2m~2p, that is 2 m ' - 2 p = 3fc + l . Considering this equation modulo 8 implies m < 4, so m G {3,4}. This proves assertion 8. Finally, if n = 0 then the equation is d2 = 2m + 3e which only has the solution (m, £, d) — (4, 2, 5) by Lemma 5 in [37]. This completes the proof of (!)• 2) Considering the equation modulo 4 implies £ is odd, and considering the equation modulo 8 implies rn = 2. If n is even then we may write n — 2s and the equation becomes (d + 2 p s ) ( d - 2p2) = -3e. As we've done many times before in these arguments we eliminate d: 4ps = 3e + l. It follows from Theorem 1.2 in [3] that s € {0,1,2}. If s = 0 or 1 then we are in case (i) or (ii) of the lemma, respectively. So assume s = 2. The equation can be factored as ( 2 p + l ) ( 2 p - l ) = 3' and since we are to have gcd (2p + 1,2p — 1) = 1 it follows 2p — 1 = 1 and hence there are no solutions. N o w assume n is odd. Considering the equation Chapter 4. Diophantine Lemmata 162 modulo 3 implies p = 1 (mod 3). It suffices to show 3 and 5 cannot divide n to complete the proof. If 3 divides n then the equation can be written as d2 = 4x3 - 3e, where x = pnl3. Changing variables we may write this as — ^ 3 - 2 4 • 3" where £ = 6t + a, 0 < a < 5 is odd. By the tables in Appendix B there are no rational points on these elliptic curves of the desired form. If 5 divides n then a similar change of variables leads us to the equation 16d\2 _ [4x\5 _ 8 a where x = pnlb and a G {1,3, 5, 7, 9}. Again, there are no solutions of the de-sired form to these hyperelliptic curves as shown in Chapter 5. This completes the proof of (2). 3) If n = 0 then the equation is d2 = 3e - 2m. Considering the equation modulo 3 it follows rn. must be odd. It is easy to check that the only solutions with £ < 2 are (m,£,d) G {(1,1,1), (3,2,1)}. As for the case when £ > 3 it follows from [19] that the only solutions are (m,*,d)G {(1,3,5), (5,4, 7)}. From now on we assume n > 1. It follows from the equation that i > 3 since the left-hand side is d 2 + 2rnpn > 1 + 10 = 11. Furthermore, since m > 2 then considering the equation modulo 4 implies £ even, and further considerations modulo 8 imply m > 3. Assertion: If m > 2 and Pmax(n) > 3 then the equation d2 + 2mpn = 3e has no solutions. Chapter 4. Diophantine Lemmata 163 Since £ is even we may factor the equation (d-3k)(d + 3k) = - 2 " y \ where £ = 2k. This leads to the two cases fd±3k = ±2 fd±3k = ±2pn \d -f 3k = +2m~xpn \d =F 3 fc = -f 2 m _ 1 . Eliminating d in the first case gives 3fc _ 2m-2pn = 1 ( and by Theorem 1.2 in [3] there are no solutions, since Pmax(n) > 3. Eliminat-ing d in the second case gives 3k - p n = 2m~2. If Pma.x(n) > 5 then by Theorem 1.5 in [6] it follows that k = 1, i.e. ^ = 2, which contradicts £ > 3. Now assume PmBX(n) = 3. We return to the original equation and write it as d2 = 2mx3 + 3e, where x = p r ' / 3 . The solutions to this equation correspond to the rational points on the elliptic curves Y2 = X3 + 2 2 a 3 6 of the form ya+spn/'A 2ad X = — - 2 t — a n d Y = W where m = 3s + a, £ = 6t + b and 2a, b e {0, 2,4}. From the tables in Appendix B we see there are no such points. This proves the assertion. Finally, we consider the solutions with n = 2a for a > 0. Factoring the equation, as we did in the proof of the assertion above, we are in one of two cases: 3fc _ 2 ™ - y - = i o r 3fc _ pn = 2 T "- 2 , (4.4) Chapter 4. Diophantine Lemmata 164 where k = 1/2. In the first case of (4.4), Theorem 1.2 of [3] implies n G {1,2}. If m = 3 then we are in case (ii)(b) or (ii)(b) of the lemma. So assume m > 4. Considering the equation modulo 4 implies k is even, i.e. 4 | so we may factor the equation as (3 s + l ) ( 3 s - l ) = 2 m - V , where k — 2s, i.e s = £/4. We are in one of two cases: f 3 " + l = 2 m _ 3 p n f 3 s ± l = 2 m " 3 [ 3 s - 1 = 2 \ 3 s = F l = 2p n . The first case clearly has no solutions (under our assumptions on p and n). Eliminating 3 s in the second case gives ± 1 = 2m~4 - p n , from which it follows from Catalan's Theorem that n = 1, and so p = 2m~4 ± 1 . It follows from assumptions on p that rn > 6. Therefore, the equation in (4.4) becomes 3 f c - 2m~2p= 1, where m > 6 and 4 | k (look modulo 16). Considering the equation modulo 5 implies 2m~2p = 0 (mod 5), thus p = 5 and we are in case (ii)(c) of the lemma. Now suppose we are in the second case of (4.4). If n = 1 then we are in (ii)(c) of the lemma. So assume n = 2" with a > 1. If m = 3 then the equation is 3 f c — pn = 2 which only has the solution (p, n, k) = (5, 2, 3) in even n (see [19]). So assume rn > 4. Considering the equation modulo 8 implies m > 5 and k is even, i.e. 4 | £. Factor the equation as ( 3 ' + p s ) ( 3 J - p s ) = 2 m " 2 , where k = 2t and n = 2s. It follows that J 3 4 +p* = 2 m ~ 3 \ 3 * - p 5 = 2, and so by eliminating ps we get 3 f — 2 m _ 4 = 1. By Catalan's Theorem (t, m) G {(1,5), (2,7)}, and solving for p, n,m and £ we get (p, n, m, *?) = (7, 2, 7,8). This completes the proof of (3). • Chapter 4. Diophantine Lemmata 165 Lemma 4.9 1. The solutions to d2 - 2rn3e = pn with m > 2, £ > 0 and n > 1 satisfy one of the following: (i) n — 2 and (a) p = 2 m " 2 3 f - 1, m > 3, and d = p + 2, (b) p = 3e - 2 m - 2 , m>3,andd = 2m~l + p, (c) p = 2m~2 -3e,m> 5, and d = 2rn~1 - p, (ii) n = 3,and(p,m,e,d) G {(7,1, 2,19), (13, 2,1,47), (17, 7,0,71), (19,1,9, 215), (73,15,2,827)}, (iii) n = 4, and (p,m,£,d) <E {(5,3,3, 29), (7, 7,4,13)}, (iv) n = 6, and (p,m,£, d) — (5, 9,1,131), (v) P„n„(n) >7,m<5and£>l, (vi) n = 1. 2. TTze solutions to d2 + 2m3e = p" with m > 1, £ > 0 and n > 1 satisfy one of the following: (i) n = 2 and (a) p = 2m~23e + l,m>3,andd = p-2, (b) p = 3e + 2 m - 2 , m > 3, and d = p - 2 • 3e, (ii) n = 3and(p,m,£,d) G {(5,2,0,11), (7,1,3,17), (13, 2,5, 35), (73,4,7,595), (97,3,4,955), (193,4,4, 2681), (1153, 5, 5, 39151)}, (iii) n = 4and{p,m,£,d) e {(5,5,1,23), (5,6, 2, 7), (7,6,1,47), (17,7,2,287)}, (iv) Pmm{n) > 7 andm < 5, (y) n = 1. 3. The solutions to d2 + pn = 2rn3e with rn > 1, £ > 0 and n > 1 satisfy one of the following: Chapter 4. Diophantine Lemmata 166 (i) n = 3and (p,m,£,d) G {(23,12,1,11), (5,1, 5,19), (7,9,0,13), (23,3,7,73), (47, 5,11,2359)}, (ii) Pm\n(ri) > 7 and m < 5, (iii) n = 1. Proof. 1) The case when £ = 0 is treated in [37], so we may assume £ > 1. We break the proof up into the following sequence of "assertions". Assertion 1: If m > 6 and Pmax(n) > 7 then there are no solutions. Write the equation as 2™tf(l)1 + y<I = d2 where q = Pmax(n) > 7 and y = pnlq. This is a particular form of the equation 2 m 3 V + y" = d2. Let E,lAc be the elliptic curve associated to a solution (x, y, d) = (a, b, c) of this equation as described in case (v) of [5]. It follows form Lemma 3.3 in [5] that Ea,b,c h a s conductor f 3» if m = 6 N(EaAc) = \ " [6p i f rn > 7. and corresponds to a cuspidal newform of weight 2, and level {3 if rn = 6 6 if m > 7. This gives a contradiction since there are no cuspidal newforms of weight 2 at these levels. This proves assertion 1. Assertion 2: The only solutions with 3 | n are the ones with n G {3,6} as stated in the lemma. Applying the change of variables pn/3 j X = n and D 22s 32* 2 3 s 3 3 i ' Chapter 4. Diophantine Lemmata 167 to the equation d2 = pn + 2 m 3* where m = 6s + a and £ = 6t + b with 0 < a, b < 5, gives the equation The rational points of the desired form on these elliptic curves can be found by inspection of the tables in Appendix B and are as follows: (X, D, a, b) = {(17/4, 71/8,1,0), (7,19,1, 2), (19/9, 215/27,1,3), (13,47,2,1), (25/4,131/8, 3,1),(73/16, 827/64,3,2)}. (p, n, TO, e, d) = {(17, 3, 7,0, 71), (7,3,1, 2,19), (19, 3,1,9, 215), (13, 3, 2,1,47), (5,6, 9,1,131), (73,3,15, 2, 827)}. This proves assertion 2. Assertion 3: There are no solutions with 5 | n. Solutions to d2 = pn + 2m3* with 5 | n correspond to rational points on D2 = Xb + 2"3fe, 0 < a, b < 9, with x coordinate of the form We show in Chapter 5 that no such points exist. Assertion 4: The only solutions with n even, 3 \ n, are the ones with n e {2,4} as stated in the lemma. Considering the equation d2 = 2m3e + pn modulo 8 implies m > 3. Since n is even we may factor the equation as D2 = X3 + 2 a 3 b . These pull back to the following solutions of the original equation: X = 2 2 . 5 3 2 t • (d + pk)(d - Pk) = 2m3' where n — 2k. It follows that we are in one of two cases Chapter 4. Diophantine Lemmata 168 Eliminating d in the first case gives 2m~ 23 e — pk = 1, in which it follows from [3] that k = 1 or 2. However, considering this equation modulo 3 implies k is odd, therefore k = 1, and so p = 2 m _ 2 3 , ? — 1. In the second case, eliminating d gives 3 ezfP k = 2m- 2. (4.5) If Pmax(k) > 5 then it follows from Theorem 1.5 in [6] that £ = 1. Recall 5 j n by assertion 3, so we may assume Pmax(k) > 7. However, assertion 1 implies in this case that m < 5. It is now easy to see that there are no solutions with P m a 3 1 (n) > 5. So we must have that k = 2 a , a e N . If k = 1, i.e. n = 2, then we are are in case (i) of the lemma. So suppose k = 2 a > 1. If the equation in (4.5) is 3^  + pk = 2m~ 2 then observe, by local considerations, that m > 5 is even and £ is odd. Factoring the equation, as we've done many times over, leads us to see there are no solutions. So equation (4.5) must be 3 e — pk = 2m~ 2. If m = 3 then the equation is 3 e — pk = 2 which has only the solution (p, £, k) = (5, 3, 2) (see [19]). So assume m > 4. Considering the equation modulo 4 implies £ is even and so we may factor as usual: ( 3 ' / 2 + p f c / 2 ) ( 3 ' / 2 - p f c / 2 ) = 2 m - 2 ) from which it follows that 3 /^2 _|_ pk/2 _ 2'"-3 3 * / 2 _ p f e / 2 = 2 ; and so, eliminating p f e / 2 we get 3 ^ 2 — 2 m ~ 4 = 1. By Catalan's Theorem (m,£) £ {(5,2). (7 ; 4)}, only one of which pulls back to a solution of the original equa-tion; (p,n,mf£-d) = (7,4,7,4,113). This proves assertion 4. Finally, if n is odd it clearly follows from the equation d? — 2 rn3 e = pn that p = 1 (mod 12). 2) Assertion 1 in the proof of (1) above also holds for this case, that is, there are no solutions if both m > 6 and P m a x (n) > 7. The case when £ = 0 is Chapter 4. Diophantine Lemmata 169 treated in [37] so we may assume £ > 1. The solutions with 3 | n correspond to points on the elliptic curves y2 = x3 - 2a2>b of the form pn/3 x = 22*32*' where 0 < a,b < 5, m = 6s + a and £ = 6t + b. It follows from the tables in Appendix B that the only such points have n = 1 and correspond to the seven solutions in part (ii) of the lemma. The solutions with 5 | n correspond to points on the hyperelliptic curves y2 = x5 - 2a3b of the form x = z 2^ 1' where 0 < a, b < 9, m = 10s + a and £ = lOt + b. It follows from Chapter 5 that there are no rational points on these curves of this form. We break the rest of the proof up into two cases depending on the parity of n. Assertion 5: The only solutions with n even are the ones with n G {2,4} as stated in the lemma. Since n is even we may factor the equation as {d + pk)(d-pk) = -2rn3e, where n = 2k, it follows that we are in one of the two cases d-pk = -2 \d±pk = 2-3e or < d + pk = 2m~13e, \d + pk = 2m~K Eliminating d in the first case gives pk — 2m~23i = 1 which has no solutions for k > 3 by [3], thus k = 1 or 2. If k = 1 then p = 2m~23e + 1 and we are in case (ii)(a) of the lemma. If k = 2 then we may factor the equation as (p+l)(p-l) = 2m-23e, Chapter 4. Diophantine Lemmata 170 where, as usual, eliminating p gives ± 1 = 3e - 2™ 4 , so (m,£) e {(5,1), (7,2), (6,1)} by Catalan. Thus, (p,n,m,£) e {(5,4,5,1), (7,4,6,1), (17,4, 7,2)}. In the second case, eliminating d gives / = 3* + 2 m " 2 . We may assume 3,5 { fe as these cases were already considered above. Sup-pose Pmax(k) > 7, then from Theorem 1.5 in [6] it follows that fi = 1, and so m > 7. However, we have already seen that there are no solutions with -Pmax(^) > 7 and m > 7. Therefore k, thus n, must be a power of 2. If = 1 then we are in (i)(b) of the lemma. If fe = 2 a > 1 then considering the equation pk = 3e + 2 m ~ 2 modulo 3 implies rn > 4 and even, and modulo 4 implies £ is even. As usual, factoring and applying Catalan gives (p, n, m, £) = (5,4, 6, 2). This proves assertion 5. Finally, if n is odd and (n, 15) = 1 then either n = 1 or Pmax(n) > 7 (and m < 5 as we've argued using [5] many times before). This proves (2). 3) As we've seen in (1) and (2) there are no solutions with both m > 6 and P m a x ( n ) > 7. The solutions with 3 | n correspond to rational points on the elliptic curves y2 = x3 + 2 a 3 b of the form X = 22s32t' and these can be determined by consulting the tables in Appendix B. Simi-larly, the solutions with 5 | n correspond to rational points on the hyperelliptic curves y2 = x5 + 2a3b of the form X = 2 2 s 3 2 T ' of which there are none by Theorem 5.1. The solutions with n even must have m = 1 (looking modulo 3) and £ = 0 (looking modulo 4), thus there are no solutions. This completes the proof of (3). • Chapter 4. Diophantine Lemmata 171 Lemma 4.10 1. The solutions to d2 - 2m = 3 V with m > 1,£ > 0 and n > 1 satisfy one of the following: (i) n = 1 and (a) p = 2 m / 2 ^ + \ £>i,d = 2 - / 2 + 1, (W P = 3* ± 2 m / 2 + 1 , * > 1, d = p ± 2 m / 2 , (c) £ = 0,m odd. (ii) n = 2 and W (p,m,£,d) € {(5,6, 2,17), (7,8,4,65)}, p = 2 m - 2 - 1, TO > h,£ = 0 and d = p + 2, (iii) n = 3and (p,m,£,d) = (17,7,0,71). 2. The solutions to d2 _ 2m = _sepn with m, £ > 0 and n > 1 satisfy one of the following: (i) n = 1 and («) (p,m,£,d) e {(5,6,1,7), (7,10, 2, 31)}, (W p = 2 m / 2 + 1 - 3*, m > 4 even, £ > 1, and d = ±(2m/2 - p), (c) £ = 0 and m > 5 odd. (nj n = 3 and (p,m,£,d) <E {(5,12,1,61), (7,9, 0,13)}. 3. 77ze solutions to d2 + 2m = 3 V rxiffn m > 1, £ > 0 and n > 1 satisfy one of the following: (i) n = 1, (ii) n = 2 and (a) {p,m,£,d) = (5,4,0,3), (b) p = ^"V1 i m l m > 3 odd, and d = 3 £/ 2p - 2, Chapter 4. Diophantine Lemmata 172 (iii) n = 3 and {p,m,i,d) e {(5,2,0,11), (11, 9,1,59), (17,15,2,107), (19, 7,1,143), (67, 5,3, 8549), (73,2, 9,1871)}, (iv) -Pmax(n) > 7 and me {1, 3, 5}. Proof . 1) The case when £ = 0 is treated in [37] so we only need to consider £ > 1. In this case it follows that m is even. As in the proofs of the other lemmata it follows from [5] there are no solutions with both Pmax(n) > 7 and m > 6. Also, there are no solutions in the case when 3 | n (resp. 5 | n) since solutions would correspond to {2, oo}-integral points on elliptic curves (resp. hyperelliptic curves) of a particular form of which there are none by Appendix B (resp. Theorem 5.1). Assertion 1: The solutions with Pmax(n) > 7 must have £ = 1. Since m is even we can factor the equation as (d+2k)(d- 2k) = 3epn, where m = 2k. One of the following cases must hold: jd + 2k = 3epn jd±2k = 3e \ d - 2 f e = l , \d^2k=pn. Eliminating d in the first case gives 3epn — 2k+l = 1 which has no solutions with P m a x ( " ) > 7 by Theorem 1.2 in [3]. Eliminating d in the second case gives 3 e - p n = ± 2 k + \ (4.6) from which £ = 1 follows by Theorem 1.5 of [6]. This proves assertion 1. Assertion 2: There are no solutions with Pm a x (??-.) ^  7. It follows from remarks above that such solutions must have m 6 {2,4} and £ = 1. Furthermore, it follows from (4.6) that pn = 2m/2+l + 3 > and so pn — 7 or 11, which contradicts Pmax(n) > 7. This proves assertion 2. Assertion 3: The solutions with n even are the ones with n = 2 as stated in the lemma . Chapter 4. Diophantine Lemmata 173 Considering the equation modulo 4 implies £ is even, so we may factor the equation as (d + 3 ^ V l / 2 ) ( d - 3 ^ V / 2 ) = 2 m , from which it follows that ( d ± 3 f / y / 2 = 2m-1 | d q F 3 ' / y / 2 = 2. Eliminating d give "&zl2pnl2 — 2m~2 — 1 and so n e {2,4} by Theorem 1.2 of [3]. Furthermore, since m is even we may factor the right-hand side of this equation as from which it follows that 2 ( m - 2 ) / 2 ± j = 3^/2 2 ( m - 2 ) / 2 -j- x = p « / 2 ^ It follows from the first equation and Catalan that (m,£) € {(4,2), (6,2), (8,4)}, and plugging these into the second equation gives p — 5 or 7. This proves assertion 3. Finally, we need to consider the case when n = 1. Following the proof in assertion 1, where the fact that m even was used to factor the equation, we obtain the two cases: 3<p - 2 m / 2 + 1 = 1 or 3 ' - p = ± 2 m / 2 + 1 . This completes the proof of (1). 2) Clearly there are no solutions with m < 3. The case when £ = 0 is treated in [37] so we only need to consider £ > 1. In this case it follows that m is even and m > 4. Furthermore, n is odd. As in the proofs of the other lemmas it follows from [5] there are no solutions with both Pmax(n) > 7 and m > 6. Also, the only solution in the case when 3 | n is (p, n, m, £, d) = (5,3,12,1,61) since solutions would correspond to {2, oo}-integral points on elliptic curves which we can determine with the use of Appendix B. Chapter 4. Diophantine Lemmata 174 Assertion 4: The solutions with Pmax(n) > 5 must have £ = 1. Furthermore, there are no solutions with PmatX(n) > 7. This follows by a similar argument as in assertions 1 and 2 of part (1) above. If 5 | n then solutions to d 2 - 2 m = —3pn would correspond to {2,oo}-integral points on hyperelliptic curves y2 — x5 + 2 2 h 3 4 with x of the form x = (—3p™/5)/(22 f c) and by Theorem 5.1 there are no such points. The only case left to consider is n = 1. Since m is even we may factor the original equation as ( d + 2 m / 2 ) ( d - 2 m / 2 ) = -3ep, and so one of the following cases must hold: d + 2 m / 2 = tfpn ( d ± 2m/2 = ± 3 e or < d - 2™/2 = _ 1 ( [ d + 2 m / 2 = Tpn_ Eliminating d in the first case gives 3ep — 2 m / 2 + 1 — 1 where m/2 + 1 must be even. By factoring the left-hand side we see the only solutions are (p, m, £) e {(5,6,1), (7,10, 2)}. Eliminating d in the second case gives p = 2rn/2+l - 3e. This completes the proof of (2). 3) For the case when £ — 0 see [37]. As usual, we can apply [5] to show there are no solutions with both Pmax(n) > 7 and rn > 6. Also, local consider-ations at 3 imply m is odd. The case when 3 | n (respectively 5 | n) corresponds to finding {2, oo}-integral points on elliptic curves (respectively hyperelliptic curves) so we can use the tables in Appendix B (respectively Theorem 5.1) to show the only solutions are the ones as stated in the lemma. In the case when 2 | n we must have £ is even also and so we may factor the equation. A result of Bennett [3] shows n = 2 or 4, and a result of Cohn [19] shows n ^ 4. This completes the proof of (3). • Lemma 4.11 1. The solutions to 3d2 — 2 m = pn with m > 0 and n > 1 satisfy one of the following: Chapter 4. Diophantine Lemmata 175 (i) n = 3 and (p, m, d) = (11,8,23), (ii) Pmax(n) > 7 or n even, and m = 1, f/n) n = 1. 2. The solutions to 3d2 - 2m = —pn with m > 0 and n > 1 satisfy one of the following: (i) n = 3 and (p, m, d) = (5, 7,1), (ii) ra = 1. 3. The solutions to 3d2 + 2 m = pn with m > 0 and n > 1 satisfy one of the following: (i) n = 3and (jp, rn, d) = (11,3, 21), (ii) Pmax (n) > 7 and m = 1, (iii) n = 2 and m G {0,1}, (iv) n = 1. 4. TTie solutions to 3d2 - 2mpn = - 1 m > 0 and n > 1 sah's/i/ m e {0, 2} and n G {1,2}. 5. The solutions to 3d2 — 2mpn = 1 with m > 0 and n > 1 satisfy m G {0,1} and n G {1,2}. Proof. In the first three cases there are no solutions with Pmax(n) > 7 and m > 2 by Theorem 1.2 of [5]. 1) First consider the case when m = 0, from which it follows that n is odd. There are no solutions with n > 4 by Theorem 1.1 of [5], and there are no solutions with n = 3 as shown in [18] (alternatively, this case corresponds to finding integral points on the Mordell curve y2 = x3 + 27). Thus n = 1 in this case. In what follows we may assume m > 1. If 3 | n then the equation describes an elliptic curve (whose minimal model is of the form y2 = x3 + 2 a 3 3 , 0 < a < 5) and solutions correspond to {2, oo}-integral points on the elliptic curve of a certain form. Using the ta-bles in Appendix B we conclude that (p, m, d) = (11,8, 23) is the only solution in this case. Similarly, if 5 | ra then the equation describes a hyperelliptic curve Chapter 4. Diophantine Lemmata 176 (whose minimal model is of the form y2 = x 5 + 2 a 3 3 , 0 < a < 9) and so we may apply results of Chapter 5 to conclude there are no solutions. Finally, if n is even then considering the equations modulo 4 implies m = 1. The proofs of (2) and (3) are very similar. 4) Considering the equation modulo 4 and 8 implies m e {0,2}. Suppose m = 0, then Theorem 1.1 of [5] implies P m a x ( ^ ) < 3. If 3 | ra then the equation can be written as (9d)2 = (3p n / 3 ) 3 - 27 which, by the tables in Appendix B, has no solutions of the desired form. If 4 | n then the equation can be written as "id2 = x4 — 1 which has no solutions by [22]. Therefore, ra = 1,2. As for the case m = 2, Theorem 1.2 of [5] implies Pmax(n) e {2,3,5}. If 5 | n then the equation can be written as (2 4 3 3 d) 2 = ( 2 2 3 p n / 5 ) 5 - 2 8 3 5 which has no solutions by Theorem 5.1. If 3 | ra then the equation can be written as (2 23 2d) 2 = (2 • 3 p n / 3 ) 3 - 2 4 3 3 which has no solutions by the tables in Appendix B. If 4 | ra then by Theorem 1.2 of [7] there are no solutions. Therefore, n = 1, 2. 5) Considering the equations modulo 4 implies ra?, € {0,1}. Suppose m = 0 then Theorem 1.1 of [5] implies -Pmax(ra) < 3, and considering the equation modulo 3 implies n odd. Thus, n = 1 or 3 | n. If 3 | ra then the equation can be written as (9d)2 = (3p"/3)3-|-27, and since y2 = x 3 + 27 is a rank 0 elliptic curve with only one nontrivial point (x, y) = (—3,0), the equation has no solutions of the desired form. Therefore n = 1. As for the case m = 1, Theorem 1.2 of [5] implies Pmax(n) 6 {2,3,5}. If 5 | ra then the equation can be written as (2 43 3d) 2 = ( 2 2 3p n / 5 ) 5 + 2 8 3 5 which has no solutions by Theorem 5.1. If 3 | ra then the equation can be written as (2 2 3 2 d) 2 = (2 • 3j tW 3 ) 3 - 2 4 3 3 which has no solutions by the tables in Appendix B. If 4 | ra then there are no solutions since the equation 3d 2 = 2x 4 + 1 has only the trivial solution d = x = 1 (see [47]). This completes the proof of the lemma. • Chapter 5 Rational points on y2 = x5 ± 2a3f3 In this chapter we are concerned with finding all the rational points on the genus 2 hyperelliptic curves y2 = xb ± 2a313 where a and j3 are integers. The results obtained here were used in the proofs of the Diophantine lemmata of Chapter 4. 5.1 I n t r o d u c t i o n a n d S t a t e m e n t o f R e s u l t s A celebrated theorem of Faltings states that a curve C of genus > 2 has only finitely many rational points: #C(K) < oo for K a number field. For fixed a and (3 the curve C : y2 = x5 ± 2a3^ is of genus 2 and so has finitely many rational points. We wish to determine all such points, i.e. C(Q). It suffices to only consider the cases 0 < a., (3 < 9 since two curves y2 = x5 + A and y2 = x5 + B are Q-isomorphic if A/B is a tenth power. Unfortunately, there is one curve we cannot say anything about, namely y2 = x5 — 2 3 3 9 . We believe there are no (finite) rational points on this curve but are unable to prove this at this time. Of course, it can be shown that there are no integral points on it (see [71]). Keeping this curve aside for the time being we wi l l prove the following theorem. T h e o r e m 5.1 Let a and (3 be integers such that 0 < a, (3 < 9, and e € {±1}. Suppose (a,P,e) ^ (3,9,-1). If C : y2 = xb + e2°3^ contains a (finite) rational point (x, y) then a, (3, e, x, y are one of those listed in Table 5.1. 177 Chapter 5. Rational points on y2 = x5 ± 2Q3'3 178 c c a P e C(Q)\ {oo} a P e C(Q) \{oo} 0 0 1 ( - 1 , 0 ) , ( 0 , ± 1 ) 6 6 1 (0,±216) 0 1 1 ( l . ± 2 ) 6 8 1 (0, ±648) 0 2 1 (0,±3) 8 0 1 (0,±16) 0 4 1 (0, ±9) , (-2, ±7) , (3, ±18)) 8 2 1 (0,±48) 0 5 1 (-3,0) 8 4 1 (0,±144) 0 6 1 (0,±27) 8-, 6 1 (0,±432) 0 8 1 (0, ±81), (18, ±1377) 8 8 1 (0, ±1296) 1 0 1 ( " 1 , ± 1 ) 0 0 -1 (1,0) 1 5 1 (3, ±27) 0 5 - 1 (3,0) 1 8 1 (7, ±173) 1 2 - 1 (3, ±15) 2 0 1 (0 ,±2) , (2 ,±6) 1 4 - 1 (3, ±9) 2 2 1 ( 0 , ± 6 ) , ( - 2 , ± 2 ) 3 8 - 1 (9, ±81) 2 4 1 (0, ±18), (-3, ±9) , (6, ±90) 4 0 - 1 (2, ±4) 2 5 1 (-3, ±27) 4 2 - 1 (10, ±316) 2 6 1 (0,±54) 5 0 - 1 (2,0), (6, ±88) 2 8 1 (0,±162) 5 4 - 1 (6, ±72) 3 0 1 (1,±3) 5 5 - 1 (6,0) 3 1 1 (1,±5) 5 6 -1 (9, ±189) 4 0 1 (0,±4) 5 8 - 1 (18, ±1296) 4 1 1 ( l , ± 7 ) , ( - 2 , ± 4 ) 7 4 - 1 (33, ±6255) 4 2 1 (0,±12) 8 1 - 1 (4, ±16) 4 3 1 (-2, ±20) 8 5 - 1 (12, ±432) 4 4 1 (0,±36) 4 5 1 (6, ±108), (-2, ±4) 4 6 1 (0,±108) 4 8 1 (1, ±324), (9, ±405) 5 0 1 (-2,0), (2, ±8) 5 1 1 (-2, ±8) 5 2 1 (1, ±17), (-2, ±16) 5 5 1 (-6,0), (-2, ±88) 6 0 1 (0,±8) 6 2 1 (0, ±24), (4, ±40) 6 4 1 (0, ±72), (12, ±504) Table 5.1: Theorem 5.1: A l l points on C : y2 = x5 ± 2°3 / 5 Chapter 5. Rational points on y2 = x5 ± 2Q3 / 3 179 5.2 B a s i c T h e o r y o f J a c o b i a n s o f C u r v e s In this section we outline the basic theory of Jacobians of curves with a focus on computing in the Jacobian using M A G M A . The reader we have in mind is one who is familiar with the theory of, and computing with, elliptic curves and wants to start computing in Jacobians. We end this section with a dis-cussion of Chabauty's technique for bounding the rational points on genus 2 curves and using its implementation in M A G M A . The reader already familiar with this material can skip directly to Section 5.3. By a hyperelliptic curve we shall mean a curve C (with a model) of the form y2 = f(x), where / (x) is a polynomial of degree 2g+1, with distinct roots, and with coefficients in a field k of characteristic ^ 2. Here g is a positive integer, the genus of the curve C. We wi l l mostly be interested in the case of genus 2 curves over number fields k (especially k = Q), however in stating the basic theory we won't restrict ourselves to genus 2 just yet. When studying hyperelliptic curves one is chiefly concerned with deter-mining the set of fc-rational points on C, denoted by C(k). This is the set of points (x, y) on C with x, y G k. A celebrated theorem of Faltings says that if g > 2 and A; is a number field, then this set is finite. Thus, one can hope to write down the set C(k) explicitly. Faltings theorem clearly does not hold for genus 1 curves (elliptic curves). For example, it is well known that the elliptic curve E : y2 = x 3 + x + 1 has infinitely many Q-rational points. Some examples of rational points on E are: (0, ±1) , (1/4, ±9 /8 ) , (72, ±611), (-287/1296, ±40879/46656), (43992/82369, ±30699397/23639903). In the elliptic curve case the rational points on C form a finitely generated abelian group, so one is interested in determining the group structure of C(k), called the Mordell-Weil group. In our example above, E(Q) ~ Z, with genera-tor P = (0,1), and the points listed above are P, 2P, SP, 4P and 5P. For curves C of genus > 2 the set C(k) does not form a group. However, C(k) can be embedded into a finitely generated abelian group J(k) called the Jacobian oiC/k (also called the Mordell-Weil group of C). We wi l l briefly sketch how the Jacobian is constructed from C and state some of the basic facts that we wi l l use. Chapters. Rational points on y2 = x 5 ± 2°3^ 180 5.2.1 Basic Setup Let k be the algebraic closure of k. By a point on C we mean a pair (x, y) of elements in k satisfying y2 = f(x) or one other element; the point at infinity, denoted oo . Let C(k) denote the set of all points on C. We can define an action of Aut(A;/fc) on C(k) as follows: if a G Aut(fc//c) and P G C(k) then Pn = (xa, y"). The set of fc-rational points on C can be defined as C(fc) = { P 6 C(fe) :Pa = P for all a G Aut(fc/fc)}. 5.2.2 Divisors The divisor group of C is the free abelian group generated by the points of C. Thus a divisor D of C is a finite formal sum of the form Pec(fc) where the rnp are integers (only finitely many of which are non-zero). The degree of D is deg(D) — 2~2pmP- li mp > 0 for all P then we write D > 0 and call £> an effective divisor. The divisors of degree 0 form a subgroup of Div(C) which we denote by Div°(C) = {D G Div(C) : deg (D) = 0}. We can define an action of Aut(fc/fc) on Div(C) in the obvious way p We say that D is defined over A: if Da = D for all a G Aut(k/k). Note that if D = 2~2,Pmp(P) then to say D is defined over k does not mean all P G C(fc), it just means that Aut(A;/fc) permutes the P in the appropriate way. The group of divisors defined over k is denoted by Divfc(C) and similarly for Div°(C). 5.2.3 Principal Divisors and Jacobian The (affine) coordinate ring k[C] of C is defined to be the quotient ring k[C} = k[x,y]/(y2-f(x)), Chapter 5. Rational points on y2 = xh ± 2a313 181 which is an integral domain. The field of fractions k(C) is called the function field of C. We can think of the function field as the set of all rational functions p(x, y)/q(x, y), with q not divisible by y2 — f(x), where we identify two such functions if they agree at all points on C. Over k we can similarly define k[C] and k(C). If P = (u, v) is a point on C then the local ring of C at P is subring of k(C) consisting of functions defined at P; k[C]P = {F e k{C) : F = g/h for some g,he k[C] with h{P) + 0}. This is a discrete valuation ring with (normalized) valuation denoted by ordp, maximal ideal denoted Mp generated by ip , a uniformizer (see, [69] chapter 2). We extend ordp to k(C) by ordp(g/h) — ordP(g) - ordp(h). The point of all this is that for g s k(C) we can write for some h 6 k(C) such that h(P) ^ 0, and this can be done at each point P on C. The order of g at P is ordp(g) and if ordp(g) > 0 then g has a zero at P; if ordp(o) < 0 then g has a pole at P. For the hyperelliptic curve C the uniformizer can be explicitly determined, it depends on the point P = (u, v) as follows { x — u if 7j 9^ 0 y if v = 0 (see, for example [67]). A function g £ k(C) has only a finite number of zeros and poles (see, for example [67]) so we can associate to g a divisor div(0)= oldP(9)(P)-pec(fc) A divisor of this form is called principal and the set of all principal divisors is denoted Pnn(C) = {d\y(g):gek(C)}. This is a subgroup of Div(C), since div(/) + div(<?) = div(/g). In fact it is a subgroup of Div°(C). We define the Jacobian of C to be the quotient J(C) = Div°(C)/Prin(C). Chapter 5. Rational points on y2 = x5 ± 2€"313 182 This is clearly an abelian group. This is going to be the main object we are concerned with. We wi l l see in a little bit that there is a more natural way to view this object when C has genus 2. If D\ and D2 are degree 0 divisors then we write D\ ~ D2, and say D\ and D2 are linearly equivalent, if D\ — D2 e Prin(C). For D <E Div°(C) we write [D]^ (or just [D]) for the element of J(C) represented by D. We can extend the action of Aut(kfk) to J(C) in the natural way. Then Jfe(C) is defined to be subgroup of J(C) fixed by Aut(kfk). When it is clear as to what curve we are referring, we shall denote J(C) and Jfc(C) by J[k) and J(k), respectively. 5.2.4 Geometric representation of the Jacobian In the case when C is an elliptic curve, say E, it is well known that there is a bijection between E(k) and J(k). To be more specific, the Riemann-Roch theorem tells us that every element of J(k) has a unique representative of the form (P) - (oo), so the bijection E(k) —> J(k) is given by P *-> [(P) -(oo)]. In this case, the points on E(k) form a finitely generated abelian group (with identity oo), and the group operation turns out to have a geometric description; P + Q + R = oo iff P, Q, and R are co-linear (with tangency requirements if the points aren't all distinct). This is sometimes stated as "an elliptic curve is its own Jacobian". Let C be a hyperelliptic curve of genus g. If P ~ (xo, yo) is a point on the curve then so is P' = (xo, -yo)- The points P and P' are zeros of the function x — XQ, which has a double pole at oo. Thus the divisor (P) + (P') — 2(oo) is principal, that is - ( P ' ) ~ (P) - 2(oo). It follows that each element of J(k) can be represented in the form r D = ] T ( P ) - r ( o o ) i=l with the following condition satisfied: if the point P,; = {xi,yi) appears in D, then the point P[ = (x-, — j/-) does not appear as one of the Pj for j ^ i. This implies, in particular, that the points of the form (x, 0) appear at most once in D. It follows from Riemann-Roch that each element of J(k) can be represented uniquely by such a divisor with the additional condition that r < g. Such divisors are called reduced. Chapter 5. Rational points on y2 = x5 ± 2 a3' 3 183 Now let's restrict out attention to the case when C has genus 2. In this case we then have that every element of J(k) has a unique (reduced) representa-tive of the form D = {P) + (Q) - 2(oo), where Q ^ P' (note P = oo or Q = oo is allowed). We denote the class of such a divisor as {P, Q}. Thus J(k) = {{P,Q}-.P,Qe C{k),Q + P ' } . (5.1) The group operation on J(k) can be described geometrically, much in the same way as for elliptic curves. The identity is O = {oo, oo} and ~ { P , Q } = {P ' ,Q '} . (5.2) Let {P i , Qi} and {P2, Q2} be two points in J(k). There isa uniqueM(x) G k[x] of degree 3 such that y = M(x) passes through the four points P i , Q\, P2, Q2-This curve intersects C at another 2 points P 3 and Q 3 and so {Pi,Qi} + {P2,Q2} + {P3,Q3} = 0. In other words { P i , Q i } + { P 2 , Q 2 } = { P ^ , g 3 } . (5.3) 5.2.5 2-torsion in the Jacobian From the identity 5.2 it follows that the elements of the form {(#], 0), (62,0)}, where 6\ and 62 are distinct roots of f(x), are of order 2 in J(k). Also, elements of the form {(0,0),oo}, where 0 is a root of f(x), are of order 2. These are precisely all the 2-torsion elements. Thus, there are = 1 5 elements of order 2 in J(k). 5.2.6 Rational Points The group Aut(k/k) acts on J(k) as follows {P,Q}° = {P°,Q°}. Chapter 5. Rational points on y2 = x5 ± 2Q3^ 184 The set of rational points on the Jacobian is the set J{k); the subset of J(k) fixed under the action of Aut(/c//c) on J(k). That is, J(jfc) = {{P, Q] e J(fc) : (P 1 7 , QC T) = (P, Q) or (Q, P ) for all a € Aut(fc/fc)}. It follows that an element {P, Q} e J(k) is rational if either (i) P,Q e C(fc),or (ii) P and Q are defined over a quadratic extension k(y/d) of /c and conju-gate over k(Vd), As an example consider the curve C : y2 = x5 + 1 over the base field k — Q. Some points in C(Q) are oo, (0, ±1) , (-1,0), and so we have the following eight elements in J(Q) O, {oo, (0,1)}, {oo, (0, -1)}, {oo, (-1,0)}, {(0,1), (-1,0)} {(0, -1 ) , (-1,0)}, {(0,1), (0,1)}, {(0, -1 ) , (0, -1)} . The element {oo, (—1,0)} is the only element of order 2 in J(Q), since x = — 1 is the only rational root of x5 + 1. Using (5.3) we can compute {oo, (0,1)} + {oo, (-1,0)} = { 0 0 , 0 0 } + {(0,1), (-1,0)} = {(0,1), (-1,0)}. Over the quadratic field Q(i) (where i = v7—l) we have (1 + i, ± ( - 1 + 2i)), (1 - i, ± ( 1 + 2i)) G C(Q(i)) which gives two more points in J((Q>): {(1 + i, - 1 + 2i), (1 - i , - 1 - 2t)}, {(1 + i, 1 - 2i), (1 - i, 1 + 2i)}. Notice that, for example, {(1 + i, 1 - 2i), (0,1)} is in J(Q) but not J(Q), since (1 + 2 ,1 — 2i) and (0,1) are not quadratic conjugates over Q. 5.2.7 Structure of the Jacobian: The Mordell-Weil theorem By construction (as a quotient of a free abelian group) the Jacobian is an abelian group. In fact, the Mordell-Weil theorem states that J(k) is finitely generated in the case when k is a number field. Thus, we can write it as J(k) =. J(k)tors x II Chapter 5. Rational points on y2 = x5 ± 2a313 185 where J(k)tars is the torsion subgroup of J(k) (which is finite) and r is the rank of J(k). Computing the torsion subgroup J(k)t0rs is a computationally straightforward task. J(k)tors embeds into J (F P ) for each prime p for which C has good reduction (p does not divide the discriminant of / ) . The finite groups J(¥p) are easy to compute and so piecing together the information at different primes we can usually, in practice, determine the structure of J(k)tors. This procedure is not effective but does work quite well in many situations. There is a crude effective procedure involving the height function of J(k) which can be used to compute J(k)tors. For all the curves we wi l l be considering, we wi l l use M A G M A to compute the torsion subgroup. There is no known effective procedure for computing the rank, however there are a number of heuristics for computing bounds on the rank. In prac-tice one can usually bound the rank r by doing a 2-descent, and then find enough independent points in J(k) which meets this bound, thus determin-ing the rank. This w i l l be the case in all the curves we consider (except for y2 = x5 - 2 3 3 9 , where we obtain a rank bound of 1 but can't find a point on the Jacobian). Coming back to the example we considered above, namely y2 — x5 + 1 over Q, we found ten elements in J(Q). It can be determined that a rank bound for J(Q) is zero and that the torsion subgroup has size 10, thus we have found J(Q) completely. It follows that the only integral solutions to y2 = x 5 + 1 are (0, ±1) and (—1,0). Of course this is certainly well known; it is a special case of Catalan's theorem. 5.2.8 Computer Representations of Jacobians Any element {(u\,v\), (u2,v2)} 6 J(k) can be represented uniquely by a pair of polynomials (a(x),b(x)) € k[x}2, where a(x) = (x — u\)(x — 112) andy = b(x) is the unique line through [u\, v\) and (u2, v2) (take y — b(x) to be the tangent line to C if (u\,vi) = (u2,v2)). This is equivalent to requiring f(x) — b(x)2 be divisible by a(x). In the case when the point of J(k) is of the form {00, (u, v)} then a(x) = x — u and b(x) = v. The identity O = {00,00} gets represented as (1,0). If we let kd[x] denote the set of polynomials of degree at most d then we have a injection <f> : J(k) —> k2[x] x ki[x], Chapter 5. Rational points on y2 = x5 ± 2°3^ 186 where the image is the set of all (a, b) such that a is monic and a | / — b2. A n algorithm for adding two elements in J(k) by adding their corre-sponding images (ai,6i), (02,62) in k2[x] x ki[x] has been given by Cantor [14]. The rational points J(k) on J(k) correspond to polynomials with rational coefficients (over k), that is, <f> restricts to 4> • J(k) <—> k2[x] x k\[x], 5.2.9 Some Examples (Using M A G M A ) Let's come back to our example y2 = x5 + 1 (where k = Q). The elements in J(Q) and their corresponding representations are as follows. O = {00,00 {00,(0,1) {00 , (0 , -1) {00, (-1,0) {(0,1), (-1,0) {(0, -1) , ( -1 ,0) {(0,1), (0,1) {(0,-1), (0,-1) {(l + i , - l + 2 i ) , . ( . l - i , - l - 2 i ) {(l + i , l - 2 i ) , ( l - - i , l + 2i) (1.0) (x , l ) ( x , - l ) (x + 1,0) (x 2 + X , X + 1) (x 2 + x, —x — 1) ( x 2 , l ) ( x 2 , - l ) ( x 2 - 2 x + 2 , 2 x - 3 ) ( x 2 - 2 x + 2 , -2x + 3). As another example consider the curve y2 = x 5 + 2 2 3 4 over Q. Some points Chapter 5. Rational points on y2 = x5 ± 2 a3^ 187 on J(Q) and their corresponding representations are as follows. {oo,(0,18)}.—(x, 18) {oo, (-3,9)} i — • (x + 3,9) {oo, (6,90)} i — • ( x - 6 , 9 0 ) { ( 0 , 1 8 ) , ( 0 , 1 8 ) } ^ ( x 2 , 1 8 ) {(-3,9), (-3,9)} i — • (x 2 + 6x + 9,45/2x + 153/2) {(0,18), (6, -90)} i—> (x 2 - 6x, -18x + 18) {(-1 + Vlli, 2 + 4VTTi ) , ( - l - Vlli, 2 - 4Vlii)} i—> (x 2 + 2x + 12,4x + 6). We now show how M A G M A can be used to find the structure of J(Q). > _ < x > : = P o l y n o m i a l R i n g ( R a t i o n a l s ( ) ) ; > C : = H y p e r e l l i p t i c C u r v e ( x ~ 5 + 2 " 2 * 3 ~ 4 ) ; J : = J a c o b i a n ( C ) ; > T,mapTtoJ:=TorsionSubgroup(J); > T ; > {mapTtoJ (t) :t i n T}; A b e l i a n Group i s o m o r p h i c t o Z/5 D e f i n e d on 1 g e n e r a t o r R e l a t i o n s : 5*P[1] = 0 { ( x , 1 8 , l ) , (x-2,-18,2), (x"2,18,2), ( x , - 1 8 , l ) , (1,0,0) ] This tells us that J{Q)t0rS — 2/5 and is generated by (x, 18), i.e. the el-ement {oo, (0,18)}. A l l of J(Q)tors is also listed (elements in M A G M A are listed as triples (a(x), b(x), deg a)). A l l that remains is to determine the rank r of J(Q) and (if possible) the r free generators. We can use a 2-descent to compute an upper bound f on the rank, then search for independent points in J(Q) and hope we get f of them, thus verifying f is the rank of J(Q). > r:=TwoSelmerGroupDatat(J);r; > R : = R a t i o n a l P o i n t s ( J : B o u n d : = 1 0 0 0 ) ; > B:=ReducedBasis(R); B; Chapter 5. Rational points on y2 = xb ± 2Q3' 3 188 1 [ (x~2 - 6*x, -18*x + 18, 2) ] We get an upper bound of 1 on the rank and we found a torsion-free ele-ment, thus J(Q) has rank 1. Therefore J(Q) ~ Z /5 x Z. Note, we can't conclude that A — (x2 — 6x, —18x +18) generates the free part, it could be a multiple of the generator. Let's suppose the generator of the free part is Q and that A — nQ for some integer n. Then taking (canonical) heights we get h(A) = n2h(Q). If A is not a generator then n > 2 and so h(G) < \h{A). So we just need to search for points on J(Q) up to canonical height \h(A) to find the generator. In M A G M A we can search for points by naive height h. Letting HC be the height constant of J(Q), i.e. the maximum difference between the canonical and naive height, we have to search up to the bound to find a generator. > HC : = H e i g h t C o n s t a n t ( J : E f f o r t : = 2 ) ; H C ; > A:=J![x"2 - 6*x, -18*x + 18]; > hA:=Height(A);hA; > newbound:=Exp(hA/4+HC);newbound; > R:=RationalPoints(J:Bound:=newbound) ; B:=ReducedBasis(R) ; B; 0. 333877813949881712480190389291 7. 08937355470437938278274010122 1303.54 53297 63808017141157 63662 [ (x"2 - 6*x, -18*x + 18, 2) ] Therefore A is indeed a generator of the free part of J(Q). Thus J(Q) = ((x, 18)) x ((x2 - 6x, -18s + 18)) ~ Z /5 x Z. Other possible choices for the free generator are A + nV, where V = (x, 18) and n any integer, these can be listed as follows. Chapter 5. Rational points on y2 = x5 ± 2 a3^ 189 > [ n * P + A : n i n {1 . . 4} ] ; [ ( x " 2 - 3 * x - 1 8 , 9 * x + 3 6 , 2 ) , ( x ~ 2 + 2 * x + 1 2 , - 4 * x - 6 , 2 ) , ( x " 2 - 6 * x , - 1 2 * x - 1 8 , 2 ) , ( x - 6 , - 9 0 , 1) ] 5.2.10 Chabauty's theorem Theorem 5.2 (Chabauty [16]) Let C be a curve defined over genus g > 1 defined over a number field k. If the Jacobian of C has rank less than g, then C(k) is finite. This result is superceded by Falting's work which gives the same conclu-sion without a condition on the rank. However, the methods of Chabauty can be used, in some situations, to give a sharp upper bound on the cardinality of C(k), hence allowing us to determine the set C(k). In our situation, C is a genus 2 curve of rank 1 and we are interested in the set of rational points C(Q). Consider C(Q) as contained in J(Q) via the embedding P i—^  {P,oo}. Suppose we have already found the torsion and free-generator of J(Q): J(Q) = J(Q)tors X (V). The basic idea is to pick an odd prime p for which C has good reduction; i.e. C = C (mod p) is a curve of genus 2 over ¥ p . Let V be the reduction of V mod p, and let m be the order of V in J(¥p). Then the divisor T = mV is in the kernel of reduction. Anything in J(Q) can be written uniquely in the form U + n • T, n G Z , where U is an element in the finite set {B + i • V : B <E J{Q)tors and l<i<m - 1}. Fix U as a member of this set. The question is for how many integers n can U + n • T be of the form {P, oo}? It turns out that this only happens if n is a root of a power series over Z p (the power series wi l l depend on U). A the-orem of Strassman can be used to bound the number of p-adic roots to this power series and hence one can find an upper bound £(U) on the number of Chapter 5. Rational points on y2 = x5 ± 2Q3' 3 190 integers n for which U + n • T is of the form {P, oo}. Summing these bounds £(U) over the finitely many U we get a bound on the number of possible el-ements of the form {P, oo} , hence a bound on the cardinality of C(Q). If this bound matches the number of known points we have found on the curve then we know we have found all the rational points. For a thorough account of Chabauty's method the reader is directed to [15] (or [25] for a similar pro-cedure using differential forms) . Let us consider the task of finding all the rational points on the curve C 3 : y2 = a;5+ 3. First we input the curve into M A G M A and search for rational points. > _ < x > : = P o l y n o m i a l R i n g ( R a t i o n a l s ( ) ) ; > C : = H y p e r e l l i p t i c C u r v e ( x ~ 5 + 3 ) ; > R a t i o n a l P o i n t s ( C : B o u n d : = 1 0 0 0 ) ; {@ (1 : 0 : 0 ) , (1 : - 2 : 1 ) , (1 : 2 : 1) @} One can check that increasing the search bound does not produce any more points. So we would like to show C 3 (Q) = { 0 0 , (1, 0), (1, ±2 )} . > J:=Jacobian (C); > r:=TwoSelmerGroupData(J);r; > T,mapTtoJ:=TorsionSubgroup(J); > T; > R:=RationalPoints(J:Bound: = 1 0 0 0);B:=ReducedBasis(R);B; 1 A b e l i a n Group of o r d e r 1 [ (x - 1, 2 , 1) ] Thus J(Q) has rank 1 and trivial torsion. Also, (x — 1, 2) is a possible gen-erator. By the procedure outlined in the previous section we can verify that P = (x - 1, 2) is indeed a generator. With a generator in hand we can now apply Chabauty at a prime > 7 to find an upper bound on the size of C(Q). In fact, what Chabauty returns is a bound on half the number of non-Weierstrass points; Weierstrass points are the points (x, 0) and the point at 0 0 , all of which Chapter 5. Rational points on y2 = x5 ± 2"3 /3 191 are easy to find. In this example the only Weierstrass point is oo. Since we know two non-Weierstrass point on our curve, we are done if Chabauty re-turns the value 1. The function "Chabauty" actually returns an indexed set of tuples < (x,z, v, k) > such that there are at most k pairs of rational points on C whose image in P 1 under the x-coordinate map are congruent to (x : z) modulo pv, and such that the only rational points on C outside these con-gruences classes are Weierstrass points. We can just get a bound by using the prefix # on the command. > P :=B[1];"\\ > #Chabauty (P,7) > #Chabauty(P,11) > #Chabauty (P,17) > #Chabauty(P,19) 3 3 8 1 Thus, applying Chabauty's method at the prime 19 is enough to show that we have found all the rational points on C 3 . It is worth noting that Strassman's theorem bounds the number of p-adic roots, not just the integer roots, so it seems likely that the bound returned wi l l be strictly greater than the number of rational points. This is what happened in the previous example for the primes 7, 11 and 17. For these primes the procedure could not decide whether the extra p-adic solutions were actually rational solutions. These p-adic points on the curve, which are not rational, are affectionately called "ghost" solutions; see [13]. As a second example let us consider the curve C 3 2 4 : = y2 = x 5 ± 3 2 4 . (Note 324 = 2 23 4.) This is the example we worked through in the previous section. We showed J 3 2 4 (Q) ~ Z /5Z x Z with torsion generator (x, 18) and free generator ( x 2 - 6 x , -18x+18). A simple search reveals the following points in C"324(Q): {oo,(0, ±18), (-3, ±9) , (6, ±90)}. Chapter 5. Rational points on y2 = x5 ± 2Q3 / 3 192 If we try to apply Chabauty's method we find that the smallest bound re-turned is 4, which occurs at the primes 7, 31,139, and 191. This is not enough to conclude we have found all the points, but it does show there is at most one other pair of rational points on the curve. It may happen that trying larger primes wi l l succeed in a bound of 3 but this simply becomes computational costly. So, how do we proceed? Well, we do have additional information given to us at the smaller primes, M A G M A returns p-adic information about these supposed "ghost" solutions, so it may be possible to piece information to-gether at different primes to conclude no other rational point can exist. We refer to this as "multiple-prime" Chabauty and consider some examples in Section 5.5. 5.3 D a t a f o r t h e c u r v e s y2 = x5 ± 2a3/B Let A = ± 2 a 3 ^ , 0 < a. fi < 9, and CA denote the curve y2 - x 5 + A. For each value of A we can use M A G M A to compute the torsion group JA {Q)tors and a rank bound r j on the Mordell-Weil group JA(Q) of CA (via a 2-descent). Furthermore, we use M A G M A to try to find YA linearly independent points in JA(Q) thus concluding the rank is exactly VA- We have already successfully done this for A = 2 2 3 4 in Section 5.2.9 and moreover we found a generator for the free part of JA(Q)-For most of the two hundred curves we consider this works out quite well in determining JA(Q>)- However, in some cases M A G M A was unable to find a non-torsion point, simply because its height is just beyond the search range. In each case, Michael Stoll [73] was able to find such points for us. In Tables 5.2 through 5.9 we list the results of the computations performed by M A G M A . Here r is the rank bound determined by M A G M A by doing a 2-descent (in the cases A — -2- 3 3 , - 2 5 3 7 we use the results of Stoll [72] to get a sharper bound, this is included in brackets). #LI is the number of linearly independent points found in JA (Q) by searching in M A G M A (and in some cases the data provided by Stoll [73]). CU(Q)i< n o w n is the set of known points on CA(Q) including the point at infinity oo. For curves of rank 0, we have CA(Q) = CA(Q)known- For the curves of rank 1, we include in column p the first prime for which Chabauty returns a bound on #CA{Q) which is equal to the number of known points in CA(Q)- This verifies we have found CA(Q) Chapter 5. Rational points on y2 = x5 ± 2°t313 193 exactly. In the case that two primes appears in column p , we were unable to find a single prime for which the Chabauty computation was successful in determining C A ( Q ) . However, a multiple-prime Chabauty argument at the two primes works in these cases. This wi l l be done in Section 5.5. Also, there we wi l l discuss the curves of rank 2. As shown in Tables 5.2 through 5.9 we have successfully determined the rank except in four cases: AG { 2 5 3 9 , - 3 9 , - 2 3 3 9 , - 2 4 3 6 } . In the case - 2 3 3 9 we have a rank bound of 1 but are unable to find a point in J ( Q ) . If a point does exist it can be shown (under Birch and Swinnerton-Dyer) to be just beyond the reach of computing at this time [73]. Thus, at this time we are unable to determine the rank. In the three other cases M A G M A has returned a rank bound of 2 but was unable to find any non-torsion points. We now show, in these cases, the rank is 0. Let A £ {2 5 3 9 , - 3 9 , - 2 4 3 6 } , and C(AD) denote the twist y2 = d(x5 + A) of CA - Over K — Q(Vd) these two curves are isomorphic from which it follows rkJA(K) = r k J A ( Q ) + r k J { f ( Q ) . Taking d = — 3 we get the following, where the curve C^ is Q-isomorphic to the one listed. The bounds for rkJA(K) were computed in M A G M A by using a 2-descent, and the ranks r fcJ^(Q) were computed above. A U A rkJA(K) r k J A d ) ( Q ) 2 5 3 9 y2 = x 5 - 2 5 3 4 < 2 2 - 3 9 y2 = x5 + 3 4 < 2 2 • - 2 4 3 6 y2 = x5 + 2 43 < 2 2 Thus r k J A { Q ) = 0 in eachef these cases. To summarize, in the case when that rank is < 1 we have now shown (using classical Chabauty implemented in M A G M A ) that C(Q) = C ( Q ) k n o w n as listed in the tables except possibly in the cases1 A € {2 2 3 4 , 2 5 , 2 G 3 2 ,2 6 3 3 }. J And when A = —2339 since we can't determine the exact rank in this case, as men-tioned above. Chapter 5. Rational points on y2 = x5 ± 2Q3^ 194 C: y2 = x 5 + 2 a 3^ a /? r #LI rank J(Q) C W k n o w n \ OO 0 0 0 0 0 Z/10 ( - 1 , 0 ) , ( 0 , ± 1 ) 0 1 1 1 1 z (1 ,±2) 11 0 2 1 1 1 Z /5 x Z (0,±3) 11 0 3 0 0 0 0 0 4 2 2 2 Z /5 x Z 2 (0, ±9) , (-2, ±7) , (3, ±18) 0 5 1 1 1 Z/2 x Z (-3,0) 29 0 6 0 0 0 Z /5 (0,±27) 0 7 0 0 0 0 0 8 1 1 1 Z /5 x Z (0, ±81), (18, ±1377) 17 0 9 1 1 1 Z 19 1 0 2 2 2 Z 2 ( - 1 , ± 1 ) 1 1 2 2 2 Z 2 1 2 0 0 0 0 1 3 0 0 0 0 1 4 1 1 1 z 19 1 5 1 1 1 z (3, ±27) 19 1 6 0 0 0 0 1 7 1 1 1 z 11 1 8 2 2 2 z 2 (7, ±173) 1 9 1 1 1 z 31 2 0 1 1 1 Z /5 x Z (0 ,±2 ) , (2 ,±6 ) 19 2 1 0 0 0 0 2 2 1 1 1 Z/5 x Z ( 0 , ± 6 ) , ( - 2 , ± 2 ) 61 2 3 1 1 1 Z 19 2 4 1 1 1 Z /5 x Z (0, ±18), (-3, ±9) , (6, ±90) 29,59 2 5 1 1 1 Z ( -3 , ±27) 29 Table 5.2: Data for y2 = xb + 2 Q 3 / 3 Chapter 5. Rational points on y2 = x5 ± 2° f3 / 3 195 C : y2 = x5 + 2Q f3 / 5 a /? r #LI rank J(Q) C(Q) k nown \ OO 2 6 1 1 1 Z /5 x Z (0 ; ±54) 11 2 7 0 0 0 0 2 8 0 0 0 Z /5 (0,±162) 2 9 1 1 1 Z 29 3 0 1 1 1 Z (1 ,±3) 13 3 1 1 1 1 Z (1 ,±5) 31 3 2 0 0 0 0 3 3 1 1 1 z 11 3 4 1 1 1 z 11 3 5 0 0 0 0 3 6 0 0 0 0 3 7 0 0 0 0 3 8 1 1 1 z 19 3 9 1 1 1 z 11 4 0 0 0 0 Z /5 (0,±4) 4 1 2 2 2 z 2 ( l , ± 7 ) , ( - 2 , ± 4 ) 4 2 0 0 0 Z /5 (0,±12) 4 3 1 1 1 Z (-2, ±20) 19 4 4 0 0 0 Z /5 (0,±36) 4 5 2 2 2 Z 2 (6, ±108) 4 6 1 1 1 Z /5 x Z (0,±108) 61 4 7 1 1 1 Z 19 4 8 1 1 1 Z/5 x Z (0,±324),(9,-±405) 47 4 9 1 1 1 Z 17 Table 5.3: Data for y 2 = x 5 + 2 a 3 / 3 (con't) Chapter 5. Rational points on y2 = x5 ± 2a313 196 C : y2 = x5 + 2Q'3 / 3 a r #LI rank J ( Q ) C(Q)known \ OO 5 0 1 1 1 Z /2 x Z (-2,0), (2, ±8) 5 1 2 2 2 Z 2 (-2, ±8) 5 2 3 3 3 z 3 ( 1 , ± 1 7 ) , ( - 2 , ± 1 6 ) 5 3 1 1 1 z 19 5 4 0 0 0 0 5 5 1 1 1 Z /2 x Z (-2, ±88), (-6,0) 11 5 6 0 0 0 0 5 7 1 1 1 z 5 8 1 1 1 z 5 9 2 0 6 0 0 0 0 Z /5 (0,±8) 6 1 1 1 1 Z 19 6 2 1 1 1 Z/5 x Z (0, ±24) , (4, ±40) 29,59 6 3 1 1 1 Z 7,29 6 4 1 1 1 Z/5 x Z (0, ±72), (12, ±504) 7 6 5 0 0 0 0 6 6 0 0 0 Z /5 (0,±216) 6 7 1 1 1 Z 17 6 8 0 0 0 Z /5 (0,±648) 6 9 0 0 0 0 7 0 1 1 1 z 11 7 1 1 1 1 z 11 7 2 1 1 1 z 31 7 3 0 0 0 0 7 4 2 2 2 z 2 Table 5.4: Data for y 2 = x 5 + 2 a 3^ (con't) Chapter 5. Rational points on y2 = x5 ± 2Q3 / 3 197 C: y2 = x5 +2a30 0i /? r #LI rank J(Q) C(Q)known \ 0 0 7 5 0 0 0 0 7 6 1 1 1 z 19 7 7 1 1 1 z 19 7 8 0 0 0 0 7 9 1 1 1 z 41 8 0 0 0 0 Z /5 (0,±16) 8 1 1 1 1 z 29 8 2 1 1 1 Z/5 x Z (0,±48) 29 8 3 0 0 0 0 8 4 0 0 0 Z /5 (0,±144) 8 5 1 1 1 Z 11 8 6 0 0 0 Z /5 (0,±432) 8 7 1 1 1 Z 29 8 8 0 0 0 Z /5 - (0,±1296) 8 9 0 0 0 0 9 0 0 0 0 0 .9 1 1 1 1 z 11 '*9 2 0 0 0 0 9 3 1 1 1 z 59 9 4 0 0 0 0 9 5 1 1 1 z 29 9 6 1 0 1 z 11 9 7 0 0 0 0 9 8 0- 0 0 0 9 9 0 0 0 0 Table 5.5: Data for y 2 = x 5 + 2° 3^ (con't) Chapter 5. Rational points on y2 = x5 ± 2"3^ 198 C : y 2 = x5 - 2 Q 3 / 3 a /? r #LI rank J(Q) C ( Q ) k n 0 w n \ OO 0 0 1 1 1 Z/2 x Z (1,0) 11 0 1 0 0 0 0 0 2 0 0 0 0 0 3 1 1 1 z 17 0 4 1 1 1 z 11 0 5 0 0 0 Z /2 (3,0) 0 6 1 1 1 Z 19 0 7 1 1 1 Z 17 0 8 0 0 0 0 0 9 2 0 1 0 0 0 0 0 1 1 0 0 0 0 1 2 2 2 2 z 2 (3, ±15) 1 3 2(0) 0 0 0 1 4 1 1 1 z (3, ±9) 13 1 5 1 1 1 z 11 1 6 2 2 2 z 2 1 7 1 1 1 z 11 1 8 0 0 0 0 1 9 1 1 1 z 7 2 0 0 0 0 0 2 1 1 1 1 Z ' 17 2 2 0 0 0 0 2 3 0 0 0 0 ' 2 4 0 0 0 0 Table 5.6: Data for y 2 = x 5 - 2 a 3^ Chapter 5. Ra tional poin ts on y2 = x 5 ± 2° 3^ 199 C : y2 = x5 - 2a3 / ? a /? 7* #LI rank J(Q) C(Q)known \ OO P 2 5 0 0 0 0 2 6 0 0 0 0 2 7 1 1 1 z 17 2 8 1 1 1 z 41 2 9 0 0 0 0 3 0 1 1 1 z 11 3 1 1 1 1 z 11 3 2 0 0 0 0 3 3 1 1 1 z 11 3 4 1 1 1 z 19 3 5 0 0 0 0 3 6 0 0 0 0 3 7 0 0 0 0 3 8 1 1 1 z (9, ±81) 17 3 9 1 0 4 0 1 1 1 z (2, ±4) 29 -4 1 1 1 1 z 11 4 2 1 1 1 z (10, ±316) 11 4 3 0 0 0 0 4 4 1 1 1 z 19 4 5 1 1 1 z 17 4 6 2 0 4 7 0 0 0 0 4 8 0 0 0 0 4 9 0 0 0 0 Table 5.7: Data for y2 = x 5 - 2"3^  (con't) Chapter 5. Rational points on y2 = x5 ± 2°3' 3 200 C : y2 = x5 - 2a'3^ a r #LI rank J(Q) C(Q)known \ OO P 5 0 1 1 1 Z /2 x Z (2,0), (6, ±88) 11 5 1 0 0 0 0 5 2 1 1 1 z 19 5 3 1 1 1 z 11 5 4 2 2 2 z 2 (6, ±72) 5 5 1 1 1 Z /2 x Z (6,0) 11 5 6 2 2 2 Z 2 (9, ±189) 5 7 3(1) 1 1 Z 11 5 8 1 1 1 Z (18, ±1296) 11 5 9 0 0 0 0 6 0 1 1 1 z 19 6 1 0 0 0 0 6 2 0 0 0 0 6 3 0 0 0 0 6 4 0 0 0 0 6 5 1 1 1 z 17 6 6 1 1 1 z 41 6 7 0 0 0 0 6 8 1 1 1 z 41 6 9 1 1 1 z 17 7 0 1 1 1 z 11 7 1 1 1 1 z 11 7 2 1 1 1 z 11 7 3 0 0 0 0 7 4 2 2 2 z 2 (33, ±6255) Table 5.8: Data for y 2 = x 5 - 2 a 3 / J (con't) Chapter 5. Rational points on y2 = x5 ± 2"3 /3 201 C : y2 = x5 - 2a3 / 3 a r #LI rank J(Q) C(Q)known \ OO P 7 5 0 0 0 0 7 6 1 1 1 z 29 7 7 1 1 1 z 11 7 8 0 0 0 0 7 9 1 1 1 z 11 8 0 0 0 0 0 8 1 1 1 1 z (4, ±16) 19 8 2 1 1 1 z 61 8 3 0 0 0 0 8 4 0 0 0 0 8 5 1 1 1 z (12, ±432) 29 8 6 0 0 0 0 8 7 1 1 1 z 29 8 8 0 0 0 0 8 9 0 0 0 0 9 0 0 0 0 0 9 1 1 1 1 z - - 11 9 2 0 0 0 0 9 3 1 1 1 z 11 9 4 0 0 0 0 9 5 1 1 1 z 11 9 6 1 1 1 z 29 9 7 0 0 0 0 9 8 0 0 0 0 9 9 0 0 0 0 Table 5.9: Data for y2 = x5 - 2a3f) (con't) Chapter 5. Rational points on y2 = x5 ± 2 a3' 3 202 In fact, we wi l l show that C(Q) = C(Q)known in these cases as well. The first case is dealt with using the results of Stoll in the next section. The last two cases are dealt with in Section 5.5 by applying a multiple-prime Chabauty argument. The case A = 2 5 is dealt with using results on ternary diophantine equations, which we also do in Section 5.5. 5.4 T h e f a m i l y o f c u r v e s y2 = x5 + A We take a digression in this section to mention some general results pertaining to our curves. The curves of interest in this chapter are a part of the family of curves where A ^ 0 is a rational number. Since CA and CR are isomorphic over Q \i A/B isa tenth power we wi l l assume that A is an integer and tenth-power free. Except for some fixed values of A not much is known about the rational points on this family of curves in general. However, recently Michael Stoll (see [75]) has announced some very interesting results regarding the number of rational points on these curves, in the case when the Jacobian JA(Q) of CA has rank 1. Before stating his results we'll fix a bit of notation. Let CLA be the number of trivial points on CA(Q); points (x,y) e CA(Q) with xy — 0, or the point at infinity oo. Non-trivial points occurs in pairs: (x,y), (x, —y), so we let UA be half the number of non-trivial points. Equivalently, UA is the number of nontrivial points with positive y coordinate. Then #CA{Q) = 2UA + OIA, and CIA is given by CA : y2 = x5 + A, dA < 4 3 2 1 if A is neither a square nor a fifth power, if A is a fifth power, A ^ \ , if A is a square, A^l, We have already seen that C\ has rank zero and that #C\ (<Q>) = 4. Thus for A ^ 1 we have #CA{Q) < 2nA + 3. In [75] Stoll proves the following, where r A denotes the Mordell-Weil rank of the Jacobian JA(Q) of CA-Chapter 5. Rational points on y2 = x5 ± 2™3^ 203 Theorem 5.3 (Stoll) Let A ^ 0 be an integer such that r& = 1. Then < 2 and consequently #CA(Q) < 7. More specifically he proves the following theorem, using a refinement of the method of Chabauty and Coleman. Here p is an odd prime and vp denotes the p-adic valuation. Theorem 5.4 (Stoll) Let A ^ 0 be an integer such that TA = 1. 1. ifvp{A) G {1,3,7,9}/or some p ^ 11,13 then nA < 1. 2. ifvp(A) = 5 for some p ^ 3,5 then HA < 1, ifvz(A) = 5 f«en TM < 2. 3. z/v P(A) G {2,4,6,8}/or some p ^ 2,3,7, or Z/TJ3(A) G {6,8}, or ifv7{A) G {2,6,8} then nA < 1, otherwise ifv3(A) G {2,4} fnen nA < 2, 4. if A = 1 (mod 3) fnen < 1, and if A = - \ (mod 3) f/ten < 2, 5. r / A = 1,3,9 (mod 11) fnen nA < 1. In the case that A = ± 2 A 3 6 the upper bounds on #CA (Q) obtained by Stoll matches the number of known points on CA (Q) in the following cases: A G {3, 3S, 2 2 3 4 ,2 3 , 3,2 43 3, 2 43 8, 2 63 4, - 2 • 34, - 2 3 3 8 , - 2 5 3 8 , -2 83}. Thus, our results in the previous section are superceded by Stoll's results, except in one case. Notice that in the case A = 2 2 3 4 = 182 we were unable to find a single prime at which the Chabauty bound is sufficient to determine C 1 8 2 (Q), thus Stoll's results now give us C 1 8 2 (Q). The curve C 1 8 2 has 7 rational points, so the bound in Theorem 5.3 is sharp. In fact, as shown in Stoll, this is the unique curve that attains this bound. One may have noticed, in the tables of the previous section, the only tor-sion groups that arose were Z / 2 Z , Z / 5 Z , and, in the single case A = 1, Z/10Z. This is also governed by a general result. It follows from results of Poonen [58] that the torsion of JA (Q) is as follows. 1. If A is neither a square nor a fifth power then JA{Q)tOT8 = 0. Chapter 5. Rational points on y2 = x5 ± 2"3^ 204 2. If A = a2 for some integer a ^ 1 then JA(Q)torS = {{(0, a), oo}, {(0, -a), oo}, {(0, a), (0, a)}, {(0, -a) , (0 , -a)} ,O}~Z/5Z. 3. If A = 65 for some integer 6 / 1 then J ^ Q W , = { {(-6,0),oo} ,0} ~ Z/2Z. 5.5 Proof of Theorem 5.1 We have verified in Section 5.3 that Theorem 5.1 holds for all A except A £ {2 23 4,2 5, 2 63 2, 2G33}. In the last section we verified, using a result of Stoll, the case A = 2 23 4. In this section we show that for A £ {2 5,2 63 2,2 63 3} the set of known points C(Q)known listed in the tables of Section 5.3 are precisely all the rational points these curves. In the last two cases we do this by applying a multiple-prime Chabauty argument. Such an argument is scarcely found in the literature, indeed I only know of only two places it is applied: [59] and [13]. In what follows, we view CA(Q) C JA(Q) via the embedding (xo,yo) {(x0,y0),oo}. 5.5.1 A = 2632 Let us first consider the case A — 2 63 2, where CA(Q)knoWn = {oo, (0, ±24), (4, ±40)} and ^ ( Q ) = ({(0, -24), oo}) x ({(4, -40), oo}) ~ Z/5Z x Z. L e t T = {(0,-24), oo} and P = {(4,-40), oo} be the generators for the torsion and free-part respectively. Considering the reduction modulo 29, : J(Q) -+ J(F 2 9 ) =i Z/30Z x Z/30Z. The reductions of T and P, denoted by T2g and P 2 9, have orders 5 and 30 respectively. We can input this into M A G M A as follows. Chapter 5. Rational points on y2 = x5 ± 2a313 205 > _ < x > : = P o l y n o m i a l R i n g ( R a t i o n a l s ( ) ) ; > C : = H y p e r e l l i p t i c C u r v e ( x ~ 5 + 2 ~ 6 * 3 ~ 2 ) ; J:=Jacobian(C) ; > T:=J! [ x , - 2 4 ] ] ; P:=J! [x-4,-4x-24 ] ; > J29:=BaseChange(J,GF(29)) ; > T29:=J29!T; P29:=J29!P; > O r d e r ( T 2 9 ) ; O r d e r ( P 2 9 ) ; 5 30 It follows that the image of the Mordell-Weil group is </>29(J(Q)) — ( P 2 9 ) x (P 2 9 ) ^ Z / 5 Z x Z/30Z (it is a simple computation to check T 2 9 £ (P 2 9 )) . A rational point (xo,yo) on CA{Q) has image of the form { (x7J ,y7j ) , 0 0 } in J ( F 2 9 ) so we determine conditions on the integers a and b such that the element aT + bP has this image. > f o r a i n [0 . . 4] do; for> f o r b i n [0..29] do; f o r I f o r > i f (a*T29+b*P29) [3] l e 1 th e n ; f o r | f o r | i f > print(<a,b,a*T29+b*P29>); f o r I f o r I i f > end i f ; f o r | f o r > end f o r ; for> end f o r ; <0, 0, (1, 0, 0)> <0, 1, (x + 25, 18, 1)> <0, 29, (x + 25, 11, 1)> <1, 0, (x, 5, 1)> <1, 10, (x + 7 , 3, 1) > <2, 8, (x + 1, 16, 1) > <3, 22, (x + 1, 13, 1)> <4, 0, (x , 24, 1)> <4, 12, (x + 7 , 26, 1)> This tells us that the image of aT + bP is {(xTJ, yo), 00} for a and b satisfying the following congruences. Chapter 5. Rational points on y2 = x5 ± 2Q3' 3 206 a (mod 5) 6 (mod 30) <hz{aT + bP) 0 0 {oo,oo} 0 1 {(4,18),oo} 0 29 {(4,11), oo} 1 0 {(0,5),oo} 0 18 {(22,3),oo} 2 8 {(28,16),oo} 3 22 {(28,13), oo} 4 0 {(0,24),oo} 4 12 {(22,26),oo} Our five known rational points on C ( Q ) are in the residue classes {oo, oo}, {(4,18),oo}, {(4, l l ) ,oo } , {(0,5),oo} and {(0, 24), oo}, to show there are no other rational points it suffices to show two things: (i) each coset of J (Q ) /ker<^ 29 contains at most one rational point, (ii) there are no rational points in the other four residue classes. The first of these follows from the fact that the differential killing the Mordell-Weil group modulo 29, ZJ — x + 18, does not vanish on any of the residue classes. This is the Coleman-Chabauty part of the argument. A s for (ii) we repeat the computations above with p = 59 and get the following classes of (o (mod 60), b (mod 60)): (0,0), (0,1), (0, 7), (0,11), (0,30), (0,49), (0, 53), (0, 59), (1, 0), (1, 36), (2,44), (3,16), (4,0), (4, 24). Considering b modulo 30, we see the four extraneous classes which appeared at the prime 29 do not appear here. Thus, these four classes do not contain a rational point. Therefore, for A = 2 6 3 2 C ^ ( Q ) = _ { o o , ( 0 , ± 2 4 ) , ( 4 , ± 4 0 ) } . I would like to thank Michael Stoll for his help with this argument. 5.5.2 A = 2 6 3 3 In this case there are no known finite points on CA{Q)- Using M A G M A we find JA(®) = (P)-%, Chapters. Rational points on y2 = x 5 ± 2°3^ 207 where P = {x 2 -24x+88,116x-584}. Making a call, in M A G M A , to Chabauty at the prime 7 we find that there are at most two rational points on C. Simi-larly, we get the same information at the prime 29. In particular, applying the same type of computations as above, aP lies in a residue class of the form {r, oo} modulo 7 only when a = 0, 2, 3 (mod 5), and aP lies in a residue class of the form {r, oo} modulo 29 only when a = 0,1,4 (mod 5). Thus the only rational points on CA lie in the coset of J(Q)/kerc/>7 which contains oo. The differential killing the Mordell-Weil group modulo 7, to = x, does not vanish on any of the residue classes thus each coset contains at most one rational point, and so oo is the only rational points on CA(Q)-5.5.3 A = 2 5 In this case we can apply results from the theory of ternary diophantine equa-tions to get our result. A n y rational solution to the equation y2 = x 5 + 2 5 is of the form (x, y) — (a/e2, b/eb) for some a, b, e e Z with (a, e) = (b, e) = 1. Thus a, b, e is a solution to b2 = a 5 + (2e 2) 5. (5.4) Let g = (a.b), then g2 divides (2e 2) 5, but (g,e) = 1, so g2 | 2 5 . Therefore, g = 1,2,4. Since 5.4 has no solutions with a, b, 2e2 pairwise coprime (see Dar-mon and Merel [28]) then g ^ 1. Also, 5 ^ 4 since otherwise 2 | (b, e), a contradiction. It must be the case that g = 2, and so dividing the equation through by 2 5 we have 2(6/8)2 = (a/2) 5 + {e2f, where b/8, a/2, e are pairwise coprime. By a result of Bennett and Skinner (see Theorem 4.4) the only solutions are with (a/e 2, b/e5) = (2, ±8) , (-2,0). 5.5.4 R a n k > 2 cases We've shown in Section 5.3 that the curves of the form y2 = x 5 + 2 a 3 / 3 , whose Mordell-Weil group has rank > 2, correspond to the following values of a and (3: (a, P) = (0,4), (1,0), (1,1), (1,8), (4,1), (4,5), (5,1), (5, 2), (7,4). Chapter 5. Rational points on y2 = x5 ± 2 Q3 / J 208 Similarly, the curves of the form y2 = x5 — 2 Q 3 / 3 with rank > 2 are the follow-ing: (a,/3) = (1,2), (1,6), (5,4), (5,6), (7,4). For these fourteen remaining curves the classical method of Chabauty can-not be applied to bound the number of rational points since the rank is not smaller that the genus. In this case, we would need to use covering methods. In this method, finitely many curves Di are constructed which are unramified covers of C, 4>i : Di —> C. In such a situation, there is a number field K such that C(Q) C (J, 4>i(Di(K)). Hence, determining K-rational points on all Di wi l l allow us to determine all Q-rational points on C. The covering curves Di that typically arise have genus 17 and thus it seems we have made the prob-lem harder. However, Di may possess maps down to some elliptic curve E, for which the Elliptic Curve Chabauty method may be applied. This method is described by Bruin in [9], [10], [11], and much of the method has been imple-mented in M A G M A by Bruin (some of which is still unavailable in the current release [12]). The methods, though implemented, require a high level of so-phistication on the part of the user. Bruin verified the results for these final fourteen cures for us [12]. It is interesting to note that one of these curves can be taken care of using a result of Bruin [10]. A rational point (X, Y) on the curve Y2 = X5 + 2 has the form X = x/s2, Y = y / s 5 for integers x,y, s such that (x, s) = (y, s) = 1. The equation can then be written as y2 = x5 + 2(s 2 ) 5 where we are now interested in coprime integer solutions x, y, s. It follows from [10] (see also [36]) that the only solutions are with (x,y, s2) = (—1, ±1 ,1 ) . These pull back to the solutions (X, Y) = (—1, ±1) . Chapter 6 Classification of Elliptic Curves over O with 2-torsion and conductor 2ap2 As we mentioned before we have broken up our attempt to classify curves of conductor 2ap2 into two stages. In the first stage, we showed if there is an elliptic curve of conductor 2°p2 then p must satisfy one of finitely many explicitly determined Diophantine equations, and we have explicit formulae for the coefficients of the elliptic curve. A l l this information is given in the theorems of Section 3.1.1. The second stage is to refine the theorems of Section 3.1.1 by using Diophantine lemmata of Chapter 4. It is stage two that is the focus of this chapter. In Section 6.1 we state the classification theorems for curves of conductor 2ap2. The novelty of these theorems is that, given a prime p, it is straightfor-ward to check whether there are any elliptic curves of conductor 2ap2 (with 2-torsion), and to determine all such curves. Of course, for small values of p (say p < 17) one could (and should!) consult the tables of Cremona. For larger values, however, we believe the work in this chapter w i l l prove valuable. 6.1 Statement of Results Let p be a prime number and N an integer satisfying the inequalities p > 5, and 0 < N < 8. In what follows, we announce nine theorems which describe, up to Q-isomor-phism, all the elliptic curves over Q, of conductor 2Np2, having a rational 209 Chapter 6. Classification of Elliptic Curves of conductor 2ap2 210 point of order 2 over Q. Each theorem corresponds to a value of N. The re-sults obtained are presented in the form of tables analogous to those of [26] and [37]. Each row consists of an elliptic curve of Q realizing the desired con-ditions. The columns of the table consist of the following properties of E: 1. A minimal model of E of the form where the a; are in Z. If N > 2, we can choose a model such that ai = ag = 0. In the statements of these theorems we omit the columns corresponding to these coefficients. 2. The order IT2I of the group T2 consisting of Q-rational 2-torsion points of E. 3. The factorization of the minimal discriminant A of E. 4. The j-invariant of E. 5. The Kodaira symbols of E at 2 and p. Also appearing in the table are the letters of identification (A,B,...) for each elliptic curve. The curves which are labeled by the same letter are linked by an isogeny over Q of degree 2 or a composition of two such isogenies. Moreover, they are numbered in the order of how they are to be determined. As in Chapter 3 we wi l l use the following notation. a. For each elliptic curve E over Q, we denote by E' the elliptic curve over Q obtained from E by a twist by y—T-b. Given an integer ra which is a square in Z we denote, for the rest of this work, by y/n the square root of ra satisfying the following condition: y2 + a\xy = x3 + a^x2 + a4x + ag, y/n = 1 mod 4 if n is odd -v/ra > 0 if ra is even . (6.1) Theorem 6.1 The elliptic curves E defined over Q, of conductor p2, and having at least one rational point of order 2, are the ones such that one of the following conditions is satisfied: Chapter 6. Classification of Elliptic Curves of conductor 2ap2 211 1. p — 7 and E is Q-isomorphic to one of the elliptic curves: minimal model T2 3 A A l [1,-1,0,-2,-1] 2 -15 3 7 3 A2 [1,-1,0,-107,552] 2 -15 3 7 9 BI [1,-1,0,-37,-78] 2 2553 7 3 B2 [1,-1,0,-1822,30393] 2 2553 7 9 2. p =17 and E is Q-isomorphic to one of the elliptic curves: minimal model T2 3 A C l [1,-1,1,-1644,-24922] 4 273J 17 2 1 7 8 C2 [1,-1,1,-26209,-1626560] 2 18863J 17 177 C3 [1,-1,1,-199,510] 4 33J 17 177 C4 [1,-1,1,-199,-68272] 2 - 3 33 1 7 4 1 7 i o 3. p — 64 fs fl square and E is Q-isomorphic to one of the elliptic curves: ai a2 a4 a 6 1^21 A J Kodaira A l 1 Py/p— 64— 1 4 - P 2 0 2 P 7 ( P - 1 6 ) J P io;iT A2 1 p\Jp— 64 — 1 4p2 2 - P 8 (256-p) J I 0 ; I 5 4 p VP — 04 P 2 Theorem 6.2 The elliptic curves E/Q of conductor 2p2 with a rational point of order 2 are the ones such that one of the following conditions is satisfied: 1. p = 7 or 17 and E is Q-isomorphic to one of the curves in the table in Appendix C. 2. 1 p = 2k + 1, w/zere k > 5, and .E zs Q-isomorphic to one of the elliptic curves: ai a2 (24 • a6 A J Kodaira A l 1 p(2p-l ) - l p^p-l) 0 2 2 f c - 8 p 8 (2"p+l):J hk-8', I2 4 16 2 2 f c - 8 p 2 A2 1 p(2p-l)- l -p3(p-l) -p 4(p-l)(2p-l) 2 f c - V (2t+'1p+l):1 4 4 16 2fe-'ip BI 1 -p(p-2)-l -p'J(p-l) 0 2 2 f c - 8 p 8 (p2-2*) 3 l 2 f c - 8 ; I2 4 16 2 2 t - 8 p 2 B2 1 -p(p-2)-l p2(p-l) -p3(p-l)(p-2) _ 2 f c - 4 p 1 0 4 4 16 2k-4p4 These are Fermat primes; it is necessary that k be a power of 2 for 2fe + 1 to be prime. Chapter 6. Classification of Elliptic Curves of conductor 2°p2 212 \T2\ = 4 for Al and Bl, and |T 2 | = 2 for the other two. 3. 2 p = 2q - 1, where q > 5 is a prime, and E is Q-isomorphic to one of the elliptic curves: a2 a.4 a 6 A j Kodaira CI 1 p(2p+l)-l 4 pJ(p+l) 16 0 22,-8 p 8 22,,-Sp2 C2 1 p(2p+l)-l 4 -pJ(p+l) 4 -p"(p+l)(2p+l) 16 2 " " V (2''+"p+l)J 2<>~ "p l<,-4;i; DI 1 -p(p+2)-l p"(p+l) 0 2 2,-8 p 8 (pz+2«y 12,-8; I2 4 16 22,,-bp-2 D2 1 -p(p+2)-l -p'(p+l) pJ(p+l)(p+2) 2 „ - 4 p 1 0 (P'+2«+-iy i Q - 4 ; i 4 4 4 16 2'l-4p4 \T2\ = 4for CI and DI, and |T 2 | = 2 for the other two. 4. there exists m > 7 such that p — 2m = d2,for some d = 1 (mod 4), and .E zs Q-isomorphic to one of the elliptic curves: ai a 2 0 4 a 6 A J Kodaira E l 1 pd-1 4 - 2m ~ y 0 2 2 m " 1 V (p-2'"-*)a 2 2 m - 1 2 p l 2 m - 1 2 ; Ii E2 1 pd-1 4 2 m - 4 p 2 2m-6p3d _ 2 m - 6 p 8 ( 2 ™ + 2 _ p ) 3 2 ™ - 6 p 2 T • T* 771— 6 j -L2 |T 2 | is 2/or tofn curves. 5. there exists m > 7 such that p + 2m = d2,for some d = 1 (mod 4), and i? is Q-isomorphie-to one of the elliptic curves: a i « 2 (24 a6 A j Kodaira FI 1 p d - 1 4 2m - 6 p 2 0 2 2 m - 1 2 p 7 (p+2'"-2 ) 3 2 2 ( m - 6 ) p l 2m-12;I i F2 1 p d - 1 4 _ 2 m - 4 p 2 _ 2 m - 6 p 3 r f 2 m - 6 p 8 ( p + r + 2 ) J 2"-<V I7H — 6 i 2^ |T 2 | zs 2/or both curves. 6. there exists m > 7 swcn zTzat 2 m — p = d2,for some d = .1 (mod 4), and £ z's Q-isomorphic to one of the elliptic curves: a\ a2 a4 <76 A i Kodaira GI 1 - p d - 1 4 2m - 6 p 2 0 2 2 m - 1 2 p 7 2 2 m - 1 2 p l2rn-12; IJ G2 1 - p d - 1 4 _ 2 m - 4 p 2 2 m - 6 p 3 r f 2 m - 6 p 8 ( 2m + 2 _ p ) J 2 , n - 6 p 2 JTM — 61 2^ 2These are Mersenne primes; it is necessary that q be a prime for 2q — 1 to be prime. Chapter 6. Classification of Elliptic Curves of conductor 2°p2 213 |T 2 | is 2 for both curves. 7. there exists m>7 such that 2 " p ~ 1 z's a square integer, say pd2 = 2 m - 1 with d = 1 (mod 4), and E 1 z's Q-isomorphic to one of the elliptic curves: « i « 2 a 6 A j-invariant Kodaira HI 1 -pd-1 4 2 m ~ 6 p 0 _ 2 2 m - 1 2 p 3 ( l _ 2 ' " - ' ! ) a 2 2 „ . - 1 2 l 2 m - 1 2 ; I H H2 1 - p d - 1 4 - 2 — V 2 m - 6 p 3 ( 2 „ , + 2 _ 1 ) 3 2 " - 6 I r „ _ 6 ; I I I 11 1 p2 d - l 4 2 m - 6 p 3 0 _ 2 2 m - 1 2 p 9 ( 1 - 2 " -2 ) 3 2 2 m - 12 l 2 m - 1 2 ; H I * 12 1 p2 d - l 4 _ 2 , n - 4 p 3 - 2 7 " - 6 p 5 ( i 2 m - 6 p 9 ( 2 " ' + 2 _ j ^ 2 m - 6 I m - 6 ; I H * |T21 z's 2 /or a/7 four curves. 8. there exists m > 7 swc« f/zaf 2 " ^ + 1 z's a square integer, say pd2 = 2m + 1 loz'f/t d = 1 (mod 4), and E is Q-isomorphic to one of the elliptic curves: Ol a 2 CI4 a 6 A j-invariant Kodaira Jl 1 p d - 1 4 2 T " - G p 0 2 2 m - 12p3 ( 2 " ' - 2 + l ) J 2 2 m - 1 2 l 2 m - 1 2 ; III J2 1 p d - 1 4 - 2 ™ - < p _ 2 m - 6 p 2 ( i 2 m - 6 p 3 ( 2 " ' + 2 + l ) - J 2 m - 6 U - 6 - . I I I KI 1 p2 d - l 4 2 m - 6 p 3 0 2 2 m - 1 2 p 9 ( 2 ™ - 2 + 1 ) 3 2 2 r n - 12 l 2 r n - 1 2 ; H I * K2 1 p2 d - l 4 - 2m - " p 3 _ 2 , n - 6 p 5 d 2 m - 6 p 9 ( 2 " ' + 2 + 1 ) ^ 2 m - B i , „ - 6 ; H i * |T 2 | z's 2/or all four curves. Theorem 6.3 The elliptic curves E/Q of conductor 4p2 with a rational point of order 2 are the ones such that one of the following conditions is satisfied: 1. p = 5 and E is Q-isomorphic to one of the elliptic curves in the table in Ap-pendix C. 2. p — 4 is a square and E is Q-isomorphic to one of the elliptic curves: 0-2 04 I T 2 I A j-invariant Kodaira A l pJp-4 -P2 2 2V 256 (p- l ) ': i p IV; 1^  A2 -Ipyjp - 4 P3 2 - 2 V p4 ( 1 6 - p ) 3 P 2 IV*; 15 Chapter 6. Classification of Elliptic Curves of conductor 2ap2 214 Theorem 6.4 The elliptic curves E/Q of conductor 8p2 with a rational point of order 2 are the ones such that one of the following conditions is satisfied: 1. p = 5,7,17,23 or 31 and E is Q-isomorphic to one of the elliptic curves in the table in Appendix C. 2. p — 2m is a square for m = 4 or 5 and E is Q-isomorphic to one of the elliptic curves: a2 (24 A j-invariant Kodaira A l p^p - 2m _ 2 m - 2 p 2 2 2 m p 7 2 1 2 - 2 ' " ( p - 2 ' " -2 ) a P I J, HI*; II A2 -2psjp - 2m P 3 _ 2 m + 6 p 8 2b~m(2,"+/1-p)J P 2 | T 2 | fs 2 for both curves. 3. p + 32 is a square and E is Q-isomorphic to one of the elliptic curves: a2 0 4 \T2\ A j-invariant Kodaira BI PVP + 32 23p2 2 2 1 0 p 7 4 ( p + 8 )J P B2 . : 7- - 2 - 2 " p 8 -2(p+128)-i n*; 15 — 2p^/p + 32 P 4 Theorem 6.5 The elliptic curves E/Q of conductor 16p2 with a rational point of order 2 are the ones such that one of the following conditions is satisfied: 1. p = 5,7,17, 23 or 31 and E is Q-isomorphic to one of the elliptic curves in the table in Appendix C. 2. p = 2k 4-1, where k > 4, and E is Q-isomorphic to one of the elliptic curves: a2 «4 m A j-invariant Kodaira A l -p(2p - 1) ( P - 1 ) P 3 4 22k+4p8 (2"p+l)J 2 2 f c - 8 p 2 T* • T* 1 2 f c - 4 ' 1 2 A2 2p{2p- 1) p2 2 2 f c + 8 p 7 (2k+«p+\f 2"-"p T* • T* A3 -2p(p+l) P 2 ( P " 1 ) 2 2 2 4 f c " V (p+22h-'f 2 2 ( 2 f c - 8 ) j , T* • T* A4 -2p(p - 2) P4 2 _ 2 f c + 8 p i o (2k+"-p2)'J 2k-«p'< II* ;IJ 3. p = 2q — 1, where q > 3 is a prime, and E is Q-isomorphic to one of the elliptic curves: Chapter 6. Classification of Elliptic Curves of conductor 2ap2 215 a-2 0,4 \T2\ A j-invariant Kodaira BI - p ( 2 p + l ) (P + 1)P3 4 2 2 , + 4 p 8 (2''p+i)* 2 2 , , - 8 p 2 T* • T* i 2 7 - 4 ' 1 2 B2 2p(2p+l) P 2 2 2"+V (2"+"p+l)a 2<'-'1p2 T*-T* V 1 B3 - 2 p ( p - l ) P 2 (P+1) 2 2 _ 2 4 , - 4 p 7 ( p _ 2 ^ - " ) J 2 2 ( 2 , - 8 ) p T* • T* M c ; - 1 2 i M B4 -2p(p+2) P 4 2 2 « + V (p2 + 2"+") - ) 4. p — 2 m fs « square for m = 2 or m > 4, and E fs Q-isomorphic to one of the elliptic curves: a 2 0,4 A j-invariant Kodaira C l -pVp ~ 2m -2m-2p2 22mp7 (p-2">-*)3 2 2 , „ - l 2 p II > l 2 m - 8 i ^ l C2 2p^p - 2m P 3 - 2 m + 6 p 8 ( 2 " . + 2 _ p ) . 2"-bp 2 IS) Im -2i I2 IT2I is 2 for both curves. 5. p + 32 is a square and E is Q-isomorphic to one of the elliptic curves: a2 0 4 |T 2 | A j-invariant Kodaira DI -Ps/p + 32 2 3p 2 2 2 1 0 p 7 4(p+8f P T*. T t 12> 1l D2 2p^p + 32 P 3 2 2 n p 8 2(p+2') J P 2 T*-T* 1 3 ' 1 2 6. there exists an odd integer m > 7 such that p + 2rn is a square and E is Q-isomorphic to one of the elliptic curves: 02 0,4 m\ A j-invariant Kodaira E l -Pv/p + 2m 2 m - 2 p 2 2 22mp7 (p+2"- 2 ) - 1 2 2 „ - l 2 E2 2PA/P + 2m P 3 2 2m + 6 p 8 (P+2m+Jy 2"—Hp2 7" • T* lm-2i 1 2 7. £/zere exists an odd integer m > 7 such that 2m — p is a square and E is Q-isomorphic to one of the elliptic curves: a2 <Z4 |T 2 | A j-invariant Kodaira Fl PsJ2™ - p 2 m - 2 p 2 2 - 2 2 m p 7 ( p - 2 " -2 ) 3 2 2 ( « . - 6 ) p T* • T* 1 2 m - 8 ' xl F2 -2psJ2m - p - p3 2 2 m + 6 p 8 (2"'+2-p);i 2 m - 6 p 2 T* • 1* 1 m - 2 i 1 2 8. there exists m > 7 SMC/J f/jflf 2 m p 1 fs a square integer and E is Q-isomorphic to one of the elliptic curves: Chapter 6. Classification of Elliptic Curves of conductor 2ap2 216 a-2 a.4 |T 2 | A j-invariant Kodaira G I p ^ ¥ 2m~2p 2 - 2 2 m p 3 ( 1 _ 2 " ' -2 ) 3 2 2 m - 1 2 G2 ~P 2 2m + 6 p 3 ( 2 m + 2 _ ] ) 3 2"- 6 i m - 2 ; n i H I 2 m - 2 p 3 2 - 22 r r y (1-2'"- 2) 3 2 2 m - 1 2 12m-8 i m H2 -P3 2 2 m+6 p 9 ( 2 " ' + 2 - l ) 3 2m-6 Im-2J m 9. f/rere exists m > 5 swc/i 2 m p + 1 fs a square integer and E is Q-isomorphic to one of the elliptic curves: a2 0 4 \T2\ A j-invariant Kodaira 11 2 m " 2 p 2 2 2 > 3 (2"-2 + l ) 3 2 2 T H - 12 I5 W _ 8 ;1II 12 P 2 2 m+6 p 3 (2"'+2 + l ) : ! i m - 2 ; n i JI 2m"2p3 2 22mp9 (2T"-2 + l ) 3 2 2 m - 1 2 i 2 m - 8 ; m J2 p3 2 2m+6p9 (2"'+2 + l ) 3 2 m - 0 i m - 2 ; n r Theorem 6.6 TTze elliptic curves E/Q of conductor 32p 2 with a rational point of order 2 are the ones such that one of the following conditions is satisfied: 1. p — 7 and E is Q-isomorphic to one of the elliptic curves in the table in Ap-pendix C. 2. p > 5 and E is Q-isomorphic to one of the elliptic curves: (i) p = 1 (mod 4), a2 0 4 \T2\ A j-invariant Kodaira A l 0 -P2 4 2 6 p 6 1728 III;IS A 2 0 22p2 2 - 2 1 2 p 6 1728 A 3 6p p2 2 2 V 2 3 3 3 11 3 T* . 7* A 4 —6p P2 2 29 p 6 2 3 3 3 11 3 7*. 7* BI 0 ~P 2 26 p 3 1728 III;III B2 0 22p 2 - 2 1 2 p 3 1728 13;III C l 0 -p3 2 2 6 p 9 1728 III;III* C2 0 22p3 2 - 2 1 2 p 9 1728 15; HI" (ii) p = 3 (mod 4), Chapter 6. Classification of Elliptic Curves of conductor 2ap2 217 a2 a4 1221 A j-invariant Kodaira DI 0 -P2 4 2 6 p 6 1728 III;15 D2 0 2 2 p 2 2 -212p6 1728 15; 15 D3 6p p2 2 2 9 p 6 2 3 3 3 11 3 T* • T* D4 —6p 2 p 2 2 9 p 6 2 3 3 3 11 3 T*. T* E l 0 P 2 -2V 1728 III;III E2 0 -22p 2 2 1 2 p 3 1728 15; n i FI 0 P3 2 - 2 6 p 9 1728 n i j i i r F2 0 - 2 V 2 21 2 p 9 1728 15; i n * 3. p - 1 zs a square and E is Q-isomorphic to one of the elliptic curves: «2 04 A j-invariant Kodaira GI 2Py/P - 1 -p2 2 2<y 64(4p - l )J P III;I* G2 -2'py/p - 1 2 2 p 3 2 - 2 1 2 p 8 64(4-p)J p* I5;i2 G I ' - 2 p ^ P " 1 - p 2 2 2 6 p 7 64(4p - l ) J P III;I* G2' 2'py/p - 1 2 2 p 3 2 - 2 1 2 p 8 64(4-p)J P 2 I5;i2 4. p — 8 z's a square and E is Q-isomorphic to one of the elliptic curves: 0-2 (24 A j-invariant Kodaira HI p V p - 8 - 2 p 2 2 2<y 64(p-2)J P III;I* H2 -2pyJp-% p 3 2 - 2 9 p 8 - 8 ( p - 3 2 )J P 2 15; 15 H I ' -P-y/P ~ 8 - 2 p2 2 2 6p 7 64(p-2)J V III;IJ H2 ' 2 P X /P - 8 p3 2 - 2 9 p 8 - 8 ( p - 3 2 )J P 2 T*- T* 5. p + 8 is a square and E is Q-isomorphic to one of the elliptic curves: a2 a.4 m\ A j-invariant Kodaira 11 Py/p + 8 2p2 2 2 6 p 7 64(p+2)J p III;I* 12 - 2 P v / p + 8 P3 2 29 p 8 8(p+32)J P 2 15; 15 11' -pVP + 8 2p2 2 2 6 p 7 64(p+2f V III;IJ 12' 2 P n / P + 8 p3 2 2 9 p 8 8(p+32)J P 2 T*. T* 1 0 ' 1 2 Theorem 6.7 The elliptic curves E/Q of conductor 64p 2 with a rational point of order 2 are the ones such that one of the following conditions is satisfied: Chapter 6. Classification of Elliptic Curves of conductor 2ap2 218 1. p = 5,7,17 and E is Q-isomorphic to one of the elliptic curves in the table in Appendix C. 2. p> 5 and E is Q-isomorphic to one of the elliptic curves: (i) p = 1 (mod 4), « 2 0,4 \T2\ A j-invariant Kodaira A l 0 -22p2 4 2 1 2 p 6 1728 15; lo A 2 0 P2 2 -26p6 1728 i i ; 15 A 3 12p 22p2 2 2 1 5 p 6 2 3 3 3 11 3 is; i$ A4 -12p 22p2 2 2 1 5 p 6 2 3 3 3 11 3 T* • T* BI 0 P 2 - 2 6 p 3 1728 II;III B2 0 -22p 2 2 1 2 p 3 1728 I 2 ;III C l 0 P3 2 - 2 V 1728 11; III* C2 0 - 2 V 2 2 'V 1728 I*,; III* (ii) p = 3 (mod 4), 0,2 0 4 m\ A j-invariant Kodaira D I 0 -22p2 4 212p6 1728 J * . T* D2 0 p2 2 - 2 G p 6 1728 i i ; 15 D3 12p 22p2 2 2 1 5 p 6 2 3 3 3 11 3 is;iS D4 -12p 22p2 2 2 1 5 p 6 2 3 3 3 11 3 1* • T* E l 0 -P 2 2 6 p 3 1728 II;III E2 0 22p 2 - 2 ] 2 p 2 1728 1*; i n F l 0 2 26p9 1728 n ; i i r F2 0 22p3 2 - 2 1 2 p 9 1728 i 5 ; i i r 3. p = 2k + 1, where k > 4, and is Q-isomorphic to one of the elliptic curves: Chapter 6. Classification of Elliptic Curves of conductor 2ap2 219 a-2 0 4 A j-invariant Kodaira G I 2p(2p- 1) 2 2 ( p - l ) p 3 2 2 f c + 1 0 p 8 (2f cp+l) 3 2 2 f c - 8 p 2 T * . T * x 0> x 2 G2 - 2 2 p ( 2 p - l ) 2 2 p 2 2k+14p7 (2f c + V+l) ; J 2 * - - i p T * . T * 1 f c + 4 ' 1 1 G3 p ( p + l ) 2 2 f c - 2 p 2 24k+2p7 ( p + 2 2 * - - ) 3 2 2 ( 2 f c - 8 ) p T* • T* - M f c - 8 ' 1 1 G4 2 2 p ( p - 2 ) 2 2 p 4 - 2 f c + 1 V ° (2t +' l -p 2 ) 3 2 f e-"p' 1 T* • T* G I ' - 2 p ( 2 p - l ) 2 2 ( p - l ) p 3 2 2 f c + 1 0 p 8 (2*p+lJ3_)l 2 2 f c - 8 p 2 1 0 ' 1 2 G2' 2 2 p ( 2 p - l ) 2 2 p 2 2 f c + 1 4 p 7 (2t+"p+l) : i T* • T* G3 ' - p ( p + l ) 2 2 f c - 2 p 2 2 4 f c + 2 p 7 (p+2 2 f e - " ) 3 2 2 ( 2 f c - 8 ) p T* • T* G4' -2 2 p(p - 2) 2 2 p 4 -2 f c + 1 V° (2f c + "-p 2 ) 3 2 f r - 4 p . l T* • T* 1 f c + 4 ' 4 |T 2 | = 4/or G2 and GI', and \T2\ = 2/or a// offers. 4. p = 2q — 1, where q > 3 is a prime, and E is Q-isomorphic to one of the elliptic curves: 0-2 a4 A j-invariant Kodaira H I 2p(2p+l) 2 2 (p+ l ) p 3 2 2 9 + 1 0 p 8 (2 ' 'p+l)J 2 2 , , - 8 p 2 J-O' 1 2 H2 -2 2 p(2p+ 1) 2 2 p 2 2 9 + 1 4 p 7 (2"+'1p+I)-) 2"->p T* • T* 1 < J + 4 I 1 1 H3 p(p - 1) 2 2 7 - 2 p 2 -24<>+2p7 (p-i2"-")-' 2 2 ( 2 , , - 8 ) p T* • T* H4 2 2 p(p+2) 2 2 p 4 2 , + 1 4 p 1 0 ( p 2 + 2 , , + . , ) a 2-1-1 pi T* • T* Lq+4 i 1 4 H I ' -2p(2p+l) 2 2 (p+ l ) p 3 2 2 "+ 1 0 p 8 (2"p+l )3 2 2 , , - 8 p 2 J^ O' x 2 H 2 ' 2 2p(2p+ 1) 2 2 p 2 2"+ 1 4p 7 (2''+"p+l)3 2''--ip T* • T* H 3 ' - p ( p - l ) 2 2 ' '~ 2 p 2 - 2 4 r ' + 2 p 7 ( p - 22 ' ' - " ) ' ! 2 2 ( 2 , , - 8 ) p T* • T* H4' -2 2 p(p + 2) 2 2 p 4 2 f '+ 1 4 p 1 0 (p2 + 2 " + 4 ) 3 2<y-4p-l T* • T* \T2\ = 4 for HI and HI', and \T2\ = 2 for all others. 5. p — 1 is a square and E is Q-isomorphic to one of the elliptic curves: a 2 a.4 A j-invariant Kodaira 11 2pv /p~ : :T p 3 2 -2 6 p 8 - 6 4 ( p - 4 )3 P 2 II; I* 12 -2 2 p 2 2 2 1 2p 7 64(4p - l ) J P 15; i t 11' -2 P v /p - 1 p 3 2 -2 6 p 8 - 6 4 ( p - 4 )3 P* II; 15 12' 2Wp - 1 -2 2 p 2 2 2 1 2p 7 64(4p - l ) J P 15; I I Chapter 6. Classification of Elliptic Curves of conductor 2ap2 220 6. there exists m > 2 such that p — 2m is a square and E is Q-isomorphic to one of the elliptic curves: « 2 0 4 \T2\ A j-invariant Kodaira JI 2p^p - 2m - 2 m p 2 2 2 2 m+V. 2 2 r , . - 1 2 p T* • T* J2 -2'py/p - 2 m 2 2 p 3 2 - 2 m + 1 2 p 8 (2"^-pf 2 " ' - 6 p 2 T* • T* 1 m + 2 > 1 2 JI' -2p^/p - 2m - 2 m p 2 2 2 2 m + 6 p 7 {r-r"-2r 2 2 m - 1 2 p T* • T* J2' 2ApyJp - 2m 22p3 2 - 2 m + 1 V ( 2 " ' +2 - p f 2 « . - 6 p 2 T* • T* 1 m + 2 i x 2 7. f/zere exists an odd integer m > 2 such that p + 2m is a square and E is Q-isomorphic to one of the elliptic curves: a2 0 4 \T2\ A j-invariant Kodaira KI 2Py/P + 2m 2mp2 2 2 2 m + 6 p 7 (p+2"'-'r 2 2 , „ - 1 2 p T* • T* K2 -2'pyfP+2m 2 2 p 3 2 2m+12p8 (P+2-+2)- 3 2 m - B p 2 T* • T* 1m+2> 1 2 K I ' -2Py/p+2m 2 m p 2 2 2 2 m + 6 p 7 ( p + 2 ' " ~2 ) a 2 2 „ . - 1 2 T* • T* K2' 2 zpv/p + 2 m 2 2 p 3 2 2»n+12p8 ( p +2" ' + 2 ) J 2 " — 6 p 2 T* • T* *-m+2i 1 2 8. zTzere exz'sfs an odd integer m > 3 swc/z £/za£ 2 m — p is a square and E is Q-isomorphic to one of the elliptic curves: (12 CI4 \T2\ A j-invariant Kodaira LI 2psJ2m - p 2mp2 2 -2 2 m +V ( p - 2 —2 ) 3 2 2 r n 1 2 p ^2m-4> II L2 -2zp^/2m - p -22p3 2 2 r n + 1 2 p 8 ( 2 » + ^ _ p ) 3 2 m - b p 2 T* • T* Am+2> 1 2 L I ' ~2p^2m - p 2mp2 2 - 2 2 m + V ( p - 2 ™ -2 ) J 2 2 , „ - 1 2 p T* • T* • • • 2 7 7 1 - 4 5 11 L2' 2zp^2m - p -22p3 2 2 m + 1 2 p 8 ( 2 ' " + 2 - p ) 3 2 m - 6 p 2 T* • T* I 7 7 l + 2 ! X2 9. there exists m > 3 such that 2"'p 1 is a square and E-is Q-isomorphic to one of the elliptic curves: Chapter 6. Classifica tion of Elliptic Curves of conductor 2ap2 221 a-2 0 4 A j-invariant Kodaira M l 2mp _ 2 2m+6 p 3 _ ( 2 m - 2 _ 1 ) 3 Zp\J V 2 2 , n - 1 2 M 2 -22p 2 m+12 p 3 ( 2 r „ + 2 _ 1 ) 3 2 m - G i™+2;ni M l ' 2mp _ 22m+6p3 - ( 2 " ' -2 - l ) 3 2 2 m - 1 2 l2m-4) HI M 2 ' -22p 2 m + 1 2 p 3 (2"'+2-l) 3 2m-6 Im+2> H I N I o „ 2 / 2 " ' - l 2mp3 - 2 2 m + V - (2" '~ 2 - l ) 3 l2m.-4; H I 2P V V 2 2 r „ - 1 2 N 2 - 2 2 p 2 ^ - 2 2 p 3 2m+12p9 ( 2m + 2 _ 1 ) 3 2 m - 6 i™+2 ;nr N I ' o „ 2 / 2 ' " - l 2mp3 - 2 2 m + V - ( 2 " -2 - l ) 3 ^2m-4> H I 2 2 m - 1 2 N 2 ' - 2 2 p 3 2 m+12 p 9 (2"'+2_l)3 2 m - « Im+2; H I |T21 = 2/or a/Z f/tese curves. 10. there exists rn > 2 swc/z i/zaf 2 m p + 1 zs A square and E is Q-isomorphic to one of the elliptic curves: 0 2 0,4 |T 2 | A j-invariant Kodaira 0 1 2mp 2 2 2 m + 6 p 3 ( 2 m - 2 _ 1 ) 3 2 2 , r , - 1 2 ^2m-4'' HI 0 2 22p 2 2 m + 1 2 p 3 ( 2 " ' +2 + l ) 3 2,..-(5 I 7 W 2 ; in or 2mp 2 2 2 7 7 i + 6 p 3 ( 2 " —2 - l ) 3 2 2 m - 1 2 I2777 . -4; in 0 2 ' 2 2 p ^ 22p 2 2 " ' + 1 V ( 2 ' "+ 2 + l ) 3 2 „ , - o 17*77+2; in PI 2mp3 2 2 2 7 f i + 6 p 9 ( 2 " -2 - l ) 3 2 2 , „ - 1 2 I2777 . -4; in P2 22p3 2 • 2 m + 1 V ( 2 " '+ 2 + l ) 3 2 " ' - e T* TTT* 1 7 7 l + 2 ' 1 1 1 PI ' - 2 p 2 ^ 2mp3 2 2 277 i+6p9 ( 2 " - 2 - l ) 3 2 2 m - 1 2 T* TTT* I 2 7 7 7 . - 4 1 1 1 1 P 2 ' 2 2 p 2 ^ 22p3 2 2 m + 1 2 p 9 ( 2 " ' + 2 + l )3 2 m - 6 1 ^ + 2 ; nr Theorem 6.8 The elliptic curves E/Q of conductor 128p2 with a rational point of order 2 are the ones such that one of the following conditions is satisfied: 1. p = 13 and E is Q-isomorphic to one of the elliptic curves in the table in Appendix C. 2. p is a prime > 5 and E is Q-isomorphic to one of the elliptic curves: Chapter 6. Classification of Elliptic Curves of conductor 2ap2 222 a2 0,4 \T2\ A j-invariant Kodaira A l 2p 2p2 2 - 2 8 p 6 27 III; 15 A2 -22p -22p2 2 2 1 3 p 6 2 5 7 3 15; 15 A l ' -2p 2p2 2 - 2 8 p 6 2 7 n i ; 15 A2' 22p -2 2 p 2 2 2 1 3 p 6 2 57 3 T*. T* BI 2p -p2 2 2 7p 6 2 57 3 III;15 B2 -2 2 P 2 3p 2 2 - 2 1 4 p 6 2 7 15; 15 BI' -2p -P2 2 2 7p 6 2 5 ? 3 n i ; 15 B2' 22p 2 3p 2 2 - 2 1 4 p 6 2 7 15; 15 3. 2p — 1 is a square, p = 1 (mod 4), p ^  13 and E is Q-isomorphic to one of the elliptic curves: a2 04 \T2\ A j-invariant Kodaira C l . 2p 3 2 - 2 8 p 8 - 1 2 8 ( p - 2 )J Hi; 15 2pV2p - 1 P» C2 - 2 W 2 p " 1 - 2 2 p 2 2 2 1 3 p 7 3 2 ( 8 p - l ) J P 15; II C l ' - 2 P v / 2 p - 1 2p 3 2 - 2 8 p 8 - 2 5 6 ( p - 2 )J P 2 n i ; 15 C2' 2 W 2 p - 1 - 2 2 p 2 2 2 1 3 p 7 3 2 ( 8 p - l ) ' J P 15; II DI 2Py/2p - 1 - P 2 2 2 7 p 7 3 2 ( 8 p - l ) J P i i ; II D2 o 2 /o -\ 2 3 p 3 2 - 2 1 4 p 8 - 1 2 8 ( p - 2 ) J in*; 15 -2'pyJ2p - 1 P 2 DI' -2psJ2p - 1 - p 2 2 2 7 p 7 3 2 ( 8 p - l ) J E ! — II; II D2' 2'pV2p - 1 2 3 p 3 2 - 2 1 4 p 8 - 1 2 8 ( p - 2 )J P 2 in*; 15 4. 2p2 — lis a square, p = 1 (mod 4), and E is Q-isomorphic to one of the elliptic curves: a2 04 \T2\ A j-invariant Kodaira E l 2Py/2p2 - 1 2p4 2 - 2 8 p 1 0 - 1 2 8 ( p 2 - 2 ) J p* n i ; I* E2 -2W2p2 - 1 -2 2 p 2 2 2 1 3 p 8 3 2 ( 8 p 2 - l ) : i P* ± 2 , ± 2 E l ' ' - 2 P x / 2 p 2 - 1 2p4 2 - 2 8 p 1 0 - 1 2 8 ( p * - 2 )a p" i l l ; I* E2' 2 2p x/2p 2 - 1 - 2 2 p 2 2 2 1 3 p 8 3 2 ( 8 p2 - l ) J P 2 15; 15 F l 2p^2p 2 - 1 - p 2 2 2 7p 8 3 2 ( 8 p2 - l ) J P 2 i i ; 15 F2 -2 2 p V / 2p 2 - 1 2 3p 4 2 - 2 1 4 p 1 0 - 1 2 8 ( p ' : - 2 ) J p" m*; I* F l ' - 2 P V V - 1 - P 2 2 2 7p 8 3 2 ( 8 p2 - l ) J P 2 i i ; 15 F2' 22pv/2p2 - 1 2 3p 4 2 - 2 1 4 p 1 0 - 1 2 8 ( p2 - 2 ) J P 4 IH*; I* Chapter 6. Classification of Elliptic Curves of conductor 2ap2 223 5. p — 2 is a square and E is Q-isomorphic to one of the elliptic curves: « 2 0 4 \T2\ A j-invariant Kodaira G I 2py/p - 2 P3 2 - 2 V - 3 2 ( p - 8 )a P 2 II; 15 G2 -2'p^p - 2 -23p2 2 214p7 128(2p- l ) J P III*;I* G I ' -2py/p- 2 P3 2 - 2 V - 3 2 ( p - 8 )J P 2 II; n G2' 2 W P - 2 -23p2 2 214p7 128(2p- l )J P III*;II H I 2Ps/p - 2 - 2 p 2 2 2V 128(2p- l )J P I I I ; I I H2 -2'psjp - 2 22p3 2 -21 3p8 -32(p-8) ' ) P 2 T*- T* H I ' -2Py/p-2 -2p2 2 28p7 128(2p- l )J P I I I ; I I H 2 ' 2 W P - 2 22p3 2 -21 3p8 -32(p-8) 'i P 2 T*- T* x2i *2 6. p" + 2 fs a square for some integer n > 1 and E is Q-isomorphic to one of the elliptic curves: 0.2 0 4 \T2\ A j-invariant Kodaira 11 2py/pn + 2 pn+2 2 27p2"+6 3 2 ( p " + 8 ) 3 P 2 " n ; i 2 n 12 -2'pyjpn + 2 2 3p 2 2 1 4p n+ 6 1 2 8 ( 2 p " + l )3 p™ m * ; i ; 11' -2p^/pn + 2 p n + 2 2 2 7p 2 n+ 6 3 2 ( p T ' + 8 ) 3 p 2 " n ; i 2 „ 12' 2zpv/p" + 2 2 3p 2 2 1 4 p n + 6 128(2p" + l )J p™ m * ; i ; J I 2p\Jpn + 2 2p2 28p"+6 128(2pT' + l ) J p" I I L I ; J2 -22Pv?pn + 2 2 2 p n + 2 2 213^271+6 3 2 ( p " + 8 ) 'J P 2 " T*. T* A 2 i 1 2 n J I ' -2Py/pn + 2 2p2 28p"+6 128(2p" + l )J p" i n r J2' 2 W P n + 2 22p"+2 2 2 1 3 p 2 n + 6 3 2 ( p " + 8 )J p 2 " T*- T* x 2 i I 2 n Theorem 6.9 T/ie elliptic curves E/Q of conductor 256p2 with a rational point of order 2 are the ones such that one of the following conditions is satisfied: 1. p — 23 and E is Q-isomorphic to one of the elliptic curves in the table in Appendix C. 2. p is a prime > 5 and E is Q-isomorphic to one of the elliptic curves: Chapter 6. Classification of Elliptic Curves of conductor 2ap2 224 a-2 0,4 \T2\ A j-invariant Kodaira A l 0 2p 2 -2V 1728 III;III A 2 0 -23p 2 2 1 5 p 3 1728 III*;III B l 0 -2p 2 2 9 p 3 1728 III;III B2 0 23p 2 - 2 1 5 p 3 1728 III*;III C I 0 2p2 2 -2V 1728 III;15 C2 0 - 2 3 p 2 2 2 1 5 / 1728 HI*; 15 D I 0 -2p2 2 29p6 1728 III;15 D2 0 23p2 2 - 2 1 5 p 6 1728 IU*; 15 E l 0 2p3 2 -2V 1728 III;III* E2 0 - 2 3 p 3 2 2 1 5 p 9 1728 III*;III* FI 0 -2p3 2 2 9 p 9 1728 III; III* F2 0 23p3 2 - 2 1 5 p 9 1728 III*;III* G I 22p 2p2 2 2 9 p 6 2 6 5 3 III;15 G2 -23p 23p2 2 2 1 5 p 6 2 6 5 3 III*;15 G I ' -22p 2p2 2 2 9 p 6 2 6 5 3 III;15 G2' 23p 23p2 2 2 1 5 p 6 2 6 5 3 III*;15 3. p-^- is a square and E is Q-isomorphic to one of the elliptic curves: 0,2 (X4 X?2\ A j-invariant Kodaira H I 2p3 2 -2V - 6 4 ( p - 4 ) 3 P 2 III;I 2 H2 - 2 V ^ - 2 V 2 2 i j y 6 4 ( 4 p - l )3 P in*; it H I ' -aV*? 2p3 2 -2 9p 8 - 6 4 ( p - 4 )3 P 2 III;15 H 2 ' 2 V ^ -2V 2 215p7 6 4 ( 4 p - l )3 P III*;It 11 -2p2 2 2V 6 4 ( 4 p - l )3 P III;It 12 23p3 2 -215p8 - 6 4 ( p - 4 ) 3 p 2 III*;15 11' - 2 V ^ -2p2 2 29p7 6 4 ( 4 p - l ) 3 P III;It 12' 2 3 V 2 T 1 23p3 2 -2 1 5 p 8 - 6 4 ( p - 4 )3 P 2 HI*; 15 4. ^ - j — z's A square and E is Q-isomorphic to one of the elliptic curves: Chapter 6. Classification of Elliptic Curves of conductor 2ap2 225 a-2 a 4 1^ 21 A j-invariant Kodaira JI 2p4 2 - 2 V ° 6 4 ( p 2 - 4 ) 3 P-I III;IS J2 - 2 V 2 2 1 5 p 8 6 4 ( 4 p2 - l ) 3 P2 in*; 15 JI ' 2 / 2 - 2 V ° 6 4 ( p 2 - 4 ) 3 P4 III;12 J2' 2 V ^ - 2 V 2 2 1 5 p 8 6 4 ( 4 p 2 - l ) 3 P2 III*; K I -2p 2 2 2 9 p 8 6 4 ( 4 p2 - l ) 3 P2 III; 15 K2 2 3 p 4 2 _ 2 i 5 p i o 6 4 ( p2 - 4 ) 3 P4 III*;II K I ' -2p 2 2 2 9 p 8 6 4 ( 4 p2 - l ) 3 P2 III;15 K2' 2V 2 _ 2 1 5 p 1 0 6 4 ( p2 - 4 ) 3 P4 in*; 13 5. PJY~ is a square and E is Q-isomorphic to one of the elliptic curves: a 2 04 \Ti I A j-invariant Kodaira L I 2p 3 2 2 9 p 8 64(p+4) 3 P2 III;15 L 2 - 2 3 p ^ 2 3 p 2 2 215p7 64(4p+l)3 P in*; i* L I ' 2p 3 2 2 9 p 8 64(p+4)3 pi III;15 L 2 ' 2 3 p 2 2 2 1 5 p 7 64(4p+l) 3 P III*;I* M l 2p 2 2 2 9 p 7 64(4p+l)3 P III;I* M 2 2 3 p 3 2 2 1 5 p 8 64(p+4) 3 P2 HI*; 15 M l ' - 2 2 P v / £ ± i 2p 2 2 2 9 p 7 64(4p+l) 3 P ni; I* M 2 ' 2 3 P x / ^ 2 3 p 3 2 2 1 5 p 8 64(p+4) 3 P2 in*; 15 6. 2-^- zs a square and E is Q-isomorphic to one of the elliptic curves: Chapter 6. Classification of Elliptic Curves of conductor 2ap2 226 a 4 122 | A j-invariant Kodaira NI 2p4 2 2V° 6 4 ( p2 + 4 ) 3 p" III;12 N2 2 V 2 2 1 5 p 8 6 4 ( 4 p 2 + l)3 P 2 in*; 1*2 N I ' 2p4 2 2V° 6 4 ( p2 + 4 ) 3 p-i III;13 N2' 2 V 2 2 1 5 p 8 6 4 ( 4 p2 + l)3 P 2 III*;IJ O l 2p2 2 2 9p 8 6 4 ( 4 p2 + l)3 P 2 III;I? 02 2 V 2 2 1 5 p 1 0 6 4 ( p2 + 4 ) 3 P 4 n r ; i ; o r 2p2 2 2 9p 8 6 4 ( 4 p2 + l)3 P 2 III;15 02' aW 1 ^ 2 V 2 2 1 5 p 1 0 6 4 ( p2 + 4 ) 3 p-i III*; I* In Chapter 8 we wi l l be interested in knowing, up to isogeny, the elliptic curves with conductor of the form 32p2 or 256p2, and their j'-invariants. We have the following corollaries to Theorems 6.6 and 6.9. Corollary 6.10 Suppose p > 5 is prime and that E/Q is an elliptic curve with a rational 2-torsion point and conductor 32p2. Then E is isogenous over Q to a curve of the form y2 — x3 4- a2x2 + C14X with coefficients given in the following table. p 0-2 0,4 j-invariant any 0 2 —p 1728 any 0 (_l)(p+l)/2 p 1728 any 0 (_ l ) (P+l ) /2 p 3 1728 7 ± 7 2 - 7 2 8000/7 7 ± 7 2 • 7 - 2 6 7 ± 7 2 2 • 7 3 - 2 6 s 2 + i , s e z 2ps - p 2 64(4p-l)a P s 2 + 8, s e Z ps - 2 p 2 64(p-2)a P s2 - 8 , s <E Z ps 2p 2 64(p+2)a P Chapter 6. Classification of Elliptic Curves of conductor 2ap2 227 Corollary 6.11 Suppose p > 5 is prime and that E/Q is an elliptic curve with a rational 2-torsion point and conductor 256p2. Then E is isogenous over Qtoa curve of the form y2 = x 3 + a2x2 + a4x with coefficients given in the follozuing table. p « 2 0,4 j-invariant any 0 ±2p 1728 any 0 ±2p2 1728 any 0 ±2p3 1728 any ±4p 2p2 2 6 5 3 23 ± 2 3 -23-39 2 • 23 5 2b3 a4057 a 23« 23 ± 2 4 -23-39 2 3 • 23 5 2b334057 i i 23« 2s 2 + 1, s G Z ±4ps 2p3 -64(p-4)3 P 4 2s 2 + 1, s e z ±4ps -2p2 64(4p-l)-s P V 2 s 2 + 1, s £ Z ±4ps 2pA 64(p2-4)- i " P^ s/2s2 + 1, s G Z ±4ps -2p2 64(4p2 - l ) a p 2 2s 2 - 1, s G Z ±4ps 2p3 64(p+4)a P 2 2s 2 — 1, s G Z ±4ps 2p2 ? V 2 s 2 — 1, s G Z ±4ps 2pA _ 64(p2+4)J P 4 \ /2s 2 - 1, s G Z ±4ps 2p2 64(4p1! + I)-5 P* 6.2 T h e P r o o f We wi l l only sketch the proof of Theorem 6.2, it should be clear from this how the proofs of the remaining theorems follow from their counterparts in Section 3.1.1 and the Diophantine lemmata of Chapter 4. Let E be an elliptic curve over Q with a rational 2-torsion point and con-ductor N = 2p2 for a fixed prime p > 5. Then E is Q-isomorphic to one of the curves in Theorem 3.2 and p satisfies one of the corresponding Diophantine equations: 1) d2 = 2mpn + 1, 2) d2 = 2rn + pn, Chapter 6. Classification of Elliptic Curves of conductor 2ap2 228 3) d2 = 2m - pn, 4) d2 = p n - 2m, 5) pd2 = 2m + 1, 6) pd2 = 2 m - l , with n > 0 and m > 7. Applying the Diophantine lemmata from Chapter 4, the solutions of these are respectively as follows: 1) (p, d, n) = ( 2 m - 2 + 1, 2p - 1,1) and (p, d, n) = {2m~2 - 1, 2p + 1,1), 2) (p,d,m,n) = (17,71,7,3), (p,d,n) = ( 2 m " 2 - l , p + 2, 2), and solutions with n — 1, 3) (p, d, m, n) = (7,13.9,3), and solutions with n = 1, 4) (p, d, n) = ( 2 m _ 2 + l , p - 2,2), and solutions with n = 1. 5) ^ is a square, 6) is a square. Thus, p must satisfy one of these conditions. Suppose p satisfies the first con-dition in 1, that is p = 27"-~2 +1, d = 2p — 1, and n = 1. Then £ is Q-isomorphic to one of y2 = x 3 + p(2p - l ) x 2 + 2 m " 2 p 3 x , y 2 = x 3 - 2p(2p - l ) x 2 + p2x, by part (1) of Theorem 3.2 (neither curve is minimal at 2). The minimal models of these curves can be computed using Corollary 2.2: 2 3 p (2p - 1) - 1 2 p 3 ( p - 1) y2 + xy = x3 + 4 X + 16 ' 2 ^ 3 , p(2p - 1) - 1 2 , -p 3 (p - 1) , - p 4 ( p - l ) ( 2 p - l ) y +xy = x + x + x+ — . Thus E is isomorphic to either A l or A2 in Theorem 6.2. Suppose that p satisfies the first condition in 4, that is, p = 2 m ~ 2 + 1, d = p — 2, and n = 2. Then a similar argument shows that E is isomorphic to either BI or B2 in Theorem 6.2. Similarly, one can verify the rest of Theorem 6.2 by considering p of each form in 1 through 6 listed above. This completes the sketch of the proof for these tables. Chapter 7 O n t h e C l a s s i f i c a t i o n o f E l l i p t i c C u r v e s o v e r Q w i t h 2 - t o r s i o n a n d c o n d u c t o r 2a32p A more appropriate title for this chapter would be "Classification of primes for which there exist elliptic curves over Q with 2-torsion and conductor 2 a 3 2 p with a G {1,2,3}", since it is the collection of primes we wi l l be study-ing, not the curves themselves. The tables in Section 3.2 provide a classifi-cation of elliptic curves of conductor 2 a 3 2 p in which the prime p must sat-isfy one of a list of Diophantine equations. In this chapter, we use the lem-mata of Chapter 4 to resolve all the Diophantine equations which occurred. Hence, we can list, rather explicitly, all the primes that can occur. In Chapter 9, we wi l l be interested in the primes for which there are no elliptic curves of conductor 2a32p, with a G {1,2,3}. Our main focus here wi l l be to "de-termine properties of this set of primes. We wi l l show that for all primes p = 317 or 1757 (mod 2040) there are no elliptic curves with 2-torsion and conductor 2 Q 3 2 p with a G {1, 2, 3}. 7.1 S t a t e m e n t o f R e s u l t s Theorem 7.1 The primes pfor which there exists an elliptic curve E/Q of conductor 18p, and having at least one rational point of order 2, satisfy one of the following: 1. pe {5,7,11,17,19,23,73}; 2. p = 2 m _ 2 3 ' ± 1 with m>7,e>0; 3. p= 2 ' " " 2 + 1 with m>7J>l; 229 Chapter 7. On the Classification of Elliptic Curves of conductor 2 Q 3 2 p 230 4. P = 3e + 2 m - 2 w i t h m>7/£>0; 5. P = 3£ _ 2m-2 w i t h m > 77 ^ > 0; 6. P = 2m - 2 _ 3<? w U h m > jf £ > Q ; 7. P = d2 + 2m3* with m>7 and £ > 0; 8. P = d2 - 2m3e with m>7and£> 0; 9. P = 2m3e - d2 with m>7and£> 0; 10. P = d2+2"1 with rn>7,£> 1; 11. P = 3d2 + 2m with m > 7; 12. P = 3d2 - 2m with m > 7; 13. P = 2m - 3d2 with m > 7; Theorem 7.2 The primes pfor which there exists an elliptic curve E/Q of conductor 36p, and having at least one rational point of order 2, satisfy one of the following: 1. {5,13}; 2. p n = d2 + 4 • 3e with £>0 even, n = lor Pmin{n) > 7; 3. pn = d2 - 4 • 3* with £ > 1 odd, n = 1 or Pmm(n) > 7; 4. pn = 4 • 3e - d2 with £ > 1 odd, n = 1 or P m i „ ( n ) > 7; 5. p" = 4_±3i £ > l odd, n = 1 or Pmm{n) > 7, p = - 1 (mod 4); 6. 4p n = 3d 2 + 1 with n G {1,2} andpn = 1 (mod 4); 7. p = 3d 2 - 4; Theorem 7.3 The primes pfor which there exists an elliptic curve E/Q of conductor 72p, and having at least one rational point of order 2, satisfy one of the following: 1. pe {5,7,13,23,29,31,47,67,73,193,1153}; 2. p = 3f±I with £ odd; Chapter 7. On the Classification of Elliptic Curves of conductor 2 a 3 2 p 231 3. pn = a? + 4 • 3e with £>1 odd, n = 1 or P m i n ( n ) > 7; 4. p n = d2 _ 4 • 3^  with l>0 even, n = lor P m i n ( n ) > 7; 5. p™ = 4 • 3 £ - d2 with £>0 even, n = 1 or P m i n ( n ) > 7; 6. p = 2 m " 2 3 f ± 1 with m e {4,5}, * > 0; 7. p n = d2 + 2m • 3e with m <E {4 ,5}, £ > 0, n = 1 or P m i n ( n ) > 7; 8. pn = d2 - 2m • 3e with m e { 4 , 5 } , l > 0 , n = l or P m i n ( n ) > 7; 9. p n = 2rn • 3e - d2 with m G { 4 , 5 } , £ > 0, n = 1 or Pmin{n) > 7; 10. pn = wif/t £ odd, p = 1 (mod 12), n = 1 or P m i „ ( n ) > 7; 22. p = 3e±4with£> 0; 22. p = 3' ± 8 with ^ > 0; 13. p = with rn <E {4 , 5} and £ odd; 14. pn = with £ odd, n=lor P m i l l (n ) > 7; 15. p = to; W. p2 = to; 27. p = 3d2 - 2rn with m e {4 , 5}; 28. p = 3d2 + 2rn with m € { 2 , 4 , 5 } . Corollary7.4 Letp>5beaprime. 2. If there exists an elliptic curve over Q with 2-torsion and conductor 18p then one of the following must hold: p = 5, p ^ 2 (mod 3), p ^ 2 (mod 5), p ^ 5 (mod 8), or p 6 and 11 (mod 17). 2. If there exists an elliptic curve over Q with 2-torsion and conductor 36p then one of the follozving must hold: p — 5, p ^ 2 (mod 3), orp^l and 5 (mod 8). Chapter 7. On the Classification of Elliptic Curves of conductor 2a32p 232 3. If there exists an elliptic curve over Q with 2-torsion and conductor 72p then one of the following must hold: p = 5, p = 29, p ^ 2 (mod 3), or p ^ 5 (mod 8). It follows that there are no elliptic curves over Q with 2-torsion and conductor 2a9p, where a £ {1,2,3}, for p satisfying p = 317 or 1757 (mod 2040) (i.e. p = 5 (mod 8),p= 2 (mod 3),p = 2 (mod 5), and p = 6 or 11 (mod 17)). By Dirichlet's theorem on primes in arithmetic progression, we have that there are infinitely many primes p for which there are no elliptic curves over Q with 2-torsion and conductor 2a9p, with a £ {1, 2,3}; since primes congruent to 317 or 1757 modulo 2040 have this property. Though this list is infinite, it misses a lot of primes with the property. Indeed, a quick search through Cre-mona's tables of elliptic curves up to conductor 130000 reveals the following list of the first few primes: 197, 317, 439,557, 653, 677,701,773,797, 821,1013,1039, 1061,1109,1231,1277,1279,1289,1301,1399,1447,1471,1493 1613,1637,1663,1709,1733. Let 5 denote set of primes which satisfy one of the forms in the statements of Theorems 7.1, 7.2, and 7.3. We would like to show that S has density zero in the set of all primes. By this we mean, if #S(X) is the number of primes in S less than X then where TT(X) is the number of all primes less than X. Determining the density of primes of the form p = is somewhat problematic. So, let S' denote set of primes which satisfy one of the^forms in the statements of Theorems 7.1, 7.2, and 7.3, except pn = Also, let S'(X) = {p £ S' : p < X}. We prove the following. Lemma 7.5 Chapter 7. On the Classification of Elliptic Curves of conductor 2a32p 233 7.2 T h e P r o o f s 7.2.1 Proof of Theorem 7.1 We proceed through the cases of Theorem 7.1 (with b = 2) and use the lem-mata of Chapter 4 to resolve the Diophantine equations that arise. Notice that in all cases we are only concerned with solutions to the Diophantine equa-tions with m > 7. In what follows, by "solvable", we mean there are solutions for which m > 7,1 > 0 and n > 1 1) According to Lemma 4.7 if d2 = 2m3epn + 1 is solvable then the prime p is of one of the following forms o m - 2 I 1 p = 2m~23e ± 1, p = ¥ — , orp= 17. 2) If d2 = 2m3e + pn is solvable then the prime p is of one of the following forms (see Lemma 4.9): d2 - 2m3e, 2m'23e - 1, 3e - 2rn~2, 21"-2 - 3e, 5, 7, 17, or 73. 3) If d? = 2T"3 f — pn is solvable then the prime p is of one of the following forms (see Lemma 4.9): 7, 23, or 2 " ^ - ^ . 4) If d2 = 2r"pn + 3e is solvable then the prime p is of one of the following forms (see Lemma 4.8): 2 m - 2 ± 3e/2_ o r 5_ 5) If d2 — 2™ + 3epn is solvable then the prime p is of one of the following forms (see Lemma 4.10): y—^, 3( ± 2 m ' 2 + \ d2 - 2m, 7, 2m~2 - 1, or 17. 3^  6) If d2 = 2 m — 3 V is solvable then the prime p is of one of the following forms (see Lemma 4.10): 2 r „ / 2 + l _ 3^ 2m _ ^2^ ^ Q r ? Chapter 7. On the Classification of Elliptic Curves of conductor 2a32p 234 7) If d? = 3epn — 2rn is solvable then the prime p is of one of the following forms (see lemma 4.10): 2 m + l + 1 d2 + 2m 3e/2 » n> 17> 1 9 ' or—#—' 8) If d2 = 3e — 2mpn is solvable then the prime p is of one of the following forms (see Lemma 4.8): 3 ^ 2 - 2 m " 2 , or 7. 9) If d2 = pn - 2rn3e is solvable then the prime p is of one of the following forms (see Lemma 4.9): 2m-23e + l, 3e + 2m~2, 17, or 2m3e + d2. 10) If 3d2 = 2rn + pn is solvable then the prime p is of one of the following forms (see Lemma 4.11): 11, or 3d2 - 2m. 11) If 3d2 = 2rn - pn is solvable then the prime p is of one of the following forms (see Lemma 4.11): 5 , o r 2 m - 3 d 2 . 12) If 3d2 = pn - 2rn is solvable then the prime p is of the form 3d2 + 2m (see Lemma 4.11). This proves Theorem 7.1. 7.2.2 Proof of Theorem 7.2 Again, we proceed through the cases of Theorem 7.2 (with b = 2) and use the lemmata of Chapter 4 to resolve the Diophantine equations that arise. In all cases, we are only concerned with solutions to the Diophantine equations with m = 2. In what follows, by "solvable", we mean there are solutions for which I > 0 and n > 1 1) If d2 = 4 • 3e + pn is solvable then the prime p is of one of the following forms (see Lemma 4.9): 13, d2-4-3e, orpn = d2-4-3e with P m i n ( n ) > 7. 2) If d2 = 4 • 3e - pn is solvable then n = 1 or P m i n (n) > 7 (see Lemma 4.9). Chapter 7. On the Classification of Elliptic Curves of conductor 2a32p 235 3) If d2 = 4pn - 3^  is solvable then n = 1 or P m i n ( n ) > 7 (see Lemma 4.8) and p = — 1 (mod 4). 4) If d2 = pn — 4 • 3e is solvable then either p = 5, n = 1 or P m j n (n) > 7 (see Lemma 4.9). 5) If 3d2 = 4pn - 1 is solvable then n 6 {1,2} (see Lemma 4.11). 6) If 3d2 = pn + 4 is solvable then n = 1 so p = 3d 2 — 4 (see Lemma 4.11). This proves Theorem 7.2. 7.2.3 Proof of Theorem 7.3 Again, we proceed through the cases of Theorem 7.3 (with 6 = 2) and use the Diophantine lemmata. In all cases, we are only concerned with solutions to the Diophantine equations with m = 2,4,5. So, by "solvable", we mean there are solutions for which m G {4, 5}, £ > 0 and n > 1 1) If d2 = 2rn3epn + 1 is solvable then the prime p is of one of the following forms (see Lemma 4.7): 2 m - 2 3 ' ± l , or 5. 2) If d2 = 4 • 3^  + pn is solvable with £ even then the prime p is of the form pn = d2 -4- 3e with n = 1 or P m i „ ( n ) > 7 (see Lemma 4.9). 3) If d2 = 2rn3e + pn is solvable then the prime p is of one of the following forms (see Lemma 4.9): ^2 _ 2m3^ 2rn~23^ 1 3^  <2rn~2 2,m~2 — 3^  or pn = d2 - 2m3e with P r a i n (n ) > 7. 4) If d2 = 4 • 3^  - pn is solvable with £ even then n = 1 or Pmj n(?i) > 7 (see Lemma 4.9). 5) If d2 = 2™3i—pn is solvable then one of the following must hold: p = 47, n = 1 or P m i n ( n ) > 7 (see Lemma 4.9). 6) If d2 = 2mpn + 3' is solvable then the prime p is of one of the following forms (see Lemma 4.9): 3 ^ + 1 f 2 m - 2 + 3 / / 2 j 5 ) o r 7 . Chapter 7. On the Classification of Elliptic Curves of conductor 2a32p 236 7) If d2 = Apn — 3e with p = 1 (mod 4) is solvable then the prime p is of the form or n = 1 or P m i n (n) > 7 (see Lemma 4.8). 8) If d2 = 4 + 3 V 1 is solvable then the prime p is of one of the following forms (see Lemma 4.10): 3 < ± 1 5 or — - — . 4 9) If d2 = 2m + 3ipn is solvable then the prime p is of one of the following forms (see Lemma 4.10): d2 - 32, 3e ± 8, or 7. 10) If d2 = 2m — 3epn is solvable then the prime p is of one of 5, 7, 23, or 31. 11) If d2 = 3epn - 2m is solvable then the prime p is of one of the following forms (see Lemma 4.10): 2 m - i + 1 3e/2 > 5> 67> or n = 1 or P m i„(n) > 7. 12) If d2 = 3e - 2rnpn is solvable then the prime p is of the form 3(l2 - 2m~2 (see Lemma 4.8). 13) If d2 — Apn - 3e, with t odd, is solvable then p = 13 or n = 1 or Pminfa) > 7 (see Lemma 4.8). 14) If d 2 = pn — 2Tn3e is solvable then the prime p is of one of the following forms (see Lemma 4.9): 2 m _ 2 3 ' + l , 3^  + 2 m " 2 , 72, 193, 1153, 5, or n = 1 or Pmm (n) > 7. 15) If 3ti 2 = 2m +pn is solvable then n - 1 and so p = 3d 2 - 2 m (see Lemma 4.11). 16) If 3d2 = 2m +pn is solvable p is either 5,13 or 29 (see Lemma 4.11). 17) If 3d2 = Apn - 1 is solvable then n 6 {1, 2} so p = ^<£±1 o r p2 = M j t l (see Lemma 4.11). 18) If 3d2 = pn - A is solvable then n = 1 so p = 3d2 + A (see Lemma 4.11). 19) If 3d2 =pn - 2rn is solvable then n = 1 so p = 3d2 + 16 or p = 3d2 + 32 (see Lemma 4.11). This proves Theorem 7.3. Chapter 7. On the Classification of Elliptic Curves of conductor 2 a3 2p 237 7.2.4 Proof of Corollary 7.4 We show that all the primes appearing in Theorems 7.1, 7.2 and 7.3 satisfy at least one of p ^ 5 (mod 8), p ^ 2 (mod 3), p ^ 2 (mod 5), or p ^ 6 and 11 (mod 17) (7-1) This wi l l prove the corollary. Theorem 7.1: Certainly the primes in (1) satisfy (7.1). Primes of the form (2) satisfy p = ±1 (mod 8) and primes of the form (3), (4) or (5) satisfy p = 1 or 3 (mod 8). Primes of the form (7) or (8) satisfy p = 1 (mod 8) and primes of the form (9) satisfy p = — 1 (mod 8). Primes of the form (10) satisfy p = 1 or 3 (mod 8) and primes of the form (11) or (12) satisfy satisfy p = 3 (mod 8). A l l that remains is to consider primes of the form (6) and (13) and show they satisfy at least one of the congruences in 7.1. Suppose p is a prime of the form p = 2m~2 — 3e with m > 7 and £ > 1. If m and £ are both even then p is a difference of squares from which we find p = 7. If rn is even and £ is odd then p = 1 (mod 3) and p = 1 (mod A), lim is odd and £ is even then p = 2 (mod 3) and p = — 1 (mod A), lim and £ both odd then p = 2 (mod 3) and p = 5 (mod 8) so we need to consider the congruence class of p modulo 5, which is as follows: if rn -2 = 1 (mod A),£ = 1 (mod 4) thenp = 2 - 3 = - 1 (mod 5); if rn -2 = 1 (mod-4), £ = 3 (mod 4) then p = 2 - 2 = 0 (mod 5); if m - 2 = 3 (mod A),£ = l (mod 4) thenp = 3 - 3 = 0 (mod 5); if m - 2 = 3 (mod 4), £ = 3 (mod 4) then p = 3-2 = 1 (mod 5). Thus, p of the form (6) satisfies one of the congruences in 7.1. Suppose p is a prime of the form p = 2 m — 3d2 with- m > 7. Modulo 3 we have 1 (mod 3) if m is even, 2 (mod 3) if m is odd. Chapter 7. On the Classification of Elliptic Curves of conductor 2a32p 238 So, assume m is odd. In this case we have 3 (mod 4) and d 2 3 (mod 4) and d 2 3 (mod 4) and d 2 1 (mod 4) and d2 1 (mod 4) and d 2 1 (mod 4) and d 2 3 (mod 5) 0 (mod 5) 0 (mod 5) 2 (mod 5) 4 (mod 5) 4 (mod 5) if m 0 (mod 5), 1 (mod 5), 4 (mod 5), 0 (mod 5), 1 (mod 5), 4 (mod 5), if m if m P = if m if rn if m Thus, the only trouble seems to occur when m = 1 (mod 4) and 5 | d, In this case the prime is of the form Some primes of this form are as follows 4517,6317, 7517, 8117,91397,103997,109397,1760477, 1818077,1994477,2042477,33197357,536675837. This is not even close to being a complete list of such primes up to 54 x 107 however we chose this collection of primes since their reductions hit ev-ery congruence class modulo 7, 11 and 13. This means, to characterize such primes locally, we have to go as far as 17. We wi l l show for p of the form (7.2) thatp ^ 6 (mod 17). In the multiplicative group U(Z/17Z) the element 2 is of order 8, and the quadratic residues are {0,1, 2,4,8,9,13,15,16}. Since m = 1 (mod 4) and 2 has order 8 in [/(Z/17Z), we consider the two case, m = 1 (mod 8) and m = 5 (mod 8), separately. Considering each possible quadratic residue in turn we have Thus, for p = 2m — 3d2 with m = 1 (mod 4) and 5 | d we have p ^ 6 and 11 (mod 17). This proves the corollary for the primes appearing in Theorem 7.1. P = 2m - 75fc2 with m = 1 (mod 4). (7.2) 2,12,5,8,14, 7,13,16, or 9 (mod 17) if m = 1 (mod 8), 15,8,1,4,10, 3,9,12, or 5 (mod 17) if m = 5 (mod 8), Chapter 7. On the Classification of Elliptic Curves of conductor 2 a3 2p 239 Theorem 7.2: If p is of the form (2), (3) or (5) then p = 1 (mod 3). If p is of the form (4) then p = 3 (mod 8). If p is of the form (6) then p = 1 (mod 3); for n = 1 this is clear, whereas for n = 2 we factor as (2p + l)(2p — 1) = 3d2 to obtain 4p = 3d\ + d?, = 1 (mod 3), where d — d\d2- Finally, if p is of the form (7) thenp = — 1 (mod 8). Therefore, the curves of conductor 36p havep = 5 or p = 1 (mod 3) or p = — 1 or 3 (mod 8). Theorem 7.3: It is easy to check that the result holds for primes in (1). If p is of the form (2), (3), (4), (10), (15) or (16) then p = 1 (mod 3). If p is of the form (5), (13), (14), (17) or (18) then p = - 1 (mod 4). If p is of the form (7) or (8) then p = 1 (mod 8). If p is of the form (6) then if £ = 0 we have p = 5 or 7, else if £ > 1 then p = 1 (mod 3) or p = — 1 (mod 4) depending on whether the sign is positive or negative. If p is of the form (9) then p = — 1 (mod 8). If p is of the form (11) then p = 1 (mod 3) if the sign is positive, whereas if the sign is negative then p = — 1 (mod 4) for £ odd and p = 5 or 17 for £ even. Finally, If p is of the form (12) then p = 1 or 3 (mod 8). Therefore, the curves of conductor 2 33 2p satisfy one of the following p = 5, p = 29, p = 1 (mod 3), or p = ± 1 or 3 (mod 8). 7.2.5 Proof of Lemma 7.5 We list the primes appearing in (7.1), (7.2), and (7.3) (except pn = d2+fm) in the following table. Unless otherwise stated £ > 0. Chapter 7. On the Classification of Elliptic Curves of conductor 2Q3 2p 240 p conditions conditions 2m-23e ± 1 m = 4, 5 or > 7 ±(d2 - 4 - 3 ^ £ o d d 3e ra = 5 or > 7 p" = ±(d 2 - 4 • 3£) n = 1 or Pmm{n) > 7, ( even ± ( 2 m - 2 - 3<) ra = 4,5 or > 7, £ > 4 = d 2 + 4 • 3* n = 1 or P m i n (n) > 7 2m.-2 + 3* TO = 4, 5 or > 7 n C i2 + 3^  P = 4 n = 1 or Pmm(n) > 7, £ o d d ±(d2 - 2m3e) m > 7 n 3 r f " + 1 P = 4 n = 1 or 2 d2 + 2m3i ra > 7 p n = d2 ± 2 m • 3* n = 1 or P m i n (n) > 7 771 = 4,5 ± ( 3 d 2 - 2m) TO = 4, 5, or > 7 pn = 2m3e _ d2 n = 1 or Pmi„(n) > 7 m = 4,5 3d2 + 2rn m = 2,4,5, or > 7 3« + l 4 £ o d d 3 d 2 - 4 We are interested in counting the number of primes of each of these forms up to X. First we observe that for the forms in the second column there are only finitely many primes satisfying the conditions with Pmm(n) > 7. Indeed, if p, £, m, n, d satisfy one of the equations then Shorey and Tijdeman ([68], page 180) implies that n is bounded by a constant, and Darmon and Granville ([27], Theorem 2) implies there are only finitely many solutions for p, £, m, n, d. From now on, we only consider the case when n = 1. This just ignores some finite (density zero) collection of primes. Also, we wi l l just bound the number of integers of each form listed in the table. This wi l l then bound the number of primes as well. If rj(X) is an upper bound on the number of inte-gers up to X satisfying one of the forms in the table then we want to show n(X) is "little-Oh" of TT(X); denoted r / ( X ) = O(TT(X)). Here ir(X) denotes the number of primes up to X and r)(X) = o(ix{X)) means l imA^oo ^pfj = 0. 1) If 2 m - 2 3 £ ± 1 < X then m, £ < clogX for a fixed constant c (i.e. c = 2 works). So there are at most r)\ (X) = c 2 log 2 X = O(TT(X)) such integers. 2) If 2 ' " ^ 2 + 1 is an integer then 2m~" 2 = - 1 (mod 3 )^- It follows that the order of 2 modulo 3e, which is 2 • 3 £ _ 1 , must divide 2(m — 2). Thus, 3^ _ 1 \m — 2, hence £ < c log m for a fixed constant c. Now if 2 " \ t + 1 < X then m < c\ log 2 X for Chapter 7. On the Classification of Elliptic Curves of conductor 2a2>2p 241 some constant c\. The number of integers of this form is bounded by rj2(X) = (ci l ogX) (c log ( C l \ogX))= o(ir(X)). 3) If ± ( 2 m " 2 - 3*) is an integer such that | 2 m _ 2 - 3e\ < X, it follows from a result of Ellison that m < c log X for some fixed constant c. Thus, r t o g + i f 2 m _ 2 _ 3 , < 0 e<l i o g 3 - , [ c 2 l o g X if 2m~ 2 - 3'> 0 for some fixed constant c 2 . Therefore, the number of primes of this form up to X is bounded above by 773 W < < & log3 [ c 3 l o g 2 X i f 2 m " 2 - 3 ^ > 0 , where Cz is some fixed constant. Thus, n-i(X) = o(n(X)). 4) If 2 m ~ 2 + 3^  < X then m,^ < c\ogX for some fixed constant c. Thus, the number of primes of this form up to X is 774(X) < c 2 log 2 X = O(TT(X)). 5) Consider the set of primes of the form p = \d2 — 2m3e\ up to X. If m and £ are even then factor to obtain P = \d + 2ml23i'2\-\d-2ml23e'2\. It follows thatp = d + 2m/23e<"2 and 1 = \d - 2ml23t>2\. Eliminate d to obtain p = 2 m / 2 r f J 3 ^ 2 ± 1. Thus, m, £ < c l o g X for some constant c, and the number of primes of this form is o(ir(X)). N o w suppose m odd and £ even; m = 2mo + 1, ^ = 2£Q. Factoring over gives p = |d - 2 m o 3^°v / 2| |d + 2 m ° 3 £ ° ^ | < X. Let e = I A/2 - 1 and F = |d + 2 r n °3 f °v / 2 | . The equation can be written as 2mo3e°eF < X. (7.3) According to Ridout [62], e cannot be too small, d e = V2- 2rno3e° > (2™o3*o)l+<5 ' Chapter 7. On the Classification of Elliptic Curves of conductor 2 a 3 2 p 242 for any b > 0, where > means "except for finitely many mo and no" (indepen-dent of X). From (7.3) it follows that F (2mo3<?o)<5 <X, that is, This implies and so, + V2{2mu3e°)1' < X. where 5\ satisfies (1 - 5)(1 + 5\) = 1. Therefore, 2m3e < X2+&2 where S2 = 25\, whence m,£ < clog A" for some fixed constant c. It now remains to bound d as a function of X. For this, we consider two cases: (i) 2m3* < X2~6\ (ii) X2~s* < 2Tn3e < X2+*\ In the first case, it follows directly from \d2 - 2m3e\ < X that d < cXx~^l2. In the second case, if d is large, say d > do := [\/2m3£] + 1, write d = do + k. Then \d2 - 2m3^| < X becomes | ^ - 2 m 3 ^ + 2d0fc + fc2| < X . from which it follows that 2d0k + k2 < X. But 2d0 > X 1 ^ 2 / 2 so /c < A " 5 2 / 2 . Thus, the number of primes of the form \d2 — 2m3e\ up to X is bounded above by max{X6^2 log 2 X, Xl~s^2 log 2 X = o(n(X)). Chapter 7. On the Classification of Elliptic Curves of conductor 2a32p 243 A similar argument works in the case when m is even, £ is odd, and in the case when both m and £ are odd. Here, we would apply Ridout's Theorem for the algebraic numbers A /3 and \/6 to get the same bound on m and £ as above. 6) If d 2 + 2m3e < X then m,i < clogX and d < \fX thus the number of primes of this form, 776 PO, satisfies V6(X) < c^/Xlog2 X = O(TT(X)). 7) A similar argument as to the one used in (5) shows that the number of primes of the form 3d 2 ± 2 m , up to X, is of order O(TT(X)). 8) If 3d 2 + 2m < X then m < cl \ogX and d < c2VX, for some fixed constants c\ and c2l thus the number of primes of this form, 77s(X), satisfies 778pO < c V x log2 X = o{ir{X)). 3e 1 9) If - J - < X then £ < c log X hence the number of such primes is of order o(TrPO). 10) The number of primes of the form 3d 2 — 4 up to X is bounded by cy/~X and hence of order O(TX(X)). 11) The argument in (5) shows that the number of primes of the form |d 2 — 4-3*| is of order o(n(X)). For the rest of the forms in the table we can assume that n = 1, as we stated at the beginning of the proof. It is then easy to see that the number primes satisfying these conditions are of order o(n(X)), since may forms reduce to the ones considered above. This completes the proof of Lemma 7.5. Chapter 8 O n t h e e q u a t i o n xn + yn = 2apz2 In this chapter, we show, if p is prime, that the equation xn + yn — 2pz2 has no solutions in coprime integers x and y with \xy\ > 1 and n > pvi2v , and if p ^ 7, the equation x™ + y n = pz2 has no solutions in coprime integers x and y with jrcj/l > 1, z even, and n > pl2p2. A modified version of the contents of this chapter has been published [4]. 8.1 I n t r o d u c t i o n Inspired by the work of Wiles [Wi95] and subsequently that of Breuil, Conrad, Diamond and Taylor [BCDT01], there has been a great amount of research centered around applying techniques from modular forms and Galois repre-sentations to Diophantine equations of the form Axp + Byq = Czr, (8.1) for p, q and r positive integers with 1/p + 1/q + 1/r < 1. We briefly outlined in Section 1.2 some of the more notable works in this area. The reader is directed to [45] for a survey, In this chapter we study the insolubility of xn + yn = 2apz2, (8.2) in coprime integers (x, y, z), for a £ {0,1}. We use the approach of [BVY04], though here we w i l l need a classification of elliptic curves over Q with ratio-nal 2-torsion and conductor 2ap2. In the case when p = 2 or 3 it is shown in [BS04] that the only solution in nonzero pairwise coprime integers (x, y, z) is 244 Chapter 8. On the equation xn + yn - 2apz2 245 (p, a, x, y, z, n) = (2 ,0 ,3 , -1 , ±11 , 5). Thus, in this chapter, we may take p to be a prime > 5. Our main results are as follows: Theorem 8.1 If n an odd prime and p > 5 a prime (p ^ 7), then the Diophantine equation xn + yn = pz2 has no solutions in coprime integers x, y and z with \xy\ > 1, z even, and n > p12p2. Theorem 8.2 Ifn an odd prime and p > 5 a prime then the Diophantine equation Xn + yn = 2PZ2 has no solutions in coprime integers x, y and z with \xy\ > 1 and n > p v i 2 p 2 . A n immediate corollary of these theorems is: Corollary 8.3 if p > 5 is a prime, then i) ifp ^ 7 the Diophantine equation xn + yn = pz2 has at most finitely many solutions in integers x, y, z, a, and n with x and y coprime, \xy\ > 1, z even and n divisible by an odd prime. ii) the Diophantine equation xn + yn = 2pz2 at most finitely many solutions in integers x, y, z, a, and n with x and y coprime, \xy\ > 1 and n divisible by an odd prime. 8.2 Elliptic Curves We always assume that n is an odd prime and (a, b, c) is an integral solution to (8.2) where a G {0,1}, \ab\ > 1. In the case that a — 0 we further assume Chapter 8. On the equation xn + yn = 2apz2 246 that c = 0 (mod 2). As in [5] we associate to the solution (a, b, c) an elliptic curve • Ea(a, b, c):Y2 = X3 + 2a+1cpX2 + 2apbnX. The following lemma, which follows from [BS04] Lemma 2.1 and corollary 2.2, summarizes some useful facts about these curves. Lemma 8.4 Let a = 0 or 1. (a) The discriminant A(E) of the curve E = Ea(a, b, c) is given by A{E) = 23a+Gp3(ab2)n. (b) The conductor N(E) of the curve E = EQ(a, 6, c) is given by N(E) = 23a+5p2 ]Jq. q\ab In particular, E has multiplicative reduction at each odd prime p dividing ab. (c) The curve Ea(a, b, c) has a Q-rational point of order 2, namely (0,0). (d) The curve Ea(a,b, c) obtains good reduction over Q(\/2ap) at all primes ideals dividing p. Over any quadratic field K, the curve Ea(a, b, c) has bad reduction at all prime ideals dividing p. (e) Ifn > 7 is prime and ab is divisible by an odd prime q, then the j-invariant j(E) of the curve E = Ea(a, b, c) satisfies ordq(j(E)) < 0. In particular, if ab ^ ± 1 then Ea(a, b, c) does not have complex multiplication. 8.3 Outline of the Proof of the main theorems To the elliptic curve Ea(a, b, c) we wi l l associate a weight 2 cuspidal newform / of level 32p2 (if a = 0) or 256p2 (if a = 1). This is done in section 8.4. Let {c i}^ j be the Fourier coefficients of / and Kf their field of definition. We wi l l refer to [Kf : Q] as the dimension of / . If / has dimension > 2 then eg £ Q for some £. We wi l l see that n must divide NormKf/Q(ce - af), for some a ^ e Z such that \af\ < £+ 1 (Proposition Chapter 8. On the equation xn +yn = 2apz2 247 8.6). This gives a bound on n in terms of £. The question then arises: How small can £ be? That is, how far must we go to find a coefficient Ci which reveals / is not of dimension 1? This is answered by a proposition of Kraus (see Proposition 8.11), from which we derive our big bound on n in the main theorem. Now suppose / is of dimension 1, that is c\, e Z for all i. We again have that n must divide NormKf/Q(ci — a^). It may happen that a and are equal from which we derive no information on n. However, the ai are all even so in the case that one of the c/s is odd, say Q , we are able to obtain a bound on n in terms of £. Again, the question arises of how small £ can be. This question is answered by another proposition of Kraus (see Proposition 8.12). The bound on n we receive in this case is much smaller than the one we obtained above. The only case that remains now is when all the coefficients c,; are even rational integers. In this case / corresponds to an elliptic curve F over Q with ratio-nal 2-torsion and conductor 32p2 or 256p2. By Lemma 8.4 (d) Ea(a,b, c) has potentially good reduction at p, we wi l l see (Proposition 8.7) that this implies F has potentially good reduction at p, i.e. p does not divide the denomina-tor of j(F). Also, by Lemma 8.4 (e) Ea(a, b. c) does not have C M , we wi l l see (Proposition 8.7) that if F has C M then we obtain a bound on n of 13. Thus, if F is an elliptic curve over Q with rational 2-torsion, conductor 32p2 or 256p2, potentially good reduction at p and without C M we wi l l not be able to derive any information on n. The question then arises; Are there any such elliptic curves? This question is answered Section 8.6. 8.4 Galois Representations and Modular Forms In this section we describe how to associate to the elliptic curve Ea(a, b, c) a weight 2 modular form. Let E-= Ea(a,b,c) for some primitive solution (a,b,c) to (8.2). We asso-ciate to the elliptic curves E a Galois representation pi : Gal(Q/Q) -> GL 2 (F„) , the representation of Gal(Q/Q) on the n-torsion points E[n] of the elliptic curve E. If n > 7 and ab ^ 1 then p% is absolutely irreducible (see [BS04] Corollary 3.1). Chapter 8. On the equation xn + yn = 2apz2 248 Let F„ be an algebraic closure of the finite field F n and v be any prime of Q extending n. To a holomorphic newform / of weight k > 1 and level N, we associate a continuous, semisimple representation unramified outside of Nn and satisfying, if f(z) = Yln°=i c " 9 n ^ o r Q : = ^ for all p coprime to Nn. Here, Frob p is a Frobenius element at the prime p. If the representation p%, after extending scalars to F n , is equivalent to pj_, for some newform / , then we say that pfh is modular, arising from / . The next lemma follows from [5] Lemma 3.3. The representation pft arises from a cuspidal newform of weight 2, level Nn(E), and trivial Nebentypus character. This lemma says that we can associate to the elliptic curve E = Ea(a: b, c) a weight 2 modular form of level 32p2 (if a = 0) or 256p2 (if a = 1). 8.5 U s e f u l P r o p o s i t i o n s In this section we collect together some results concerning the newforms of level Nn(E) from which our representation pft can arise. The proofs of these propositions can be found in [5]. The first proposition gives a relationship be-tween n and the coefficients of the newform. We wi l l use this result to obtain the bounds on n in the main theorem. P r o p o s i t i o n 8.6 Suppose n>7isa prime and E = Ei(a, b, c) is a curve associated to a primitive solution of (8.2) with ab ^ ± 1 . Suppose further that pJiV : Gal(Q/Q) - GL 2 (F„) tracep^j,(Frobp) = cp (mod u) Lemma 8.5 Suppose that n > 7 fs a prime and that ptfh solution (a, b, c) to (8.2) with ab ^ ± 1 . Put is associated to a primitive oo f =YJ C M Q rn (q := e 2 ™ ) Chapter 8. On the equation xn +yn = 2apz2 249 is a newform of weight 2 and level Nn(E) giving rise to pf and that Kj is a number field containing the Fourier coefficients of f. If q is a prime, coprime to 2pn, then n divides one of either NormKf/Q (cq ± {q + 1)) or NormKf/Q(cq ± 2r), for some integer r < y/q. In the case when the space of cuspforms of level Nn(E) contains newforms associated to elliptic curves with rational 2-torsion we wil l find the following result useful. Proposition 8.7 Suppose n ^ p is an odd prime and E = Ea(a, b, c) is a curve associated to a primitive solution of (8.2). Suppose also that E' is another elliptic curve defined over Q such that pf = off. Then the denominator of the j-invariant j(E') is not divisible by p. Finally, in the case when the space of cuspforms of level Nn(E) contains newforms associated to elliptic curves with rational 2-torsion and C M we wi l l need the following result. Proposition 8.8 Suppose n> 7 is a prime and E = Ei(a, b, c) is a curve associated to a primitive solution of(8.2) with ab ^ ± 1 . Suppose that p® arises from a newform having CM by an imaginary quadratic field K. Then one of the following holds: (a) ab = ±2r, r > 0, 2 /ABC and 2 splits in K. (b) n = 7 or 13, n splits in K and either E(K) has infinite order for all elliptic curves of conductor 2n or ab = ± 2 r 3 s with s > 0 and 3 ramifies in K. 8.6 Elliptic curves with rational 2-torsion It is possible that the modular form associated to E = Ea(a, b, c) has rational integer coefficients in which case the results of the previous section wi l l not help in eliminating the existence of such a form. In this case however, the modular form must correspond to an isogeny class of elliptic curves over Q with 2-torsion and conductor equal to the level of the modular form: 32p2 Chapter 8. On the equation xn + yn = 2°pz2 250 or 256p2. We use the classification of such elliptic curves found in Chapter 6. We restate the relevant results (Corollaries 6.10 and 6.11) of Chapter 6 in the following two propositions. Proposition 8.9 Suppose p > 5 is prime and that E/Q is an elliptic curve with a rational 2-torsion point and conductor S2p2. Then E is isogenous over Q to a curve of the form y2 = x3 + a2x2 + a4x with coefficients given in the following table. P 0-2 0,4 j-invariant any 0 -P2 1728 any 0 ( _ 1 ) ( p + l ) / 2 p 1728 any 0 ( _ 1 ) ( p + l ) / 2 p 3 1728 7 ±7 2 • 7 2 8000/7 7 ±7 2-7 - 2 6 7 ±72 2 • 7 3 - 2 6 s2 + l , s e Z 2ps -P2 64(4p-iy p s 2 + 8, s £ Z ps -2p2 64 (p -2 ) 'i P s2 - 8, s £ Z ps 2p2 64(p+2)a P Proposition 8.10 Suppose p > 5 is prime and that E/Q is an elliptic curve with a rational 2-torsion point and conductor 256p2. Then E is isogenous over Qtoa curve of the form y2 = x 3 + 1J2X 2 + a 4 X with coefficients given in the following table. Chapter 8. On the equation xn + yn = 2apz2 251 V « 2 a\ j-invariant any 0 ±2p 1728 any 0 ±2p2 1728 any 0 ± 2 p 3 1728 any ±Ap 2p2 2 6 5 3 23 ± 2 3 -23-39 2 - 23 5 2b 3 : : ) 4057 ; 5 2 3 6 23 ± 2 4 - 23 - 39 2 3 • 23 5 2b 3 a 4057 i s 236 2s 2 + 1, s e Z ±4ps 2p3 - 6 4 ( p - 4 )a P 2 2s 2 + 1, s <E Z ±4ps - 2 p 2 6 4 ( 4 p - l )a P v"2s2 + 1, s e Z ±Aps 2pA 6 4 ( p * - 4 )3 P 4 \ /2s 2 + 1, s € Z ±Aps -2p2 64(4p*- l )3 2s 2 - 1, s € Z ±Aps 2p3 64(p+4)a >> p z 2s 2 - 1, s e Z ±Aps 2p2 64(4p+l); i P V 2 s 2 - 1, s € Z ±Aps 2p4 64(p 'J +4) 3 P 4 \ /2s 2 - 1, s e z ±Aps 2p2 64(4p*2 + l)';1 p*1 The main feature of these propositions we wi l l use is that an elliptic curve E/Q with rational 2-torsion and conductor 32p2 or 256p2 either has C M or p dividing the denominator of j{E), with one exception: there are curves of conductor 32p2 when p = 7 without C M and potentially good reduction at p, namely y2 = x3 ± 7x2 + 14x and y2 = x3 ± A9x2 + 686x. It is the presence of these curves which prevents us from extending the results of Theorem 8.1 to include p =7. 8.7 T h e o r e m s 8.1 a n d 8.2 To prove Theorems 8.1 and 8.2, we wi l l combine Propositions 8.9 and 8.10 with a result of Kraus (Lemma 1 of [43]) and the proposition of Appendice II of Kraus and Oesterle [46] (regarding this last assertion, note the comments in the Appendice of [43]). We define /x(/V) = /Vn(l + y ) , • l\N ^ ' Chapter 8. On the equation xn +yn = 2apz2 252 where the product is over prime I. Proposition 8.11 (Kraus) Let N be a positive integer and f = 2^2n>i cnQn be a weight 2, level N newform, normalized so that c\ = 1. Suppose that for every prime p with (p, N) = 1 and p < u.(N)/6 we have cp e Z. Then we may conclude that cn £ Z/or all n > 1. Proposition 8.12 (Kraus and OesterU) Let k be a positive integer, x « Dirichlet character of conductor N and f = Y^n>o c " 9 n be a modular form of weight k, char-acter xfor To(N), with C,,. £ Z. Let pbe a rational prime. If Cn = 0 (mod p) for all n < p.{N)k/V2, then cn = 0 (mod p) for all n. We now proceed with the proofs of Theorems 8.1 and 8.2; in each case, from Lemma 8.5, we may assume the existence of a weight 2, level N cuspidal newform / (with trivial character), where N £ {32jD2,256p2} . If / has at least one Fourier coefficient that is not a rational integer, then, from Proposition 8.11, there is a prime / coprime to 2p with , f l W p + l ) l f N = 3 V , [64p(p+l) ifJV = 256p2. such that c; ^ Z. It follows from Proposition 8.6 that n divides Norm^- /Q(Q — a{), where a; is the ith Fourier coefficient corresponding to the Frey curve E(a,b,c). Since a; e Z (whereby a/ ^ q), and I is coprime to 2p, the Weil bounds; | Q | < 2y/i, \a^\ < £ + 1, imply that n < ( i + 1 + 2 ^ ) | K ' : S l = ( ^ + l ) 2 M , (8.4) where, as previously, Kj denotes the field of definition for the Fourier co-efficients of the form / . Next, we note that [Kf : Q] < g^N) where g^N) denotes the dimension (as a C-vector space) of the space of cuspidal, weight 2, level N newforms. Applying Theorem 2 of Martin [49] we have 5o +(32/) < < 3p 2, Chapter 8. On the equation xn + yn = 2apz1 253 and , , 9 s 256p2 +1 o g+(256p2) < — P - < 22p2 Combining these with inequalities (8.3) and (8.4), we may therefore con-clude that f ( y 8 p ( p + l ) + l ) 6 P i f /V = 32p2, n ^ \ ) , N 4 V ( 8 - 5 ) I f >/64p(p + 1) + l j i f / ^ = 256p2. It follows, after routine calculation, that ,12p2 jf /V = 39r) 2 p 12p ifAr  32p2, l p i 3 2 P ifAT = 256p^ where these inequalities are a consequence of (8.5) for p > 5. It remains, then, to consider the case when the form / has rational integer Fourier coefficients cTl for all n > 1. In such a situation, / corresponds to an isogeny class of elliptic curves over Q with conductor N. Define r = E a n d 9*= E CTi(re)^n' n>l,(n,2p) = l n>l ,(n,2p)=l where CTI (n) is the usual sum of divisors function; i.e. o\ (n) = 2~2d\n d- Lemma 4.6.5 of Miyake [Mi: 1989] ensures that /* and g* are weight 2 modular forms of level dividing 512p3. Applying Proposition 8.12 (at the prime 2) to /* - g* and using the fact that a (I) = 1 + 1, for all primes I one of the following necessarily occurs: (i) There exists a prime /, coprime to 2p, satisfying I < 128p2(p + 1) and ci = 1 (mod 2). (ii) Q = 0 (mod 2) for all prime / coprime to 2p. In the former case, since n divides the (nonzero) integer q — a;, we obtain the inequality n <l + l + 2Vl< 128p2(p + 1) + 1 + 16JVP + 1 < p2p, (8.6) where the last inequality is valid for p > 5. In the latter situation, there neces-sarily exists a curve, say F, in the given isogeny class, with a rational 2-torsion Chapter 8. On the equation xn + yn = 2°pz2 254 point. Propositions 8.9 and 8.10 then immediately imply Theorems 8.1 and 8.2. Regarding Theorem 8.1, where N = 32p2, we may apply Proposition 8.9 to conclude that, for p ^ 7, F has j-invariant whose denominator is divisible by p or C M by an order in Q(y/—I). In the former case, we get a contradiction with Proposition 8.7, thus the latter case must hold, from which it follows from Proposition 8.8 that n < 13 (note, part (a) of Proposition cannot hold in this case since we are assuming c = 0 (mod 2) and a, b, c pairwise coprime). Regarding Theorem 8.2, where TV = 256p2, we apply Proposition 8.10 to con-clude that F has j-invariant whose denominator is divisible by p or C M by an order in Q(v /--1) or Q(\ / -2)- In the former case, we again get a contradiction with Proposition 8.7, thus the latter case must hold, from which it follows from Proposition 8.8 that n < 13. Combining these observations with (8.6) and the inequalities following (8.5) completes the proofs of Theorems 8.1 and 8.2. Corollary 8.3 is an easy consequence of Theorems 8.1 and 8.2, after apply-ing a result of Darmon and Granville [DG95] (which implies, for fixed values of n > 4 and a, that the equation xn + yn = 2apz2 has at most finitely many solutions in coprime, nonzero integers x, y and z. 8.8 C o n c l u d i n g Remarks In case p S {2,3,5}, equation 8.2 is solved completely in [5], for n > 4. Further,-the equation xn + yn = 7z 2 with x, y and z coprime nonzero integers, may, as in e.g. Kraus [38], be treated tor fixed values for n. We wi l l not undertake this here. Chapter 9 O n t h e e q u a t i o n x3 + y 3 = ± p m z In this chapter we restrict our attention to determining primes p for which x3 + y 3 = ±prnzn can be shown to be unsolvable in integers [x,y, z) for all suitable large primes n. 9.1 In t roduct ion Let T denote the set of primes p for which there are no elliptic curves over Q with rational 2-torsion and conductor in {18p, 36p, 72p}. We have already seen in Chapter 7 that T is infinite, in fact it contains all primes p satisfying p = 317 or 1757 (mod 2040) (see Corollary 7.4). It is believed that T contains all primes except for a set of density zero. Corollary 7.5 is the most we can prove in this direction. The first few elements of T are 197,317,439, 557, 653,677, 701, 773,797,821,1013,1039, 1061,1109,1231,1277,1279,1289,1301,1399,1447,1471,1493 1613,1637,1663,1709,1733. In this chapter we prove the following. Theorem 9.1 Let p e T and m > 1 an integer. Then the equation xi + y3 = ±pmzn (9.1) has no solutions in coprime nonzero integers x, y and z, and prime n satisfying n > p8p and n \ rn. 255 Chapter 9. On the equation x3 + y3 = ±pmz 256 We remark that in the case that n \ m the equation can be written as x3 + y3 = zn. Kraus has treated these equations in [44]. So, in what follows, we wi l l assume n\m. As an almost immediate consequence of this theorem, we have: Corollary 9.2 Let p £ T. Then equation (9.1) has at most finitely many solutions in coprime nonzero integers x, y and z, and integers m > 1, n > 5 with n\m. 9.2 Frey Curve Let p be a prime number > 5, n a prime > 7 and m a positive integer. We con-sider a proper, nontrivial solution (a, b, c) of the equation a3 + b3 = ±prncn, i.e. pgcd(a, b,pc) = 1 1 . We suppose, without loss of generality, that the following conditions are satisfied: {— 1 (mod 4) if c is even, (9.2) 1 (mod 4) if c is odd. Darmon and Granville [27] associate to the triple (a, b, c) an elliptic curve defined over Q. It is, up to Q-isomorphism, the elliptic curve that we denote Ea,b, with equation y2 = x3 + 3abx + b3 - a 3 , (9.3) which has a point of order 2; (a — b, 0). The standard invariants 04(0, b),ce(a, b) and A(a , 6) associated with the equation 9.3 are the following: 'c4(a,b) = -2432ab c6{a,b) = 2 5 3 3 ( a 3 - 6 3 ) (9.4) A(a,b) = - 2 4 3 V m c 2 n We determine the conductor Nsa b of J E ^ . We designate by 1Z the product of the prime numbers distinct from 2,3, and p that divide c, which is to say the largest squarefree integer prime to 6p which divides c. Given an integer k and a prime number I, we denote by vi(k) the exponent of I in the decomposition of k into prime factors. Jpgcd denotes the pairwise gcd. Chapter 9. On the equation x3 + y3 = ±pmz 257 Lemma 9.3 We have (under conditions (9.2) on a, b, and c) 2 • 32pTZ if c even, b = — 1 (mod 4), Nsa,b = { 2332pTl ifc odd, v2(a) = 1 and b = 1 (mod 4), 2 2 3V& ifc odd, v2(a) > 2 and b = 1 (mod 4). Proof. We wi l l use Theorems 2.1, 2.3, and 2.4 to compute v e ( N E A B ) for all primes I. To do this we first need to move the point of order 2 to (0,0). Apply-ing the change of variables x = X + (a - b), y = Y, the curve Ea^ is Q-isomorphic to Ea.b : Y2 = X3 + 3(a - b)X2 + 3(a2 - ab + b2)X. (9.5) The invariants of this model are still as in (9.4). 1) Let I be a prime number > 5 which divides pc. As the integers a, b and pc are prime to each other we have I \ a — b and equation (9.5) is minimal at /. On the other hand, if lis a prime number > 5 and is prime to pc then vi(A(a, b)) = 0 and again equation (9.5) is minimal at I. It follows from Theorem 2.4 that [ 1 if £ divides pc, vi(NEAB) = { (9.6) [0 if £ does not divide pc. 2) We determine the exponent of 2 in N E U B-2.1) Suppose that c is even. In this case ab is odd, because pgcd(a, b,pc) = 1. We have ±prncn = (a + b)(a2 — ab + b2) and the number a2 — ab + b2 is odd. As n is > 5, we have a + b = 0 (mod 32). As a and b are odd, it follows that 4 does not divide a — b. Therefore iv2(3(a-b)) = v2(a-b) = l, \v2(3(a2 - ab + b2)) =0 . Thus, from Theorem 2.1, the value of v2(NE^A^) depends on the congruence class of 3(a - 6) modulo 8 (note v2(A) > 4 + 2n > 14). It follows from a + b = Chapter 9. On the equation x3 + y3 = ±pmz 258 0 (mod 32), a — b = 2 (mod 4) and the assumption b = — 1 (mod 4) (see 9.2) that a - b = 2 (mod 8), hence 3 ( a - 6) = 6 (mod 8). So, from theorem 2.1, V2(NE0 J = 1. 2.2) Suppose that c is odd. It follows from condition (9.2) that a is even and b=l (mod 4). We therefore have (v2(3(a-b)) = 0, \v2(3(a2 -ab + b2)) = 0, thus, from Theorem 2.1 the value of v 2 { N E ( A ^ ) depends on the congruence classes of 3(a - b) and 3(a 2 - ab + b2) modulo 4. Since 6 = 1 (mod 4) then f 1 (mod 4) if a EE 0 (mod 4), 3(a - b) = < 1-1 (mod 4) i f a = 2(mod4), and 0 0 , - l ( m o d 4 ) if a = 0 (mod 4), 3 a 2 - a6 + b2) = { ' 1 1 (mod 4) if a = 2 (mod 4). It follows from Theorem 2.1 that 2 i f a = 0(mod4), 3 if a = 2 (mod 4). 3) We now determine the exponent of 3 in N E B B • 3.1) Suppose that 3 divides c. Under this hypothesis 3 does not divide ab. From the equality a 3 + 63 = pmcn we have a = -b (mod 3) and 3 does not divide a — b or a 3 — b3. It follows that 3 divides a 2 — ab + b2. Therefore vs(3(a — b)) = 1 and v3(3(a2 -ab + b2)) > 2. The Neron type of Ea}b at 3 is then i ; with v = 2nv^(c) - 3 and v ^ ( N E A ,,) = 2 by Theorem 2.3. 3.2) Suppose that 3 divides ab. We have in this case 3 does not divide a - b or a2 — ab + b2 since gcd(a, b) = 1. Therefore v-s(3(a — b)) = 1 and v^(3(a2 — ab + b2)) = 1. The Neron type of Ea^ at 3 is thus III and again v3(NEa b) = 2 by Theorem 2.3. Chapter 9. On the equation x3 + y3 = ±pmzn 259 3.3) Suppose that 3 does not divide abc. As 3 does not divide pc, we have from the equality a? + b3 = pmcn that a = b (mod 3) and so a - b = 0 (mod 3) and 3 does not divide a2 - ab + b2. Thus v3(3(a - b)) > 2 and v3(3(a2 - ab + b2)) = 1. The Neron type of Ea^ at 3 is thus III, and we have again vs(Nsa J = 2. This completes the proof of Lemma 9.3. • 9.3 T h e M o d u l a r Galois Representation p%b Let Q be an algebraic closure of Q and Ea^[n] the subgroup of n-torsion points of Eaib(Q). Ea^n] is a vector space of dimension 2 over Z / n Z on which the Galois group Gal(Q/Q) acts naturally. We denote the corresponding mod n Galois representation on Eafi{n] by : Gal(Q/Q) - GL 2 (F„) . Let k and N(pn'b) denote the weight and conductor of pn'b respectively, which are defined by Serre in [64]. Lemma 9.4 1. k = 2. {18p if c even, b = — 1 (mod 4). 36p if c odd, v2(a) >2andb=l (mod 4), 72p ifcodd, v2{a) = 1 and 6 = 1 (mod 4). 3. The representation pnb is irreducible. Proof. 1) Recall n ^ p by assumption. If n \ c then Ea>b has good reduction at p. Otherwise, Ea^ has multiplicative reduction at n and the exponent of n in the minimal discriminant is a multiple of n. From which the-above assertion follows, see ([64], P. 191, Proposition 5). 2) Let q be a prime distinct from p and n. The curve Ea<b has multiplicative reduction at q (Lemma 9.3) and the exponent of q in the minimal discrimi-nant of Eafi is a multiple of n (see 9.4). This assertion then follows as a direct consequence of Lemma 9.3 and a proposition of Kraus [42]; see also ([64], p. 120). Chapter 9. On the equation x3 + y3 = ±pmzn 260 3) Suppose pnb is reducible. Since Ea^ has a point of order 2 there exists a subgroup of Eatb{Q) of order 2 stable under Galois G(Q/Q) . If n > 11 then the modular curve Yo(2n) does not have any Q-rational points (see [39] which uses the results of [50]), from which the lemma follows. Suppose n = 7. The modular curve Yo(14) is the elliptic curve 14al in the table of [26]. It follows that Fo(14) has a rational point of order 2 and so there corresponds two Q-isomorphism classes of elliptic curves over Q with j invariants —153 and 255 3, respectively. These are precisely the curves 49al and 49a2 in the tables of [26], each of which contains a subgroup of order 14 stable under Galois. Since Ea^ has j-invariant . _ 6912(a6)3 the lemma follows. • Given an integer N > 1 we let S2(TQ(N)) denote the C-vector space of cus-pidal modular forms of weight 2 for the congruence subgroup Fo(N). Denote by S^N) the subspace of newforms of S2{To(N)), and g^iN) its dimension as a C-vector space. See [49] for an explicit determination of g^(N). Since the representation p'nb is irreducible of weight 2 and Ea^ is modular (by the extraordinary work of Breuil, Conrad, Diamond, Taylor, and Wiles: [80], [77], [8]) there exists a newform / e 5 2 + {N(pn'b)) whose Taylor expansion is t ^ g + Yl an(f)Qn where q = e2"1 n>2 and a place B of Q of residual characteristic n such that for all prime numbers I which do not divide TINE,, H one has ai{f) = ai{Eatb) (mod B). It follows that n | N o r m X / / Q ( a ( ( / ) - a,(£•„,(,)), (9.7) where Kf denotes the field of definition of the coefficients. Chapter 9. On the equation x3 + y3 = ±pmzn 261 9.4 Proof of Theorem 9.1 We now proceed with the proof of Theorem 9.1. Let us suppose that / is a weight 2, level N cuspidal newform (with trivial character), where N e {18p,36p,72p}, corresponding, as in Section 9.3, to a nontrivial solution to equation (9.1). From Theorem 3 of [43], we may suppose that / has rational integer Fourier coefficients, provided n > p4p (in case N = 18p or 36p) or n > p8p (in case iV = 72p). This follows from 9.7 and applying Theorem 1 of [49] to obtain {5p/4 ii N = 72p. To finish the proof of Theorem 9.1 we wi l l combine the results of Chapter 7 with the Proposition of Kraus and Oesterle, see Proposition 8.12. Since the form / has rational integer Fourier coefficients am(f) for all m > 1, / corresponds to an isogeny class of elliptic curves over Q with conductor N = 18p, 32p, or 72p. Define /* and g* as in section 8.7, though this time they are both weight 2 cusp forms with level dividing 2 4 3 3 p 2 . Applying the Proposition of Kraus and Oesterle to /* — g*, and using a(l) = 1 + 1, for all primes I one of the following necessarily occurs: (i) There exists a prime I, coprime to 6p, satisfying I < 144p(p + 1) and ai(/) = l (mod2) . (ii) ai(f) = 0 (mod 2) for all prime I coprime to 6p. In the former case, since n divides the (nonzero) integer o/(/) — ai{Eab) we obtain the inequality n <l + l + 2Vl< 144p(p + 1) + 1 + 24yfp{p+ 1) < pP, where the last inequality is valid for p > 5. In the latter case, there exists and elliptic curve F, in the given isogeny class, with a rational 2-torsion point. That is, F is an elliptic curves over Q with 2-torsion and conductor l&p, 36p or 72p. It follows that p $ T. Therefore, for p G T such an F cannot exist, hence Chapter 9. On the equation x3 + y3 = ±pmzn 262 n < pAp (if N = 18p or 36p) or n < p8p (if N = 72p). This completes the proof of Theorem 9.1. 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Appendix A O n t h e Q - l s o m o r p h i s m C l a s s e s o f E l l i p t i c C u r v e s w i t h 2 - T o r s i o n a n d C o n d u c t o r 2 a3V 5 In this appendix, we provide the proof of the main lemmata used in our clas-sification of elliptic curves. In particular, we wi l l give a list of elliptic curves which contains a representative for each Q-isomorphism class of curves con-taining 2 torsion and having conductor of the form 2a3l3ps. Let E be an elliptic curve over Q of conductor 2M3LpN, with 0 < M < 8 and 0 < L, N < 2, and having at least one rational point of order 2. We may assume that E is given by a model of the form y2 — x3 + ax2 + bx, where a and b are integers both divisible by p iff ./V = 2, both divisible by 3 iff L = 2, and a, b have no other common odd divisors (see results of Chapter 2). We may also assume that this model is minimal outside of 2. From the hypothesis on the conductor of E there exist three natural numbers a, (3 and 5 such that , ! b2(a2 - 46) = ± 2 Q 3 V (A.l) We have 6 ^ 0 and the only possible divisors of 6 are 2, 3 and p. We consider the two cases: (i) 6 > 0, (ii) 6 < 0. The first case is treated in Section A . l and the second in A.2. A . l b > 0 Lemma A . l Suppose 6 > 0. Then there exists an integer d, and non-negative in-tegers m,£, and n satisfying one of the equations in the first column and E is Q-270 Appendix A.' Q-Isomorphism Classes 271 isomorphic to the corresponding curve in the second column, for some n , r 2 , r 3 £ {0,1}; except in cases 1 through 9, where ifm = l then r\ £ {1,2}. Diophantine Equation y2 = x3 + a2x2 + (14X a2 a.4 1 d2 - 2m3epn = ± 1 2 r i 3 r 2 pr3d 2 m + 2 r i - 2 ^ e + 2 r 2 p n + 2 r 3 2 d2 - 2m3e = ±pn 2Ti3T2 pT3d 2m+2r1-23e+2r2p2r3 3 d2 - 2mpn = ±3e 2ri3r2 pr3d 2 m + 2 7 ' i — 2 ^2r2pn+2r3 4 \d2 _ 2 m = ±3epn 2r'3r2 pr3d 2m.+2r1-2^2r2p2r3 5 pd2 - 2m3l = ± 1 2r*3r2 pT3+1d 2 m + 2 r i - 2 3 f + 2 r 2 p 2 r 3 + l 6 pd2 - 2m = ±3e 2r^3r2pr3+1d 2 m + 2 r i - 2 ^ 2 r 2 p 2 r 3 + l 7 3d2 - 2mpn = ± 1 2 r i 3 r 2 + V 3 ^ 2m+2n-2g2r2 + l n+2r3 8 3d2 - 2m = ±pn 2 r i 3 r 2 + V 3 d 2-m.+2r1-2^2r2 + lp2r3 9 3pd2 - 2 ' n = ± 1 2 n 3 ^ 2 + l p r 3 + l d 2 m + 2 r i - 2 £ 2 r 2 + lp2r3 + \ 10 ,d2 - 3epn = ±2m 2 r i + 13 r 2p'' 3(i 2 2 r a ^e+2r2pn+2r3 11 d2 - 3f = ±2mpn 2ri+13r2pr3d 2 2 n ^e+2r2p2r3 12 \d2 - pn = ±2m3e 2ri+13r2pr3d 22T\ £2r2pn+2r3 13 d2 - 1 = ± 2 m 3 V 2 ' ' l + l 3 r 2 pr3d 22ri 32r2p2r3 14 , pd2 - 3e = ±2m 2 r i + 1 3 r 2 p T ' 3 + 1 d 2 2 r 1 3 f + 2 r 2 p 2 r 3 + l 15 pd2 - 1 = ±2m3e 2 n + 1 3 r 2 p r 3 + 1 d 22rx<i2r2p2r3 + \ 16 3d2 - pn = ±2m 2ri+13r2+1pT3d 22rl£2r2 + lpn+2r3 17 3d2 - 1 = ±2mpn 2r>+13r2+1Pr3d 22r, 3 2 r 2 + l p 2 r 3 18 3pd2 - 1 = ± 2 m 2 r i + 1 3 r 2 + V 3 + 1 d 2 2 r ! 3 2 r 2 + l 2 r 3 + l 19 2d2 - 3 V = ± 1 2^+23T2pT3d 22r,+l3^+2r2pn+2r3 20 2d2 - 3e = ±pn 2ri+23r2pr3d 2 2 r i + l 3 « + 2 r 2 p 2 r 3 21 2d2 - pn = ±3e 2 r i + 23 r 2p r 3rf 22n+l32r 2pn+2r 3 22 2d2 - 1 = ±3epn 2ri+23r2pr3d 2 2 r 1 + l 3 2 r 2 p 2 r 3 23 ; 2pd2 -3e = ±l 2 r » + 2 3 r 2 pr3+1d 2 2 n + l 3 « + 2 T - 2 2 r 3 + l 24 2pd2 -l = ±3f 2 r i + 2 3 r 2 p r 3 + 1 d 22r ' i + l 3 2 r 2 p 2 r 3 + l 25 6 d 2 - pn = ± 1 2 r i + 2 3 r 2 + 1 p r 3 d 2 2 n +1 3 2 r 2 + l ^ n + 2 r 3 2 6 6d2 - 1 = ±pn 2r*+23r2+1pr3d 2 2 r 1 + l 3 2 r 2 + l p 2 r 3 2 7 6pd2 - 1 = ± 1 2 r i + 2 3 r 2 + 1 p r 3 + 1 d 2 2 r i + l 3 2 r 2 + l p 2 r 3 + l Remark. To avoid trivial redundancies in the list above we are free to make the following convention: m,£ and n may be zero if they appear on the right-hand side of the Diophantine equation, otherwise they must be > 1. A warning to the reader. The following proof is tedious and very repeti-tive. We have included all the details only for the purpose of completeness. Appendix A. Q-Isomorphism Classes 272 For those interested in getting an idea of the flavor of the proof, we suggest only reading a few cases. Proof . It follows from (A.l) that the only possible divisors of b are 2, 3 and p. Thus, there exist integers i, j and k such that b = 2i3jpk, and 0 < 2i < a, 0 < 2j < (3, 0 < 2k < 5. We obtain from (A.l) a2 _ 2i+2ypk = ± 2 ^ - ^ - ^ p 6 - 2 k (A.2) In what follows we consider the following twenty-seven cases: l ) z + 2 > a-r 2i,j>(3- 2j, k> 5- 2k, 2)z + 2 > a - 2z, j > 0 - 2j, k<5- 2k, 3)i + 2 > a - 2i,3>j3- 2j, k = S- 2k, 4)z + 2 > a L 2i,3<[3- 2j, k> 5- 2k, 5)i + 2 > a 7 2i,j</3- 2j, k<6- 2k, 6) i + 2 > a - 2i, j < [3 - 2j, k = 5- 2k, 7)i + 2 > a - 2z, 3 ~ (3 — 2j, k>5- 2k, 8)z + 2 > a - 2i,j = [3- 2j, k<5- 2k, 9)i + 2 > a - 2i,j = P - 2j, k = 5- 2k, 10) i + 2 < a - Ii, j > 0 - 2j, k> 6 - 2k, 11) i + 2 < a - 2i, j > [3 - 2j, k < 6 - 2k, 12) i + 2 < a -2i,j>P - 2j, k = 5 - 2k, 13) z + 2 < a -2i,j<(3 - 2j, k > 5 - 2k, 14) i + 2 < a' - 2i, j < (3 - 2j, k < 5 - 2k, 15) i + 2 < a -2i,3<0 - 2j, k = S - 2k, 16) i + 2 < Q : -2i,j = 0 - 2j, k > 5 - 2k, 17) i + 2 < a- -2i,j = f3 - 2j, k<5 - 2k, 18) i + 2 < a -2i,j = 0 - 2j, k = 5 - 2k, 19) i + 2 = a - 2%, j > 0 - 2j, k>5 - 2k, 20) z + 2 = a -2i,j > 0 -2j,k< 6 - 2k, 21) i + 2 = a -2i,j> 0 - 2j, k = 6 - 2k, 22) z + 2 = a -2i,j<0 - 2j, k> 6 - 2k, 23) z + 2 = a -2i,j < (3 - 2j, k <S - 2k, 24) z + 2 = a -2i,j < 0 - 2j, k = 5 - 2k, 25) i + 2 = a -2i,j = p - 2j, k>5 - 2k, Appendix A. Q-Isomorphism Classes 273 26) i + 2 = a - 2i, j = (3 - 2j, k < 6 - 2k, 27) i + 2 = cx - 2i, j = (3 - 2j, k = 6 - 2k. 1. We have i + 2 > a - 2i, j > (3 - 2j and k > 8 - 2k. In this case v2(a?) = a - 2i, V2,(a2) = (3 — 2j and vp(a?) — 5 — 2k so a, (3, and S are even. Therefore, v2(a) = § - i, V3(a) = f — j , and vp(a) = | - k. Let 2 a u = 22 *32 -?r)2 K P so (A.2) becomes u2 _ 23i-a+233j-Pp3k-6 = ± 1 > with 3i - a + 2 > 1, 3j - f3 > 1 and 3fc - S > 1. Let d = u, m = 3i — a + 2, I = 3j — (3, n = 3k — 6, then (d, rn, £,n,p) is a solution to d2 - 2m3epn = ±1, with m, £, n > 1. The model for E can be written y2 = x3 + 2? _ i 32 ~jp2-kdx2 + 2 J3V^-There exist six integers r i , q\, r2, q2,7-3, and g 3 such that - - « = 2 g i + r i , - - J = 2<72 + r 2 ) - - fc = 2g3 + r 3 , with r i , r 2 , r 3 € {0,1}. There are two cases to consider: l . i )Wehave (m,ri) = (1,0). Putting •7" QI 2 2(«- l)32g 2 j ) 2 g3 ' 2 3 («- 1 )3 3 ( «73 3 '23 ' we obtain the new model for E Y2 = X3 + 223r2pT3dX2 + 233e+2r2pn+2r3X which is the curve in case 1 of the lemma with r\ = 2. Appendix A. Q-Isomorphism Classes 274 l . i i )Wehave (m,ri) > (1,0) 1. Putting x = x Y - V 22<?i 32<?2^2(73 ' 23 f'i 3 3 l ? 2 p 3 g 3 ' we obtain the new model for E Y2 = Xs + 2ri3r2prsdX2 + 2 m + 2 r i - 2 3 £ + 2 r 2 p " + 2 r 3 X which is the curve in case 1 of the lemma with r\ = 0 or 1. 2. We have i + 2 > a - 2i, j > (3 - 2j and k < S - 2k. In this case i; 2(a 2) = a — 2i, ^3(a2) = (3 — 2j and t>p(a2) = A; so a, (3, and fc are even. Therefore, v2{o) = f - h V3(a) = f - j, and up(a) — | . Let a : u = — 22-%32-lp2 so (A.2) becomes u2 _ 2 3 z - a + 2 3 3 j - , a = -4-p<5-3fc with 3i - a + 2 > 1, 3j - (3 > 1 and 5 - 3fc > 1. Let ! d = u, m = 3i — a + 2,. I = 3j — P: n = d — 3k, then (d, m, ^ , n, p) is a solution to with m,£,n>l. The the model for £ can be written y 2 = x 3 + 2'-i3%-jp?dx2 + 2*3? pkx. There exist six integers v\,q\, r2, q2, r3, and q3 such that a ' /? k -'-t = 2q1+.n, - - J = 2q2+r2, - = 2q3 + r3, with r\,r2,r3 e {0,1}. There are, again, two cases to consider: 2.i)Wehave (m,n) = (1,0). Putting ' Y = X Y= 22(c?l-1)3292p2g3 ' 2 3 ^ i - 1)33<?2j93(?3 ' 1 Lexicographic order. I Appendix A. Q-Isomorphism Classes 275 we obtain the new model for E Y2 = X3 + 223r2pr3dX2 + 233e+2r2p2r3X which is the curve in case 2 of the lemma with r\ = 2. 2.ii) We have (m,r a ) > (1,0). Putting Y = x y - V we obtain the new model for E Y2 = X3 + 2ri¥2pridX2 + 2 m + 2 r ^ H i + 2 r 2 p 2 T 3 X which is the curve in case 2 of the lemma with n = 0 or 1. 3. We have i + 2 > a — 2i,j > (3—2j and k = 5—2k. In this case i>2(a2) = ct — 2i,v3(a2) = (3 — 2j so a and /3 are even. Therefore, v2(a) = ^ — i,v3(a) = § — j. Also, vp(a2) > k = 8 — 2k so vp(a?) > where £3 denotes the residue of k modulo 2. Let 22-'-32"ip-so (A.l) becomes pt3u2 - 23i~a+233J-0 = ±1, with 3i - a + 2 > 1 and 3j - (3 > 1. Let d = u, m = 3i — a + 2, £ = 3j — ft, then (d, m, ^ ) is a solution to with rn, £ > 1. The model for E can be written y2 = x3 + 22^32 -ip^dx2 + 2i3jpkx. There exist six integers r\, q\, r2, q2, r3, and q3 such that a (3 k - - i = 2q1+r1, - - j = 2q2 + r2, - = 2q3 + r 3 , Appendix A. Q-Isomorphism Classes 276 with r\, r2, r 3 G {0,1}. We have two cases to consider: 3.1) Suppose £3 = 0. 3.1.i) If (m,n) = (1,0), then putting X= * „ ,Y- V 22(9i-l)32q 2p2 g 3 ' 2 3('?i- 1)3 3 <?2p3q 3 ' we obtain the new model for E Y2 = X3 + 223T2pT3dX2 + 233e+2r2p2r3X which is the curve in case 2 of the lemma with n = 0 and r\ — 2. 3.1.ii) If (m,r x ) > (1,0), then putting X= * n , Y = V 22<?1 3292p2q3 ' 231?1 3 3 « 2 p 3 « ' we obtain the new model for E Y2 = X3 + 2riSr2pTidX2 + 2 m + 2 r ^ - 2 3 e + 2 r 2 p 2 r i X which is the curve in case 2 of the lemma with n = 0 and r\ = 0 or 1. 3.2) Suppose £3 = 1. 3.2.i) If (m, T-I ) = (1,0), then putting x v X = — - — — - — • , Y - y 22(91-1)3292^2(93-1+1-3) ' 2 3(9l- 1)3 392p3(9 .3-l+r 3) ' we obtain the new model for E Y2 = X3 + 223r2p2~r3dX2 +• 233e+2r2p3-2r3X which is the curve in case 5 of the lemma with r\ = 2 and r 3 = 1 — r 3 . 3.2.ii) We have (rn, n ) > (1,0) and r 3 = 0. Putting X = „ „ ! , , , , , Y = 2 29i 3292^2(93-l+r-i) ' ~ 2 3 9l3 3 92p 3 (<?3-l+r3) ' we obtain the new model for E Y2 = X3 + 2ri¥2p2-r3dX2 + 2 m + 2 r x - 2 3 e + 2 r 2 p 3 - 2 r 3 X which is the curve in case 5 of the lemma with r\ = 0 or 1 and 7-3 = 1 — 7 -3 . Appen dix A. Q -Isom orphism Cla sses 277 4. We have i + 2 > a - 2i, j < (3 — 2j and k > 5 — 2k. In this case V2{o?) = a — 2i, t>3(a2) = j and f p (a 2 ) = 5 — 2k so a, j, and 5 are even. Therefore, v2(a) — § ~~ ^3( a ) = o' a n d V P(° ) = f — ^- Let 2 a 22" l35p2-so (A.2) becomes 2 _ 23i-<*+2p3fc-15 _ -J-3/3-3J with Si - a + 2 > 1, /3 - 3j > 1 and 3/c - 5 > 1. Let d = u, m = 3i — a + 2, £ = /3 — 3j, n = 3k — <5, then (d, m, ^, n,p) is a solution to d 2 - 2rnpn = ±3e, with m, £, n > 1. The the model for £ can be written y2 = x3 + 22-i3ip*~kdx2 + 2i3jpkx. There exist six integers v\, q\, r2, q2, r^, and 93 such that a. j 5 - - i = 2q1.+ n, - = 2q2 + r2, - - k = 2q3 + r 3 , with r 2 , r3 6 {0,1}. There are, again, two cases to consider: 4.i) We have (m, n) = (1,0). Putting 22( 9 i - l)32 9 2 ? ,2f /3 ' 2 3(<?i-l)3392p3 93 ' we obtain the new model for E Y2 = X3 + 223r2pr3dX2 + 2 3 3 2 r 2 p n + 2 r 3 X which is the curve in case 3 of the lemma with 7*1 = 2. 4.ii) We have (m, rj) > (1,0). Putting x = ,5, „, , y = 2291 3292p2g3 ' 23<?l 3392p3Q3 ' Appendix A. Q-Isomorphism Classes 278 we obtain the new model for E Y2 = X3 + 2ri3r2pr3dX2 + 2 m + 2 r ^ - 2 3 2 r 2 p n + 2 r 3 X which is the curve in case 3 of the lemma with r\ = 0 or 1. 5. Wehavez + 2 > a-2i, j < (3-2j and k < 5-2k. In this case v2(a2) = ct — 2i, v3(a2) = j and vp(a?) = k so a, j , and k are even. Therefore, v2(a) = f - i, v3(a) = 5 , and vp(a) = | . Let a u = :—77 2 ? - * 3 M so (A.2) becomes u2 - 23*~a+2 = ±3<3~3ips~3k, with 3? - a + 2 > 1, (3 - 3j > 1 and 5 - 3k > 1. Let d = u, m = 3i — a + 2, £ = j3 — 3j, n = 6 — 3k, then (d, m, £, n, p) is a solution to d2 - 2rn = ±3epn, with m, £, n > 1. The the model for E can be written y2 = x3 + 22-^p^dx2 + 2i33pkx. There exist six integers r\,q\, r2, q2, T3, and 173 such that '^-i = 2q1+r1, ^ = 2q2 + r2, ^ = 2g3 + r 3 , with ri, r2, r3 e {0,1}. There are, again, two cases to consider: 5.i) We have (m, r x ) = (1,0). Putting 22(<y-i-l)32<,2p2g3 ' 2 3(9i- 1)3 392p393 ' we obtain the new model for E Y2 = X3 + 223T2 pT3dX2 + 2 3 3 2 r 2 p 2 r 3 X Appendix A. Q-Isomorphism Classes 279 which is the curve in case 4 of the lemma with r\ = 2. 5.ii)Wehave (m,r\) > (1,0). Putting 2 2 ( ?l 3292p2q3 ' 23''1 33l?2p l^* ' we obtain the new model for E Y2 = X3 + 2ri3r2pr3dX2 + 2m+2^-232r2p2rsX which is the curve in case 4 of the lemma with r\ — 0 or 1. 6. We haye i + 2 > a — 2i, j < f3 — 2j and k = 5 — 2k. In this case i»2(a 2 ) = a — 2i and vz(a2) — j so a and j are even. Therefore, v2(a) = § — i arid W3(a) = | . Also, vp(a2) > k = 6 — 2k so up(a) > where £3 denotes the residue of k modulo 2. Let a 1* = : ZZ 22 J32p 2 so (A.2) becomes p « u 2 _ 2 3 t - a + 2 = ± 3 P - 3 i ) with 3i - a + 2 > 1 and P - 3j > 1. Let d = u, m. = 3i — a + 2, £ = ft — 3j, then (d, m, is a solution to p t 3 d 2 - 2 m = ±3^ with m,£> 1, and the model for E 1 can be written y 2 = x 3 + 2^~l3ipt^Ldx2 + 2z3jpkx. There exist six integers r\,q\, r2, qi, r3, and 53 such that a j' k + 6 3 --i = 2qi+n, - = 2q2 + r2, =2q3 + r3, with r\, r2, r 3 e {0,1}. We have two cases to consider: 6.1) Suppose 6 3 = 0. Appendix AJ Q-Isomorphism Classes 280 6.1.i) If (m,n) = (1,0), then putting a; X = Y = y 22(gi-l)3292p2g3 ' 2 3 (9i-I)33q2p3c73 ' we obtain the new model for E Y2 = Xs + 223r2pr3dX2 + 2332r2p2riX which is the curve in case 4 of the lemma with n = 0 and r\ = 2. 6.1.ii) If (. m, n) > (1, 0), then putting x X Y = y 2291 3292p2q3 ' 23(?l 3392p3<33 ' we obtain the new model for E Y2 = X3 + 2ri3r2pr3dX2 + 2m+2r^-232r2p2r3X which is the, curve in case 4 of the lemma with n = 0 and r\ = 0 or 1. 6.2) Suppose 6 3 = 1. 6.2.i) If (m, ri) = (1,0) and r 3 = 0, then putting x y X = -. Y 2 2 ( 9 i - l ) 3 2 9 2 p 2 ( 9 3 - l ) ' 2 3 < < ? l - l)3 3 92p3(93 - l ) ' we obtain the new model for E ! Y2 = X3 + 22T2p2dX2 + 2332r2p3X which is the curve in case 6 of the lemma with r\ = 2 and r3 = 1. 6.2.ii) If (m, n) = (1,0) and 7-3 = 1, then putting y = 23(91- l )3392p393 ' 22('/l-l)32<72p293 we obtain the new model for E Y2 = X3 + 223T2pdX2 + 2332r2pX which is the; curve in case 6 of the lemma with r\ = 2, r 3 = 0. 6.2.iii) If ;(m, r\) > (1,0) and r3 = 0, then putting ! ,Y = X Y= V 22<n 3292^,2 (93-1) ' 2 3 ( ' i 3 3 9 2 p 3 ( ( « - i ) ' Appendix A: Q-Isomorphism Classes 281 we obtain the new model for E Y2 = X3 + 2 n 3r2p2dX2 + 2rn+2ri-232r2p3X which is the curve in case 6 of the lemma with r\ = 0 or 1 and r 3 = 1. 6.2.iv) If \(m,r\) > (1,0) and r 3 = 0, then putting 1 x v X — Y = 22<?i 3292p2q3 ' 2 3 <?l 33<?2p3<y3 ' we obtain the new model for E Y2 = X3 + 2ri¥2pdX2 + 2 m + 2 r i - 2 3 2 r 2 P X which is the curve in case 6 of the lemma with r\ = 0 or 1 and r 3 = 0. 7. We have i + 2 > a - 2i, j = (3 - 2j and /c > 8 - 2k. In this case u 2(a 2) = a — 2i, and vp(a2) = 8 — 2k so a and 8 are even. Therefore, u2(a) = § — i and •Up(a) = I — k. Also, ^ ( a 2 ) > j = (3 - 2j so v 3(a) > where e2 denotes the residue of j modulo 2. Let a u = — 22 " ' 3 2 p5" so (A.2) becomes 3£2-a2 - 23r-a+2p3k-s = ± 1 , with 3i - a ± 2 > 1 and 3A; - d > 1. Let d = u. m = 3i — a + 2, n = 3k — 8, then (d,m,n,p) is a solution to ; • d 2 - 2 m p " = ± l , with m, n > 1, and the model for E 1 can be written y2 = x3 + 2 ? - ^ p ^ d a ? + 2iffpkx. There exist six integers r\, q\, r 2 , g 2 , r 3 , and <?3 such that. - ~ i i = 2qi+ri, 2 = 2 g 2 + r 2 , - - A; = 2g3 + r 3 , Appendix A. Q-Isomorphism Classes 282 with r\, r2, r3 € {0,1}. We have two cases to consider: 7.1) Suppose e2 = 0. 7.1.i) If (rn,n) = (1,0), then putting f i i X = ^-, . „ „ , Y = 22(qi-l)32<;2p2<?3 ' 2 3 ( '» - 1 )3 3 "2p3g 3 ' we obtain the new model for E Y2 = X3 + 223T2pr3dX2 + 2332r2pn+2r3X which is the curve in case 3 of the lemma with £ = 0 and r\ = 2. 7.1.ii) If (m, n) > (1,0), then putting x Y - V 221l 32<72p2<73 ' 2 3 , ? i 33</2p3(j3 ' we obtain the new model for E Y2 = X3 + 2 n 3 r y ' 3 d X 2 + 2 m + 2 r i - 2 3 2 r 2 p n + 2 r 3 X which is the curve in case 3 of the lemma with £ = 0 and r\ = 0 or 1. 7.2) Suppose e2 = 1. 7.2.i) If (m, n) = (1,0) and r 2 = 0, then putting X ii X = — — — — — , Y 22(91-1)32(92-1)^293 ' 2 3 (« - 1 )3 3( f'2 - l )p 393 ' we obtain the new model for E Y2 = X3 + 2232pr3dX2 + 2333pn+2r3X which is the curve in case 7 of the lemma with r\ = 2 and r2 = 1. 7.2.ii) If (m, n) = (1, 0) and r2 = 1, then putting ^ = _ _—_ Y = 22(91-1)3292^293 ' 23(<?i-l)3392p3g3 ' we obtain the new model for E Y2 = X3 + 223pr3dX2 + 233pn+2r3X which is the curve in case 7 of the lemma with r\ = 2, r2 = 0. Appendix A. Q-Isomorphism Classes 283 7.2.iii) If (TO, n) > (1,0) and r 2 = 0, then putting v = x Y = V 22qi32(r/2-l)p2q3 ' 2 3 (« 33(<?2- l)p3<j3 ' we obtain the new model for i? . Y2 = X3 + 2Ti32pT3dX2 + 2m+2ri-233pn+2r3X which is the curve in case 7 of the lemma with r\ = 0 or 1 and r 2 = 1. 7.2.iv) If (TO, r i ) > (1,0) and r 2 = 1, then putting Y _ x Y= V 22cn 32?2p2<73 ' 2 3 '/l 3 3 «p3q 3 ' we obtain the new model for E Y2 = X3 + 2 r i 3 p r W 2 + 2 m + 2 r i - 2 3 p n + 2 r 3 X which is the curve in case 7 of the lemma with r\ = 0 or 1 and r 2 = 0. 8. We have i + 2 > a — 2i, j = (i - 2j and k < 5 - 2k. In this case u 2 (a: a — 2i, and vp(a2) — so a and fc are even. Therefore, t>2(a) = f — i and t;p(o) = | . Also, v3(a2) > j = 8 — 2j so i>3(a2) > where e2 denotes the residue of j modulo 2. Let 2^ u • J + t-2 k 22~l3 2 p2 so (A.2) becomes 3 t 2 u 2 - 2 3 i " a + 2 = ±ps-3k, with 3z - a + 2 > 1 and <5 - 3/c > 1. Let d = u, m =-3i — a + 2, n = <$ — 3fc, then (d, TO, n, p) is a solution to d2 - 2m = ± p n , with rn, n > 1 and the model for E can be written y2 = x3 + 2? _ i 3 2 p t d x 2 + 2*3 Jp f cz-Appendix A. Q-Isomorphism Classes 284 There exist six integers n , q\, r 2, qi, r%, and q% such that a 3 k --i = 2q1+n, - = 2q2 + r2, - = 2qs + r 3, with r\,r2,r3 G {0,1}. We have two cases to consider: 8.1) Suppose £2 = 0. 8.1-i) If (m,n) = (1,0), then putting 22(91-1)3292^293 ' 2 3 ( 9 i - 1 ) 3 3 ' ? 2 p 3 9 3 ' we obtain the new model for E Y2 = X3 + 22y2pTidX2 + 2332r2p2rsX which is the curve in case 4 of the lemma with £ = 0 and r\ = 2. 8.1.ii) If (m, n ) > (1,0), then putting x v v _ y _ y 22913292^293 ' 2 3 9 1 3 3 < ? 2 p 3 ' ? 3 ' we obtain the new model for E y 2 = X 3 + 2 r i 3 T V 3 d X 2 + 2m+2ri-232r2p2r3X which is the curve in case 4 of the lemma with ~£ = 0 and ry = 0 or 1. 8.2) Suppose £2 = 1-8.2.1) If (m, n ) = (1, 0) and r2 = 0, then putting X = — , — ^ 7 7 - ^ - , Y = 2 2 ( 9 1 - l ) 3 2 ( 9 2 - l ) p 2 9 3 ' 2 3 (9l - 1 )3 3 fe- 1 )p 3 < ' 3 ' we obtain the new model for E Y2 = X3 + 2H2pridX2 + 23S3p2r3X which is the curve in case 8 of the lemma with r\ = 2 and r2 = 1. 8.2.ii) If (m, ri) = (1,0) and r2 = 1, then putting x = — T T — - r - r — - — , y = 2 7 22(91-l)32g2p293 ' 23(<>1~1)33'J2p3<» ' Appendix A. Q-Isomorphism Classes 285 we obtain the new model for E Y2 = X3 + 223pT3dX2 + 233p2r3X i which is the curve in case 8 of the lemma with r\ = 2 , r 2 = 0. 8.2.iii) If ( m , r i ) > (1,0) and r 2 = 0, then putting x = x Y=  V 2 2 9 l 3 2 ( 9 2 - l ) j r , 2 ( ? 3 ' 2 3 f ' 1 3 3 ( < ? 2 - 1 )p 3 '?3 ' we obtain the new model for E Y2 = X3 + 2ri32pr3dX2 + 2 m + 2 r i - 2 3 3 p 2 r 3 X which is the curve in case 8 of the lemma with r i = 0 or 1 and r 2 = 1. 8.2.iv) If ( m , r i ) > (1,0) and r 2 = l,then putting x = x y= V 22<n 3292pigs' 2 3 f " 3 3 ' ' 2p 3 (i3 ' we obtain the new model for E Y2 = X3 + 2Ti3pr3dX2 + 2 m + 2 r ' - 2 3 p 2 r 3 X which is the curve in case 8 of the lemma with r\ = 0 or 1 and r 2 = 0. 9. We have i + 2 > a — 2i, j = [3 — 2j and k = 5 — 2k. In this case v 2 (a 2 ) = a — 2i, so a is even. Therefore t>2(a) = | - i. Also, v3(a2) > j = (3 — 2j and vp(a2) > k = d - 2k so v-i(a) > and vp(a) > where e2 denotes the residue of j modulo 2 and 6 3 denotes the residue of k modulo 2 . Let a 22~l3 2 p 2 so (A.2) becomes 3 £ V 3 t x 2 - 2 3 i - a + 2 = ± 1 , w i t h 3 i - a + 2 > 1. Let d = u, m = 3i — a + 2 , then (d, m) is a solution to 3 e 2 p £ 3 d 2 - 2rn = ± 1 , Appendix A. Q-Isomorphism Classes 286 with TO, n > 1, and the model for E can be written y2 = x3+ 2^-i32^Zpt^dx2+ 2i3jpkx. There exist six integers r\, q\, r2, q2, r3, and q3 such that a j' + £2 k + £3 - - ' i = 2 g i + r i , — - — = 2<j2 + r2, — - — = 2g 3 + r 3 , with n , r2, r 3 £ {0,1}. We have four cases to consider: 9.1) Suppose £2 = 0 and £3 = 0. 9.1.i) If (TO, r i ) = (1,0), then putting X 11 : x = ^ —A „ , y 22(gi-l)3292p293 ' 2 3 ( 9 i - 1 ) 3 3 9 2 p 3 g 3 ' we obtain the new model for E Y2 = X3 + 22¥2pr3dX2 + 2332r2p2r3X which is the curve in case 4 of the lemma with £ = 0, n = 0 and r\ = 2. 9.1.ii) If (TO, n) > (1,0), then putting x v ; Y = Y = 2291 32g2p2q3 ' 2 3 ( ' l 3 3 ' « p 3 9 3 ' we obtain the new model for E Y2 = X3 + 2ri3r2pT3dX2 + 2m+2T^H2r2p2r3X which is the curve in case 4 of the lemma with t = 0, n = 0 and r i = 0 or 1. 9.2) Suppose £ 2 = 0 and £3 = 1. 9.2.i) If (TO, r i ) = (1,0), then putting x = -, ' „ * „ y = 2 2 ( g 1 - l ) 3 2 ( ; 2 : p 2 ( g 3 - l + r 3 ) ' 2 3 ( « - ^ ^ t e - H - r s ) ' we obtain the new model for E ' y 2 = X 3 + 2 2 3 r V ~ r 3 d X 2 + 2 3 3 2 7 V " 2 r 3 X which is the curve in case 6 of the lemma with £ = 0, r\ = 2 and r% = 1 - r-3. Appendix A. Q-Isomorphism Classes 287 9.2.ii) If (m, n) > (1, 0), then putting x = . „ _„ * „ y = 2 2 g i 3 2 g 2 p 2 (g 3 - l+ r - 3 ) ' ~ 2 3 9l 3 3 9 2 p 3 ( 9 3 - l + r 3 ) ' we obtain the new model for E Y2 = X3 + 2 2 ~ r i 3 r V d X 2 + 2 r n + 2 - 2 r i 3 2 r 2 p 3 - 2 r 3 X which is the curve in case 6 of the lemma with £ = 0, r\ = 0 or 1 and r3 = l—r3. 9.3) Suppose e2 = l and e3 = 0. 9.3.i) If (m, r a ) = (1,0), then putting x=,„ , , * , „ ,y 22 ( 9 i^ l )32 (q 2 - l+r 2 ) p 2 g 3 ' 2 3 ( « - l ) 3 3 ( ' / 2 - l + r 2 ) p 3 r / 3 ' we obtain the new model for E Y2 = X3 + 2232-r2prsdX2 + 2333-2r2p2r3X which is the curve in case 8 of the lemma with n = 0, r\ = 2 and r 2 = 1 — r2. 9.3.ii) If (rn, n) > (1,0), then putting x= „ „, * , , . ; y = y 22 9 i 32 ( g 2 - l+r 2 )p2g 3 ' 2 3 « 3 3 (92 - l+ r 2 ) p 3 g 3 ' we obtain the new model for E Y2 = X3 + 22~ri32~r2pr3dX2 + 2 m + 2 - 2 r i 3 3 - 2 r 2 v 2 r 3 X which is the curve in case 8 of the lemma with n — 0, r\ = 0 or 1 and r2 1 - r2. . 9.4) Suppose e2 = 1 and £3 = 1. 9.4.i) If (m, n) = (1, 0), then putting x 11 X = "-777 Y = 22(q1-l)32(q2-l+r2)p2(q3-l+r3) ' 2 3 ( 9 1 - 1 ) 3 % 2 - l + r 2 ) p 3 ( g 3 - l + r 3 ) ' we obtain the new model for E Y2 = X3 + 2232~T2 p2~T3dX2 + 2 3 3 3 - 2 7 V ~ 2 r 3 X Appendix A. Q-Isomorphism Classes 288 which is the curve in case 9 of the lemma with r\ = 2, r 2 = 1 — r 2 and r 3 1 - r 3 . 9.4.ii) If (TO, n ) > (1,0), then putting x= „ „ . y - y ( 22qi32(7 2- l+r 2)p2(g3- l+r-3) ' 2 3 ' ? 1 3 3 ( f ' 2 - 1 + r 2 ) p 3 ( 9 3 - l + r 3 ) ' we obtain the new model for E Y2 = X3 + 22~ri32~T2p2~T3dX2 + 2rn+2-2ri33~2r2p3~2r3X which is the curve in case 9 of the lemma with r\ = 0 or 1, r 2 = 1 — r 2 and r3 = 1 - r 3 . 10. We have i + 2 < a - 2i, j > ft - 2j and A; > 5 - 2k. In this case V2{a2) = i 3-2, v3(o?) = (3 — 2j and i j p (a 2 ) = <J — 2fc so z, and 5 are even, i + 1, v3{o) = f - j , and up( a Therefore, v2{a) = f n3(a) | — j , and vp a) = f — k. Let 2 l + 1 3 f - ^ i - f c so (A.2) becomes j u 2 _ ^j-^Sk-S = ± 2 a - 3 , : - 2 ) with a - 3i - 2 > 1, 3j - /? > 1 and 3k - 5 > 1. Let d = u, TO = a — 3i — 2, £ = 3j — (3, n = 3k — S, then (cL m,£,n,p) is a solution to d 2 - 3V ( = ± 2 m , with m,£,n > 1. The model for E can be written y2 = x3 + 2 2 + 1 3 f _ J ' p f ~kdx2 + 2i3jpkx. There exist six integers r\, q\, r 2 , q2, r 3 , and c/3 such that -+ |1 = 2c/!+ri , - - j = 2q2 + r 2 , - - A; = 2g3 + r 3 , w i t h r ! , r 2 , r 3 G {0,1}. Putting x = — — , „——, y = 2 2((/ i- l+r 1 )32 ,72 p 2 9 3 ' 2 3 ( ' ? l - 1 + r l ) 3 3 ' ? 2 p 3 9 3 ' Appendix A. Q-Isomorphism Classes 289 we obtain the new model for E Y2 = X3 + 2 2 - r i 3 r V 3 d X 2 + 2 2 - 2 r i 3 £ + 2 r y t + 2 r 3 X which is the curve in case 10 of the lemma with n = 1 — r\. 11. We have i + 2 < a - 2i, j > [3 - 2j and k < 5 - 2k. In this case v2{a2) = i + 2, v3(a2) = f3 — 2jandvp(a2) = k so i, (3, and k are even. Therefore, v2{a) = | + 1,v3(a) = § - j, and v p(a) = | . Let a u = 22 + 1 32-Jp2 so (A.2) becomes u2 - 3 3 ^ = ± 2 Q - 3 ' : - y - 3 f c ; with a - 3i - 2 > 1, 3j - /? > 1 and 5 - 3fc > 1. Let d = u, m = a — 3i — 2, £ = 3j — (3, n = 5 — 3k, then (d, m, £, n,p) is a solution to d2 - 3e = ±2mpn, with m,£,n> 1. The model for £ can be written y2 = x3+ 2*+l3*-jp*dx2+ 2i3ipkx. There exist six integers r\,q\,r2lq2,r3, and q3 such that i (3 k - + l = 2 g i + r i , --j = 2q2 + r2, - = 2q3 + r 3 , w i t h r i , r 2 , r 3 G {0,1}. Putting 22(9i-l+r1)32g2p2g3 ' ~ 23('?i-l+'-i)33g2p3q3 ' we obtain the new model for E ' Y2 = X3 + 2 2 - r i 3 r 2 pTidX2 + 22-2r^3e+2r2p2r3X which is the curve in case 11 of the lemma with rj = 1 — r\. ) Appendix A. Q-Isomorphism Classes 290 12. We have i + 2 < a - 2i, j > (3 - 2j and k = 5 - 2k. In this case ^2(a 2) — * + 2, v3(o?) = (3 — 2j so i and /3 are even. Therefore, 1*2(0) = | + 1, t>3(a) = I — j. Also, i>p(a2) > = 5 — 2fc so up(a) > where £3 denotes the residue of fc modulo 2. Let u = 2 2 + 1 3 2 - ^ 2 so (A.2) becomes pt3u2 - 33j~p = ±2a~3i~2, with a - 3i - 2 > 1 and 3j - /3 > 1. Let d = u, m = a — 3i — 2, £ = 3j — P, then (d, m, ^) is a solution to p c 3 d 2 _ 3 ^ = ± 2 , n with m,£ > 1. The model for E can be written y 2 = x 3 + 2 f + 1 3 f - J p ^ d x 2 + 2iS>pkx. There exist six integers rlr q\, r2, q2, r3, and c/3 such that i (3 k + 63 - + l = 2 g i + r i , - - j = 2q2+r2, — - — = 2g3 + r 3 , with ri,r2,r3 G {0,1}. We have two cases to consider: 12.1) Suppose £3 = 0. Putting y X = — , n , Y 22( g i - i+n)3292^293 ' 2 3 ( « - 1 + r i ) 3 3 ( ' 2 p 3 < y 3 ' we obtain the new model for E Y2 = X3 + 2 2 _ r i 3 r 2 Pr3dX2 + 22-2r^3e+2r2p2r3X which is the curve in case 11 of the lemma with n = 0 and r\ = 1 — r\. 12.2) Suppose £3 = 1. Putting v = x Y - V r.O(„, _ l a . r , 1o9,.„ . . ' W m _ l J . r ^ ' 2 2 ( ( / i - l + r i ) 3 2 q r 2 p 2 ( 9 3 - l + r 3 ) ' 2 3 ( ' ' l - 1 + 7 ' l ) 3 3 9 2 p 3 ( g 3 - l + r 3 ) : Appendix A. Q-Isomorphism Classes 291 we obtain the new model for E Y2 = X3 + 2 2 " r i 3 r 2 p2-ridX2 + 22-2r^3e+2r2p3-2r3X which is the curve in case 14 of the lemma with r\ = 1 — r\ and r 3 = 1 — r 3 . 13. We have i + 2 < a — 2i, j < [3 — 2j and k > 5 — 2k. In this case V2(a2) = 1 + 2, v3(a2) = j and vp(a2) = 5 — 2k so i, j , and 5 are even. Therefore, v2(a) = | + 1, v3(a) = | , and vp(a) = | — k. Let 2 i + 1 3 2 p i - f c so (A.2) becomes U2 - p3k~S = ± 2 ° - 3 i - 2 3 / 5 - 3 i ; with a - 3i - 2 > 1, (3 - 3j > 1 and 3k - S > 1. Let d = u, m = a — 3i — 2, (. = (3 — 3j = I, n = 3k — 5, then (d, ?n ; ^, n,p) is a solution to d2 - pn = ±2m3e, with rn, £,n> 1. The model for E can be written y2 = x3 + 22 + 1 32pf- f c dx 2 + 2 ' 3 ; / x . There exist six integers r\, q\, r2, q2, r 3 , and q% such that %- + l = 2 q i + n , J- = 2q2+r2, 6--k = 2q-i + r-i, w i t h ? ' i , r 2 . r 3 G {0,1}. Putting x= „, * „ „ , y = 22( (/i-l+r1)32g 2p2g 3 ' 2 3('?l- 1+ rl)33 '/2p3g3 ' we obtain the new model for E Y2 = X3 + 2 2 - r i 3 r V ' 3 d X 2 + 2 2 - 2 r ' 3 2 r 2 p n + 2 r s X which is the curve in case 12 of the lemma with r\ = 1 — r\. Appendix A. Q-Isomorphism Classes 292 14. We have i + 2 < a - 2i, j < (3 - 2j and k < 8 - 2k. In this case V2(a2) = i + 2, V3,(a2) = j and vv(a2) = k so i, j, and k are even. Therefore, V2{a) = | + 1, vs(a) = \, and v p(a) = | . Let u = 22+132p2 so (A.2) becomes u2 - 1 = ± 2 ^ ' ^ - V" 3 f e , with a - 3i - 2 > 1, /? - 3J > 1 and <5 - 3fc > 1. Let d = u, m = a — 3i — 2, £ = (3 — 3j, n = 8 — 3k, then (d, m, £, n,p) is a solution to d 2 - 1 = ± 2 m 3 V , with m,£,n > 1. The model for £ can be written y2 = x3 + 25 + 1 3tpf dx 2 + 2i3jpkx. There exist six integers n , r/i, r 2 , g 2 , rs, and 5/3 such that l- + 1 = 2c/! + n , i = 2g2 + r-2, ^ = 2g3 + r 3 , with r i , 7 - 2 , ?"3 G {0,1}. Putting A - — , . . Y - " 22(<Ji-l+n)32(;2p2g3 ; 2 3 ( '»~ 1 + r i )3 3 9 2 J» 3 « ' we obtain the new model for E Y2 = X3 + 2 2 - ' r i 3 r V W 2 + 2 2 - 2 r i 3 2 r 2 p 2 r 3 X which is the curve in case 13 of the lemma with r\ = 1 — r\. 15. We have i + 2 < a - 2i, j < (3 - 2j and k = 8 - 2k. In this case ^ ( o 2 ) = 7 + 2 and 7j3(a 2 ) = j so i and j are even. Therefore, ?j2(a) = | + 1 and i>3(a) = | . Also, u p(a 2) > A: = 5 - 2/c so 7j p(a) > 4^p- where C3 denotes the residue of k modulo 2. Let a U ~ i n j H - t 3 22 + 1 32p 2 Appendix A. Q-Isomorphism Classes 293 so (A.2) becomes pCiu2 - 1 = ±2a~3i-23l3-3j, with a - 3i - 2 > 1 and (3 - 3 j > 1. Let d = u, rn = a — 3i — 2, £ = (3 — 3j, then (d, m, )^ is a solution to p £ 3 d 2 - 1 = ±2m3e, with m, £ > 1. The model for E can be written y2 = x3 + 2 t + 1 3 2 p t d x 2 + 2i33pkx. There exist six integers r\, q\, r2, q2, r3, and q3 such that - + 1 = 2 ^ + n , - = 2 9 2 + r 2 , — - — = 2(?3 + r 3 , with r j , r 2 , r3 G {0,1}. We have two cases to consider. 15.1) Suppose £3 = 0. Putting X = „ , , , * „ , Y V 22(9i-i+n)32</2p2(/3' 2 3('?i- 1+ ri)33'/2p3-73' we obtain the new model for E Y2 = X3 + 2 2 - r i 3 T ' 2 pr3dX2 + 22~2ri32r2p2riX which is the curve in case 13 of the lemma with n = 0 and r\ = 1 — r\. 15.2) Suppose £ 3 = 1. Putting 22(gi-i+n)32<72p2(93-i+r3) ' 2 3 ( 9 1 - 1 + r i ) 3 3 « p 3 ( f « - 1 + r 3 ) ' we obtain the new model for E Y2 = X3 + 2 2 _ n 3 r 2 p2~T3dX2 + 22-2ri32T2p3~2TiX which is the curve in case 15 of the lemma with r\ = 1 — r\ and r 3 = 1 - r 3 . 16. We have i + 2 < a — 2i, j = f3 — 2j and k > 5 — 2k. In this case 7j2(a2) = i + 2, and vp(a2) = 5 — 2k so i and 8 are even. Therefore, v2(a) = | +1 Appendix A. Q-Isomorphism Classes 294 and vp(a) = | - k. Also, v3(a2) > j = (3-2j so v3(a?) > where e2 denotes the residue of j modulo 2. Let u = 2 2 + 1 3 2 P2~ so (A.2) becomes 3 £ 2 t i 2 ~p3k~6 = ± 2 a " 3 i - 2 , with a - 3z - 2 > 1 and 3k - 6 > 1. Let d = u, rn = a — 3i — 2, n = 3k — 5, then (d, m, n,p) is a solution to 3 £ 2 d 2 - pn = ± 2 m , with m, n > 1. The model for E can be written y2 = x3 + 2 t + 1 3 ^ p 2 - f c d x 2 + 2l3Jpkx. There exist six integers r\, q\, r 2 , g 2 , r 3 , and g 3 such that ^ + l = 2 g 1 + r 1 , i ± f 2 . = 2 9 2 + r 2 , ^ - f c = 2g3 + r 3 , with r i , r 2 , r 3 G {0,1}. We have two cases to consider. 16.1) Suppose e2 = 0. Putting X = , * , , Y - y 22(qi-l+ri)3292p293 ' 2 3 ( «" 1 + r i ) 3 3 f ' 2 p 3 ' « ' we obtain the new model for E Y2 = X3 + 22~ri3r2 pTidX2 + 2 2 - 2 r i 3 2 r 2 p n + 2 r i X which is the curve in case 12 of the lemma with £ = 0 and r\ = 1 - n . 16.2) Suppose e2 = 1. Putting x v 22(<?i-l+r1)32(72-l+r2)p2<?3 ' 2 3 ( « ~ 1 + r i ) 3 3(9 2 " - 1 + 7 ' 2 ) p 393 : we obtain the new model for E Y2 = X3 + 22-ri32~r2pr3dX2 + 22-2r^33~2r2pn+2riX Appendix A. Q-Isomorphism Classes 295 which is the curve in case 16 of the lemma with r\ = 1 — r\ and r2 = 1 — r2. 17. We have i + 2 < a - 2i, j = {3 - 2j and k < 8 - 2k. In this case v2{a2) = i + 2, and vp(a2) = k so i and are even. Therefore, v2(a) = | +1 and Up(a) = | . Also, v3(a2) > j = (3 - 2j so vz(a2) > ^y 2 - where e2 is the residue of j modulo 2. Let u = 2 2 + 1 3 ~ p so (A.2) becomes 3c2u2 - 1 = ± 2 a - 3 i - y - 3 f c , with a - 3i— 2 > 1 and 5 - 3k > 1. Let d = u, m = a — 3i — 2, n = <5 — 3/c, then (d, TO, n,p) is a solution to 3 t 2 d 2 - 1 = ±2mpn, with TO, n > 1. The model for £" can be written j / 2 = x 3 + 2^+l3^p^dx2 + 2l3Jpkx. There exist six integers r\, q\, r2, q2, r3, and q3 such that i ? + £2 k - + l = 2 9 l + r 1 , J-^— = 2q2 + r2, - = 2q3 + r3, with r i , r2,r3 G {0,1}. We have two cases to consider. 17.1) Suppose £2 = 0. Putting X= „ , * „ „ ,Y V 22(91-1+7-1)3292^293' 2 3 ( 9 i - 1 + 7 ' i ) 3 3''2p 3« ' we obtain the new model for E Y2 = X3 + 22~ri3r2 pr3dX2 + 22'2ri32r2p2r3X which is the curve in case 13 of the lemma with £ = 0 and r\ = 1 - r\. 17.2) Suppose £2 = 0. Putting 22(91-1+^)32(92-1+7-2)^293 ' 2 3(9i - 1+ r i)3 3(«2 - l+'-2)p393 : Appendix A. Q-Isomorphism Classes 296 we obtain the new model for E Y2 = X3 + 22-ri32-r2pr3dX2 + 2 2 - 2 n 3 3 - 2 r 2 p 2 r s X which is the curve in case 17 of the lemma with r\ = 1 — r\ and r 2 = 1 — r 2 . 18. We have i + 2 < a — 2i, j = f3 — 2j and k = 5 — 2k. In this case V2(a2) = i + 2, so % is even. Therefore V2(a) = | + 1. Also, v 3 (a 2 ) > j = (3 — 2j and vp(a?) > k = 8 — 2k so w3(a) > 3-^2- and vp(a) > 4^p- where £ 2 denotes the residue of j modulo 2 and £3 denotes the residue of k modulo 2. Let 22"1_13 2 p 2 so (A.2) becomes 3t2pt3u2 -1 = ± 2 " - 3 * ' - 2 , with a - 3 i - 2 > 1. Let d = u, m = a — 3i — 2, then (d, m) is a solution to 3£ 2/>£ 3d2 - 1 = ± 2 " \ with m,n > 1. The model for E can be written y 2 = x 3 + 22 + 1 3 — p 2 dx 2 + 2'3> f cx. There exist six integers r\, q\, r 2 , c/2, rs, and 53 such that - + 1 = 2 < 7 i + n , = 2c/2 + r 2 , - = 2c/3 + r 3 , with r i , r 2 , r 3 e {0,1}. There are four cases to consider. 18.1) Suppose (.2 = 0 and £3 = 0. Putting x= „, * „ , y = y 22(gi- 1+n)32.72p293 ' 2 3 ( « - 1 + r i ) 3 3 9 2 p 3 9 3 ' we obtain the new model for E Y2 = X 3 +- 2 2 - r i 3 r 2 p r W 2 +- 2 2 - 2 r i 3 2 r 2 p 2 r 3 X Appendix A. <Q-Isomorphism Classes 297 which is the curve in case 13 of the lemma with £ = 0, n = 0 and r\ = 1 — r\. 18.2) Suppose £2 = 0 and £3 = 1. Putting x y X — -T7T, 7~1 TT^ 7Ti 7~i Ti Y ~ 22(qi-l+ri)3252p2( ( /3-l+r3) ' " 2 3 ( ^ - 1 + r l ) 3 3 < ?2p 3 ( «3 - l + r 3 ) ' we obtain the new model for E Y2 = X3 + 22-ri3T2p2-r3dX2 + 22~2ri32r2p3-2r3X which is the curve in case 15 of the lemma with £ = 0, r\ = 1 — r\ and n = 1 - r 3 . 18.3) Suppose £2 = 1 and £3 = 0. Putting X = T T ^ — — r z ^ r — — — r ^ ; — , Y ^ 22(gi-l+n)32(92-l+r2)p293 ' 2 3 ( ( « - l + n ) 3 3 ( < Z 2 - l + r 2 ) p 3 9 3 ' we obtain the new model for E Y2 = X3 + 22-ri32-r2pT3dX2 + 22~2ri33-2r2p2riX which is the curve in case 17 of the lemma with n = 0, r\ = 1 - r\ and T2 = 1 - T2. 18.4) Suppose £2 = 1 and £3 = 1. Putting x= „, , „, * , „, „ y = y 2 2 ( < 7 i - l + r i ) 3 2 ( q 2 - l + r 2 ) p 2 ( l 7 3 - l + r 3 ) ' 2 3 ( « - 1 + r l)3 3 ( ( '2-l + f2 )p3 ( W - l+r 3 ) ' we obtain the new model for E Y2 = X3 + 22-r'-32-r2p2~r3dX2 + 2 2 - 2 n 3 3 - 2 r - 2 p 3 - 2 r 3 X which is the curve in case 18 of the lemma with n = 1 — n , r2 = 1 - r2 and r 3 = 1 - r 3 . 19. We have i + 2 = a - 2i, j > (3 - 2j and k > 5 - 2k. In this case u 3 (a2) = (3 — 2j and vp(a2) = 6 — 2k so (3, and £ are even. Therefore, v3(a) = § - j and vp{a) = § - fc. Also, W2(a 2 ) => i + 2 = a - 2i so t/2(a) => 1 : y J - + 1 where £1 is the residue of i modulo 2. Let 2 2 + 1 3 2 - J P 2 - f e Appendix A. Q-Isomorphism Classes 298 so (A.2) becomes 2 £ l u 2 - 3 3 j - V f c ~ * = ± l , with 3j - (3 > 1 and 3k - 8 > 1. Let d = u, £ = 3j — (3 = £, n = 3k — 8, then (d, £, n,p) is a solution to 2 e i d 2 - 3 V = ± l , with £,n > 1. The model for E can be written y2 = x3 + 2 ^ + 1 3 f - J p ^ ~ k d x 2 + 2?:3Jpfca:. There exist six integers r\,q\, r2, c/2/ ^3/ arid c/3 such that i + ei f3 8 —^— + l = 2q1+r1, - - j = 2q2+r2, - - k = 2q3 + r3, with r\, r2, r3 £ {0,1}. We have two cases to consider. 19.1) Suppose ei = 0. Putting y x= „ n , y 2 2 ( 9 1 - 1 + ^ ) 3 2 9 2 ^ 2 9 3 ' 2 3 ( ' / i _ : 1 + r i ) 3 3 ' / 2 p 3 ' « : we obtain the new model for E Y2 = X 3 + 2 2 - r i 3 r V r W 2 + 2 2 - 2 ^ + 2 r ' V + 2 , ' 3 X which is the curve in case 10 of the lemma with m = 0 and r\ = 1 — r\. 19.2) Suppose ei = l . Putting v = x Y = V 2 2 (q i -1)3292^ ,293 ' 2 3 ( « ~ 1 ) 3 3 < « p 3 < ' 3 ' we obtain the new model for E Y2 = X3 + 2ri+23r2 Pr3dX2 + 2 2 r i + 1 3 * + 2 7 " 2 p n + 2 r 3 X which is the curve in case 19 of the lemma. 20. We have i + 2 = a - 2i, j > [3 - 2j and k < 8 - 2k. In this case v3(a2) = (3 — 2j and vp(a2) = k so (3, and k are even. Therefore, v3(a) = | - j , Appendix A. Q-Isomorphism Classes 299 and vp(a) — | . Also, v2(a2) => i + 2 = a - 2i so v2(a) => + 1 where ej is the residue of i modulo 2. Let 2 2 + 1 3 2 J ? 2 so (A.2) becomes 2Clu2 - 3 3 j - / ? = ±p*" 3 f c , with 3j - /? > 1 and 8 - 3k > 1. Let d = u, £ = 3j — f3, n = 8 — 3k, then (d, £, n, p) is a solution to 2 £ lo? - 3* = ±pn, with £, n > 1. The model for E can be written y2 = x3 + 2 i d r L + 1 3 f - j p t d a : 2 + 2*3? pkx. There exist six integers r\, q\, r2, q2, r 3 , and q3 such that i -\-1\ 0 k —2—+ 1 = 2 g 1 + r i , - - j = 2c/2 + r 2 , - = 2g 3 + r 3 , with r i , r 2 , r 3 € {0,1}. We have two cases to consider. 20.1) Suppose e i = 0. Putting x = „ , y 2 2( 9i - l+n) 3 2 9 2p2 9 3 ' 2 3 ( « - l + n ) 3392^93 ' we obtain the new model for E Y2 = X3 + 2 2 ' r i 3 r 2 pr3dX2 + 22-2ri3e+2r2~p2r3X which is the curve in case 11 of the lemma with m = 0 and r\ = 1 - r\. 20.2) Suppose ei = 0. Putting x= „, A , , y = y 2 2 ( 9 1 - 1 ) 3 2 9 2 ^ 2 9 3 ' _ 2 3 ( 9 l - 1 ) 3 3 ' ? 2 p 3 9 3 ' we obtain the new model for E y 2 = X 3 + 2 r i + 2 3 r 2 pT3dX2 + 2 2 r i + 1 3 e + 2 r 2 p 2 r 3 X Appendix A. Q-Isomorphism Classes 300 which is the curve in case 20 of the lemma. 21. We have i + 2 = a - 2i, j > (3 - 2j and k = 5 - 2k. In this case v3{a2) = (3 — 2j so (3 is even. Therefore, 1*3(0 ) = | — j. Also, v2(a?) >i + 2 = a — 2i and vp(a?) > k = 5 - 2k so let e\ and £3 denote the residues of i and k modulo 2, respectively. Then v2(a) > ^ + 1 and -wp(a) > = ^ 3 . . Let a u = 2 2 + 1 3 2 - J p - r so (A.2) becomes 2^pt2u2 - 3 3 j ' - ^ = ± 1 , with 3j - /3 > 1. Let d = u, £ = 3j — f3, then (d, m, I) is a solution to 2clpC2d2 - 3 e = ± 1 , with ^ > 1. The model for i? can be written y 2 = x3 + 2 l i 2 i i + 1 3 f - J p ^ d x 2 + 2i3jpkx. There exist six integers r\, q\, r2, q2, r3, and q3 such that — \-l = 2qi+n, - - 3 =2q2 + r2, — - — =2g 3 + r 3 , with r i , r2, r3 e {0,1}. We have four cases to consider: 21.1) Suppose £i = 0 and £3 = 0. Putting X = * . „ , y = 22(91-1+^)3292^293' ~~ 2 3 (< ' i - 1 + r i ) 3 3 ' ? 2 p 3 ' ? 3 ' we obtain the new model for E Y2 = X3 + 22~ri3r2 pr3dX2 + 22-2ri3i+2r2p2r3X which is the curve in case 11 of the lemma with m = 0, n = 0 and r\ = 1 — r\. 21.2) Suppose £i = 0 and £ 3 = 1. Putting X = ——, „ _ „ —, — r , Y 22(9i-l+ri)3292p2(93-l+r 3) ' 2 3 ( 9 i - 1 + r i ) 3 3 < ? 2 p 3 ( 9 3 - l + r 3 ) ; Appendix A. Q-Isomorphism Classes 301 we obtain the new model for E Y2=X3 + 22-r^p2-rMX2 + 22-2ri3<+2r 2 p3-2r 3 X which is the curve in case 14 of the lemma with m = 0, r\ = 1 — r\ and r 3 = 1 - r 3 . 21.3) Suppose e\ = 1 and £ 3 = 0. Putting *=_„, ,* „ ,Y = 22(<7i-l)32g2p2g3 ' 2 3( < Ji- 1)3 392p3g 3 ' we obtain the new model for E Y2 = X3 + 2^+23r*pr3dX2 + 22ri+13e+2r*p2r3X which is the curve in case 20 of the lemma with n = 0. 21.4) Suppose e i = l and £ 3 = 1. Putting x= „, „ * , y = y 22(gi-1)3292^2(93-1+1-3) ' 2 3 ( , ?l - l)3 3 92 J D3(9 3 - l+r3) ' we obtain the new model for E Y2 = X3 + 2 r i + 2 3 V _ r 3 d X 2 + 2 2 n + l 3 / + 2 r 2 j , 3 - 2 r 3 X which is the curve in case 23 of the lemma with r 3 = 1 — r 3 . 22. We have i + 2 = a - 2i, j < P - 2j and fc > 5 - 2fc. In this case 7j 3 (a 2 ) = j and vp(o2) = 5 — 2k so j , and 5 are even. Therefore, u3(a) = § and •Up(a) = § - A;. Also, -^(a 2) => i + 2 = a - 2i so U2(a) => + 1 where e i is the residue of i modulo 2. Let u 2 2 + 1 3 2 P 2 - f c so (A.2) becomes 2 ^ u 2 _ j 3 3 / c - 6 = ± 3 / 5 - 3 , ; with /? - 3j > 1 and Sk - 6 > 1. Let d = u, £ = P — 3j, n = 3k — 5, Appendix A. Q-Isomorphism Classes 302 then (d, £, n, p) is a solution to 2eid2 -pn = ±3e, with £,n> 1. The model for E can be written y2 = x3 + 2^+l3*p?~kdx2 + 2i3jpkx. There exist six integers r\, q\, r2, q2, r3, and q3 such that ^ i + l = 2 g 1 + n , i = 2c/2 + r 2 , 5--k = 2q3 + r3, with r i ,r2,r3 e {0,1}. We have two cases to consider. 22.1) Suppose ei = 0. Putting 22(qri-l+ri)3292p2q3 ' 23(</i-l+ri)33<72p3g 3 ' we obtain the new model for E Y2 = X3 + 22~Tl3r2pr3dX2 + 22^32r2pn+2r3X which is the curve in case 12 of the lemma with m = 0 and r\ = 1 — r\. 22.2) Suppose f.\ = 1. Putting 2 2 ( '? i - 1 )3 2 < '2p2(/3 ' 2 3 ( f ' 1 - 1 )3 3 r ' 2 p 3 ( ? 3 ' we obtain the new model for E Y2 = X3 + 2 r i + 2 3 r V W 2 + 2 2 r ' i + 1 3 2 r V l + 2 r 3 X which is the curve in case 21 of the lemma. 23. We have i + 2 = ct - 2i, j < (3 - 2j and k > 6 - 2k. In this case v3(a2) = j and f p (a 2 ) = k so j , and A; are even. Therefore, ^ (a) = | and up(d) = | . Also, ?j2(a2) => i + 2 = a — 2i so ^(a) => ^^p- + 1 where e\ is the residue of i modulo 2. Let u = , * ± a + i „ i * 2 2 + I 35p2 so (A.2) becomes 2 e i n 2 - 1 = ±3^-3jps-3k, Appendix A. Q-Isomorphism Classes 303 with (3 - 3j > 1 and 5 - 3k > 1. Let d = u, £ = (3 — 3j, n = 5 — 3k, then (d, I, n,p) is a solution to 2eid2 - 1 = ±3epn, with I, n > 1. The model for E can be written y 2 = x3 + 2 i i 2 l i + 1 3 2 p f dx 2 + 2l3jpkx. There exist six integers r\, q\, r2, q2, T3, and q3 such that %-^7p- + 1 = 2q1+r1, J- = 2q2 + r2, ^ = 2g 3 + r 3 , with r i , r 2 , r 3 G {0,1}. We have two cases to consider: 23.1) Suppose e i = 0. Putting 22(qi-i+n)32< ? 2p293' 2 3 ( « - 1 + r i ) 3 3 " 2 p 3 ' ? 3 ' we obtain the new model for E Y2 = X3 + 2 2 - r i 3 7 V ' 3 d X 2 - + 2 2 ~ 2 n 32r'2p2r3X which is the curve in case 13 of the lemma with m = 0 and r\ = 1 — r\. 23.2) Suppose t\ = 0. This is impossible since there are no solutions to the equation 2d2 — 1 = ±3epn with £ > 1 due to a local obstruction at 3. 24. We have i + 2 = a - 2i, j < [3 — 2j and k = 5 - 2k. In this case t/3(a2) = j so j is even. Therefore, v3(a) = | . Also, v2(a2) >i + 2 = a-2i and u p(a 2) > k = 5 — 2k so let e\ and e3 denote the residues of i and A: modulo 2, respectively. Then v2(a) > ^ + 1 and vp(a) > Let a U = — T : TT 2 — + 1 3 2 p 2 so (A.2) becomes 2 e V 2 t i 2 - 1 = ±3l3-3j, Appendix A. Q-Isomorphism Classes 304 with/3-3.7 > l .Le t d = u, £ = (3 — 3j, then (d, I) is a solution to 2eipe2d2 - 1 = ± 3 ' , with £ > 1. The model for E can be written y2 = x3 + 2^+lS2p^dx2 + 2i3jpkx. There exist six integers r\, q\, r2, q2, r 3 , and q3 such that — + 1 = 2qi +ru -=2q2 + r2, — - — = 2q3 + r 3 , with n , r 2 , r 3 G {0,1}. We have four cases to consider: 24.1) Suppose ei = 0 and e3 = 0. Putting 7* li X = ^r, . , „ „ , Y 22(9 l-i+n)3292j t,293' 23(<Ji-1+'-i)33q2p3<j3' we obtain the new model for E Y2 = X3 + 2 2 ~ n 3 r 2 pT3dX2 + 22-2n32r2p2r3X which is the curve in case 13 of the lemma with m = 0, n = 0 and r\ = 1 — r\. 24.2) Suppose ei = 0 and e3 = 1. Putting 22 (9 i - l+ri )32q 2 p2 ( q a - l+r 3 ) ' ~~ 2 3 (<?i -H-r i )33g 2 ? ,3 (g3- l+r 3 ) ' we obtain the new model for E Y2 = X3 + 22~ri3r2 p2~rsdX2 + 22-2ri32r2p3-2r3X which is the curve in case 15 of the lemma with m = 0, r\ = 1 — r\ and r 3 = 1 - r3. 24.3) Suppose t\ = 1 and e3 = 0. This is impossible since there are no solutions to the equation 2d2 — 1 = ±3^, with £ > 1, due to a local obstruction at 3. Appendix A. Q-Isomorphism Classes 305 24.4) Suppose ei = 1 and £3 = 1. Putting x v v _ y ... y ^ 9 2 ( f l 1 - l ) Q 2 « 2 T ) 2 ( q 3 - l + r 3 ) ' 2 2 ( q 1 - l ) 3 2 9 2 p 2 ( q 3 - l + r 3 ) ' 23(<?1 "1) 3 3 g 2 p 3 ( g 3 - l + r 3 ) ' we obtain the new model for E Y2 = X3 + 2 n + 2 3 r V - r W 2 + 22ri+132r2p3-2r3X which is the curve in case 24 of the lemma with r3 = 1 — r3. 25. We have i + 2 = a - 2i, j = 0 - 2j and k > 6 - 2k. In this case vp(a2) = 5 — 2k so 6 is even. Therefore, vp(a) = | - A;. Also, t>2(a2) > 7 + 2 = a — 2? and t;3(a2) > j = 0 — 2j so let ei and E 2 denote the residues of i and j modulo 2, respectively. Then 7j 2(a) > ^p- + 1 and 173(a) > 4^p. Let a 7i = 2 2 + 1 3 2 p 2 - f e so (A.2) becomes 2ei3e2u2 - P 3 k ~ s = ± 1 , with 3A;-<S > l .Le t d = u, n = 3k — 5, then (d, n,p) is a solution to 2 £ l 3 £ 2 d 2 - p " = ± 1 , with ?i > 1. The model for E can be written y 2 = X 3 + 2!±^ + \ 3 ^ p ^ - k d x 2 + 2i3ipkX. There exist six integers r\,q\, r 2 , q2, r3, and q3 such that —2—+ 1 = 2c/!+r-i, — J = 2q2 + r2, - - k = 2q3 + r3, with r i , 7 - 2 , 7 - 3 G {0,1}. We have four cases to consider: 25.1) Suppose ei = 0 and £ 2 = 0. Putting v = x Y = 22(91-1+^)3292^293 ' 2 3 ( « i - 1 + r i)33g2p3g 3 ' Appendix A. Q-Isomorphism Classes 306 we obtain the new model for E Y2 = X3 + 2 2 - r i 3 7 V W 2 + 2 2 - 2 r i 3 2 r y + 2 r 3 X which is the curve in case 12 of the lemma with m = 0,£ = 0 and r\ = 1 — r\. 25.2) Suppose t\ = 0 and £2 = 1- Putting 22(9i-l+ri)32(g2-l + r-2)p2g3 ' 23(f'l-1+rl)33(<?2-l+'<-2)p3(73 ' we obtain the new model for E Y2 = x3 + 2 2 - r i 3 2 " r y w 2 + 2 2 - 2 7 - i 3 3 - 2 r y + 2 r 3 x which is the curve in case 16 of the lemma with m = 0, r\ = 1 — r\ and r 2 = 1 - r 2 . 25.3) Suppose ei = 1 and e2 = 0. Putting X= „, A „ . Y - V 22(gi-l)32<72p2g3 ' 23(f'i-1)33<?27>3,'3 ' we obtain the new model for E Y2 = X3 + 2 n + 2 3 r y ' W 2 + 2 2 ' ' i + 1 3 2 r 2 p n + 2 r 3 X which is the curve in case 21 of the lemma with £ = 0. 25.4) Suppose £i = 1 and £2 = 1- Putting X= „, * , , „ , Y " 22(q1-l)32(,n-l+r2)p2q3 ' 2 3(«" 1)3 3( ('2-l+r-2)p3r /3 ' we obtain the new model for E Y2 = X3 + 2 r i + 2 3 2 ~ r ' V 3 d X 2 + 2 2 r i + 1 3 3 - 2 r 2 p n + 2 r ' 3 X which is the curve in case 25 of the lemma with r2 = 1 — r2-26. We have i + 2 = a — 2%, j = ,3 - 2j and k < 5 - 2k. In this case vp(a2) = k so k is even. Therefore, vp(a) = | . Also, v2(a2) >i + 2 = a — 2i and 7j 3(a 2) > j = [3 — 2j so let e\ and £2 denote the residues of i and j modulo 2, respectively. Then v2(a) > ^ y 1 + 1 and v3(a) > 14p. Let a u 2 2 + 1 3 2 pi Appendix A. Q-Isomorphism Classes 307 so (A.2) becomes 2 e i 3 £ 2 n 2 - 1 = ± / ~ 3 f c , with 5-3k>l. Suppose that (d, n,p) is a solution to 2 £ l 3 £ 2 d 2 - 1 = ± p n , with n > 1. Then we may write u = d, 5 — 3k — n. Then the model for E can be written y2 = x 3 + 2 ^ + 1 3 ^ p 2 d x 2 + 2i&pkx. There exist six integers r\, q\, r2, q2, r3, and g 3 such that — h i = 2(7i + r r , — j = 2g2 + r 2 , - = 2q3 + r 3 , with n , r 2 , r 3 G {0,1}. We have four cases to consider: 26.1) Suppose €\ = 0 and t.2 = 0. Putting x x = — — * „ „ , y 2 2 ( f / i - l + r-])32<j2p2g3 ' 2 3 ( « ~ 1 + r i ) 3 3 < 7 2 p 3 ' J 3 ' we obtain the new model for E Y2 = X3 + 22~ri3r2 prsdX2 + 2 2 - 2 r i 3 2 r V r 3 X which is the curve in case 13 of the lemma with m = 0,1 = 0 and 7*1 = 1 — r\. 26.2) Suppose t\ = 0 and e2 = 1. Putting x= „ , ! _ , y = y 2 2 ( q i - l + r i ) 3 2 ( , , 2 - l + r 2 ) p 2 < 7 3 ' 2 3 ( ' 7 l - l + n ) 3 3 ( g 2 - l + r 2 ) p 3 g 3 ' we obtain the new model for E Y2 = X3 + 2 2 - r i 3 2 " 7 V 3 d X 2 + 2 2 - 2 r i 3 3 ~ 2 ? V r 3 X which is the curve in case 17 of the lemma with m = 0, r i = 1 — r\ and r2 = 1 - r 2 . Appendix A. Q-Isomorphism Classes 308 26.3) Suppose ei = 1 and e2 = 0. Putting x= „, A „ , y 22(91 -1)3292^293 ' 2 3 ( '?i- 1 )3 3 ' '2]9 3 ' '3 ' we obtain the new model for E Y2 = X3 + 2r'+23r2Pr3dX2 + 2 2 r i + 1 3 2 7 V r 3 X which is the curve in case 22 of the lemma with £ = 0. 26.4) Suppose £1 = 1 and e2 = 1. Putting y x = ———-—-—-—, y 2 2 ( 9 l - l ) 3 2 ( 9 2 - l + r 2 ) p 2 9 3 ' 2 3(' 'l- 1)3 3(<'2-l+'-2)p3g3 ' we obtain the new model for E Y2 = X3 + 2ri+232"r2pr3dX2 + 22ri+133~2r2p2rsX which is the curve in case 26 of the lemma with r2 = 1 — r2. 27. We have i + 2 = a - 2i, j = f3 - 2j and k = 5 - 2k. In this case v2(a2) > i + 2 = a - 2i, v3(a2) > j = j3 - 2j and vp{a2) > k = 5 - 2k. Let e\, e2 and £3 denote the residues of i, j and k modulo 2, respectively. Then v2(a) > ^ + l, v3{a) > a n d vp{a) > Let 2 2 ^*3 2 p 2 so (A.2) becomes 2 £ l 3 £ 2 p £ 3 u 2 - 1 = ± 1 . Clearly u = 0 is a solution to this equation and this leads to the curve y2 = x3 + 2 r 3V x, where r, s, r; € {0,1,2,3}, which appears in one of the cases 13, 15, 17,18, 22, 24,26, 27 of the lemma with d = 0. The only other solution to 2 £ l 3 £ 2 p £ 3 u 2 — 1 = ± 1 has u = 1 and (ei, e2, £ 3 ) = (1,0,0). The model for E can be written y2 = x3 + 2 ^ 1 + 1 3 2 p f x 2 + 2i3jpkx. Appendix A. Q-Isomorphism Classes 309 There exist six integers r\, q\, r2, qi, r3, and q3 such that i + €\ j k — — + 1 = 2qi + rr, - = 2q2 + r2, - = 2q3 + r3, with r i , r2, r3 £ {0,1}. Putting 2 2(' 'l- 1)3 2 (/2p2 (,3 ' _ 2 3('?l- 1)3 3''2p3<73 ' we obtain the new model for E Y2 = X3 + 2 r i + 2 3 r V W 2 + 22ri+132r2p2r3X which is the curve in case 22 of the lemma with £ = 0 and n = 0. This completes the proof of the lemma. • A . 2 b < 0 Lemma A.2 Suppose b < 0. Then there exists an integer d, and non-negative in-tegers rn,£, and n satisfying one of the equations in the first column and E is Q-isomorphic to the corresponding curve in the second column, for some r\,r2,r3 £ {0,1}; except in cases 2 to 8, where if m = 1 then r\ £ {1,2}. Appendix A. Q-Isomorphism Classes 310 Diophantine Equation y2 = x 3 + a2x2 + CI4X 0,4 2 d2 + 2m3e = pn 2ri3r2pr*d _ 2 m + 2 r 1 - 23^+2r 2 ^2r 3 3 d2 + 2mpn = 3* _2" l+2r-! - 2 3 2 r 2 p n + 2 r 3 4 d2 + 2m = 3epn 2ri 3T2pT3d _2m+2ri — 2r£2r2p2r3 6 pd2 + 2m = 3e 2ri 3r2pr3+1d _2m+2r1-2^2r2p2r3 + l 8 3d2 + 2 m = pn 2 r i y2 + l p r 3 d _2m+2r1-232r2 + lp2r3 10 d2 + 3epn = 2m 2 r i + 1 3 ' ' 2 p r 3 d _22r1 ^e+2r2pn+2r3 11 d2 + 3e = 2mpn 2 r i + 1 3 r 2 p r 3 d _ 2 2 n 3 « + 2 r 2 p 2 r 3 12 d2 + p n = 2m3e 2 r i + 1 3 r 2 p r 3 d — 2 2 r i 3 2 r 2 p n + 2 r 3 13 d2 + l = 2m3epn 2ri+13r2pr3d _ 2 2 r l 3 2 r 2 p 2 r 3 14 pd2 + 3e = 2m 2 r i + 1 3 r 2 p r 3 + 1 d _ 2 2 n 3 « + 2 r 2 p 2 r 3 + l 15 pd2 + 1= 2rn3e 2ri+13r2pr3+1d _ 2 2 . i 3 2 r 2 p 2 r 3 + l 16 3d2 + pn = 2m 2 n + i 3 r 2 + i p r 3 d — 22 ' 1 3 2 r 2 + l p " + 2 r 3 17 3d2 + 1 = 2mpn 2ri+13T2+1pr3d _ 2 2 r 1 3 2 r 2 + l p 2 r 3 18 3pd2 + 1 = 2m 2ri+13r2+1pT3+1d _ 2 2 n 3 2 r 2 + l p 2 r 3 + l 20 2d2 + 3e = pn 2ri+23r2pT3d _ 2 2 r i + l 3 ^ + 2 7 - 2 p 2 r 3 21 2d2 + pn = 3* 2 n + 2 3 r 2 p ' ' 3 d _ 2 2 r i + 1 3 2 r 2 ^ n + 2 r 3 22 2d2 + 1 = 3epn 2 r i + 2 3 r 2 p ' ' 3 d _ 2 2 r i + l 3 2 r 2 p 2 r 3 24 2pd2 + 1 = 3* 2 r i + 2 3 r 2 p r 3 + 1 d _ 2 2 r 1 + l 3 2 r 2 p 2 r 3 + l 26 6d2 + 1 = pn 2 r i + 2 3 r 2 + l p r 3 d _ 2 2 j ! + l 3 2 r 2 + l p 2 r 3 27 6pd2 + 1 = 1 2 r i + 2 3 r 2 + i p r 3 + 1 d _ 2 2 r i + l 3 2 r 2 + l p 2 r 3 + l The proof of this lemma is entirely analogous to that of lemma A . l . The only change that needs to be made is that b, i.e. a 4 , is now negative, and the minus sign on the right-hand side of the Diophantine equations changes to a plus sign. We've kept the numbering of rows in the table the same as the pre-vious lemma. This allows one to see the analogy between the two lemmata. Of course, some of the rows don't appear, say for example, the row analogous to row 1. This row would have equation d2 + 2m3epn = 1, but this has no solu-tions except with d = m = n = £ = 0, and the corresponding curve is already contained in row 2. Appendix B T a b l e s o f 5 - i n t e g r a l P o i n t s o n E l l i p t i c C u r v e s . In this section, we present tables listing all the S-integral points on curves of the form y2 = x3 ± 2 a 3 6 , where 5 = {2, 3, oo}. These results are used in the proofs of the Diophantine lemmata of Chapter 4 (in the case when 3 divides n). For the reader interested in a very brief account of the theory behind com-puting ^-integral points on elliptic curves, we sketch this in the the first two sections. The tables are presented in Section B.3. B . l 5 - i n t e g r a l p o i n t s o n E l l i p t i c C u r v e s Let S be a finite set of primes (places) including the place at infinity; S = {pi,..., ps-i, oo}. The set of S-integers of Q is Zs := {x G Q : \x\p < 1 for all p £ S}, where, for p finite, the \x\p's are the usual (normalized) p-adic absolute values of Q, and for p infinite, |x|oo is the usual archimedean absolute value of Q. In other words, a rational number is an S-integer if the only primes in its denominator are those in 5. Let E be the elliptic curve over <Q> given by the following equation, in long Weierstrass form, y2 + a\xy + a3y = x3 + a2x2 + a\x + a^, ai e Z. The set of integral points of E(Q) is E(Z) = {P G E(Q) : x(P) G Z} 311 Appendix B. Tables of S-integral Points on Elliptic Curves. 312 and the set of S-integral points of E(Q) is E(ZS) = {Pe E(Q) : x(P) e ZS}. The fact that x(P) G Z (resp. Zs) implies y(P) 6 Z (resp. Z s ) , provided y(P) £ Q, is straightforward to check using the equation defining E. Siegel proved in 1929 that the number of integral points on an elliptic curve over a number field is finite and Mahler generalized this result to S-integral points in 1934 (see [Sil:1989]). However, the methods they used to prove these results were not effective, which means that they did not yield an algorithm to find all of the points. In 1968, Baker gave an effective upper bound on the size of integral points, based on his work on linear forms in complex logarithms, thus, theoretically, producing an algorithm to find all integral points. In some cases, this led to the complete determination of sets of solutions to a given elliptic Diophantine equation. However, the bounds one obtains using Baker's work are usually astronomical, typically at least of size 10 2 0 or so, which makes naive search-ing for all points impossible. In his thesis [dW:1989], de Weger developed a technique using lattice basis reduction (LLL algorithm) to reduce the bounds obtained from Baker's work. This resulted in an algorithm to find integral points on elliptic curves which works well in practice (though, one needs to deal with computations in various complicated number fields). This method does not make use of the underlying group structure of the elliptic curve. That, combined with the need to consider complicated number fields, led Lang and Zagier to suggest a way to work directly on the elliptic curve. Moreover, this new approach can be generalized to apply to 5-integral points as well. We discuss this approach in the next section. B.2 C o m p u t i n g 5 - i n t e g r a l p o i n t s o n E l l i p t i c C u r v e s Let 5 = {pi , . • - ,ps-i, oo} and consider the Mordell-Weil group E(Q) of the elliptic curve E : y2 = x 3 + ax + b, Appendix B. Tables of S-integral Points on Elliptic Curves. 313 over Q. Recall that E(Q) is a finitely generated abelian group so it can be written as E(Q) =• E(Q)tors x Zr, where E(Q)tors is the (finite) torsion subgroup of E(Q) and r the rank of E(Q). The method of Lang and Zagier requires that we know generators for E(Q), so, let P\,...,Pr be generators for the free part E(Q). Every rational point P G E(Q) has a unique representation r P = T + ^ n , P i , »=i where n„; £ Z and T G E(Q)tors. In the case when P is an 5-integral point we want to show N := max{\ni\} < N2 for an effectively computable constant JV2 depending only on E and 5. Thus, all 5-integral points on E(Q) are contained in the finite set r {T + J2mPi--Q< \nz\< N,T e E{Q)tors}, and can be determined, provided N2 is small enough. We briefly sketch the details involved in finding the upper bound A 2 . For all the details the reader should consult [GPZ:1996]. Let P = (x, y) G E(Q) be an 5-integral points and choose p G 5 such that \x\p — max{\x\q : q G 5}. It is straightforward to show - A < C2e-C*N2 (B.l) for effectively computable constants C2 and C3 which depend only on a, b, 1 — 1/2 #5, and the generators Pi. Obtaining a lower bound on \x\p together with (B.l) would then give and upper bound on N. If the upper bound on N is quite large then an application of de Weger reduction could bring this bound down to a more manageable level. In practice this is usually the case. Appendix B. Tables of S-integral Points on Elliptic Curves. 314 Lower bounds on \x\p~ are obtained by estimating a linear form in ellip-tic logarithms. In the case when p = oo such estimates were done by David [Da:1995]. In the case when p = Pi G S, estimates for lower bounds of p-adic elliptic logarithms in general are not known. However, if the rank of E(Q) is at most 2 then such a bound was obtained by Remond and Urfels [RU:1996]. Gebel, Petho and Zimmer [GPZ:1996] applied these lower bounds to find all oMntegeral points on Mordell 's curves y2 — x3 + k, with |fc| < 104, and rank at most 2. Their algorithms were implemented in the S I M A T H package, and have since made their way into the M A G M A package. We wi l l use M A G M A to generate the tables in the next section. B.3 T a b l e s o f ^ - i n t e g r a l p o i n t s o n t h e c u r v e s y2 = x3 ± 2a3b For S = {2,3, oo}, the following tables list the 5-integer points on curves of the form y2 = x3 ± 2a3b. It is easy to check all these curves have rank < 2. The points were found using the M A G M A package. If (XQ, yo) is a point on the curve then so is (XQ, —yo), thus, in the tables, we list only the points with non-negative y coordinate. These curves always contain the point at infinity, oo, so, it suffices to list only the finite points in the tables. Values (a, b) are left absent from the table if there are no finite S-integral points on the curve. Appendix B. Tables of S-integral Points on Elliptic Curves. 315 y2 = x 3 + 2 a 3 6 a b S-integral points \{oo} 0 0 (2,3), (0,1), (-1,0) 0 1 (1,2), (-23/16,11/64) 0 2 (0, 3), (-2,1), (3,6), (6,15), (40,253), (-15/16,183/64) 0 3 (-3,0) 0 4 (0,9) 1 0 (-1,1), (17/4, 71/8) 1 2 (7,19) 1 3 (3,9), (-15/4, 9/8), (19/9,215/27), (5745/16,435447/64) 1 5 (-5,19) 2 0 (0,2) 2 1 (-2, 2), (13,47) 2 2 (0, 6), (4,10), (12,42), (-3,3), (105/4,1077/8) 2 3 (6,18), (-3, 9), (-2,10), (33/4,207/8), (366, 7002) 2 4 • (0,18) 3 0 (-2,0), (2,4), (1, 3), (-7/4,13/8), (46,3121) 3 1 (-2,4), (25/4,131/8), (8158, 736844), (1,5), (10/9,136/271), (10, 32), (-23/9, 73/27), (478/81,11044/729), (505/256, 23053/4096) 3 2 (-2,8), (73/16,827/64) 3 3 (-6,0) 3 5 (-2,44) 4 0 (0,4) 4 1 (1,7) 4 2 (0,12) 4 4 (0, 36), (-8, 28), (9,45), (72,612) 5 2 (1,17) 5 5 (-47/9,2359/27) Table B . l : S-integral points on y2 = x 3 + 2 a3 f t Appendix B. Tables of S-integral Points on Elliptic Curves. 316 y2 = x3 - 2 a 3 6 a b S-integral points \{oo} 0 0 (1,0) 0 3 (3,0) 0 4 (13,46) 0 5 (7,10) 1 0 (3,5) 1 2 (3, 3), (57/4,429/8) 1 3 (7,17) 2 0 (2,2), (5,11),(106/9,1090/27) 2 4 (10,26) 2 5 (13,35) 3 0 (2,0) 3 2 (6,12), (33/4,177/8), (1942/9,85580/27) 3 3 (6,0), (10, 28), (33,189) 3 4 (18,72), (153/16,963/64), (657/4,16839/8), (9, 9), (22,100), (1809, 76941),-(-54,396), (97, 955) 3 5 (70,584) 4 1 (4,4), (28,148), (73/9, 595/27) 4 3 (12,36) 4 4 (193,2681) 5 2 (9,21) 5 5 (1153,39151) Table B.2: S-integral points on y2 = x3 - 2 a 3 6 Appendix C T a b l e s o f Q - I s o m o r p h i s m C l a s s e s o f C u r v e s o f C o n d u c t o r 2ap2 w i t h S m a l l p. In the theorems of Chapter 6, we classified curves up to primes p which sat-isfied some family of Diophantine equations. There were some extraneous small primes and corresponding curves that did not fit into any family and we referred to the tables in this appendix for a list those extra curves. Let me emphasize that the following tables list the E X T R A curves that are not contained in the tables of Chapter 6, they do N O T list all the curves of the indicated conductor. In these tables a% and a 4 are the coefficients of the curves as we have found them (by applying the Diophantine lemmata to the tables of Chapter 3 Section 3.1). The minimal model of the curve is also included as 5-tuple of coefficients (a[ ,a'2,a'3,a'4, a'6). It is sufficient to include just the minimal model in the table but we thought we should include the a 2 and a 4 for the sake of the reader who wishes to verify these results. 317 Appendix C. Tables with small p 318 Conductor: TV = 2p2 P ai j-invariant minimal model 7 -7 12, 2 7 7 2 5a43 3 2 « 7 3 1,1,0,220,2192 7 2-7-13 - 7 5 9.3 76 1,1,0,-1740,22184 17 2 - 17-71 175 5fa733l3 217 6 1,1,1,-32663,-1583717 17 -17-71 2 517 3 2 2 1 7 3 1,1,1,-29773,-1989473 Table C l : Extraneous curves of conductor 2p 2. Conductor: N = 22p2 P a2 a<i j-invariant minimal model 5 2-5-11 5 5 -24 109 3 f 0,-1,0 , -908,-15688 5 - 5 • 11 - 5 2 2l 4 31 3 5 a 0,-1,0,-1033,-12438 Table C.2: Extraneous curves of conductor 2 2 p 2 . Appendix C. Tables with small p 319 Conductor: N = 23p2 P 0.2 C14 j-invariant minimal model 5 5-9 2 2 - 5 3 2 4 3 3 7 3 5 2 0,0,0,-175,-750 5 - 2 - 5 - 9 5 2 22 3 3 1 0 7 3 5 0,0,0,-2675,-53250 5 2 - 5 - 3 5 4 22 3 3 1 3 3 5 4 0,0,0,325,-4250 5 5-3 5 2 21 1 3 3 5 0,0,0,-50,125 5 - 5 5 2 1 1 0 , 1 , 0 , - 3 , - 2 5 - 5 2 5 3 2 1 1 0 , - 1 , 0 , - 8 3 , - 8 8 5 2 • 5 5 2 417 3 0,1,0,-28,48 5 2 - 5 2 5 3 2 417 3 0,-1,0,-708,7412 7 7-15 2 3 . 7 3 22 3 3 1 9 3 7^ 0,0,0,-931,-10290 7 - 2 - 7 - 1 5 7 2 2-33 13 3 23 3 7 .0,0,0,-14651,-682570 7 2 - 7 - 9 7 4 2 33 5 9 3 7 4 0,0,0,-2891,47334 7 7-3 2 2 . ? 2 24 3 3 7 0,0,0,49,-686 7 - 7 - 5 2 3 -7 2 - 22 7 0,1,0,-16,1392 7 2 - 7 - 5 - 7 3 2-11" T* 0,1,0,-1976,32752 17 17 2 2 -17 2 4 5 3 0 , - 1 , 0 , - 2 8 , - 1 2 17 172 2 2 • 173 2 4 5 3 0,1,0,-8188,-107904 17 - 2 • 17 17 2 2 5 3 13 3 0, -1 ,0 , -368, -2596 17 - 2 • 172 173 2 2 5 3 13 3 0,1,0,-106448,-13392656 23 23-3 2 3 • 23 2 22 3 3 5 3 23 0,0,0,2645,-73002 23 - 2 - 2 3 - 3 - 2 3 3 2 - 33 5 3 7 3 23 2 0,0,0,-18515,-754354 31 -31 2 3 • 31 2 22 2 3 3 31 0,-1,0,7368,74780 31 2-31 - 3 1 3 2-973 31 2 0,-1,0,-31072,643692 31 -31 2 3 -31 _ 2 2 7 3 0,-1 ,0 , -72,380 31 31 2 2 3 • 31 3 _ 2 2 7 3 0,1,0,-69512,-10626560 31 2-31 -31 2•127 3 0,-1,0,-1312,18732 31 - 2 - 3 1 2 - 3 1 3 2•127 3 0,1,0, -1261152, -545434592 Table C.3: Extraneous curves of conductor 23p2. Appendix C. Tables with small p 320 Conductor: JV = 24p2 P 0.2 a 4 j-invariant minimal model 5 - 5 - 9 2 2 - 5 3 2"3J 7 J 5 2 0,0,0,-175,750 5 2 - 5 - 9 5 2 2^3J107 J 5 0,0,0,-2675,53250 5 - 2 • 5 • 11 5 5 -2"109J 5« 0,1,0,-908,15688 5 5-11 - 5 2 21 "31 J 5 3 0,1,0,-1033,12438 5 - 2 - 5 - 3 5 4 2'':3-,13•;, 0,0,0,325,4250 5 - 5 • 3 5 2 2'13J 5 0,0 ,0 , -50 , -125 5 5 5 2 1 1 0 , - 1 , 0 , - 3 , 2 5 5 2 5 3 2 n 0,1,0,-83,88 5 - 2 • 5 5 2 4 17 3 0 , - 1 , 0 , - 2 8 , - 4 8 5 - 2 5 2 5 3 2 4 17 3 0,1,0,-708,-7412 7 7- 13 2 7 - 7 2 5J 43 J 2 6 . 7 3 0,1,0,3512,-133260 7 - 2 - 7 - 1 3 - 7 5 23 .70 0,1,0,-27848,-1475468 7 - 7 - 3 2 2 - 7 2 2'3i l 7 0,0,0,46,686 7 7 -5 2 3 - 7 2 - 22 7 0 , -1 ,0 , -16 , -1392 7 - 2 - 7 - 5 - 7 3 2 1 1 ° 0, -1 ,0 , -1976, -32752 7 - 7 - 3 2 4 • 7 - 3 3 5 3 0,0,0,-35,98 7 7 2 - 3 2 4 . 7 3 - 3 3 5 3 0, 0,0,-1715,-33614 7 2 - 7 - 3 - 7 3 3 5 3 17 3 0,0,0, -595,5586 7 - 2 • 7 2 - 3 - 7 3 3 3 5 3 17 3 0,0,0,-29155,-1915998 17 - 2 - 1 7 - 7 1 17 5 5 ° 7J 3 1 J 2-17B 0,1,0,-522608,100312660 17 17-71 2 5 • 17 2 5J23J43- :' 22.173 0,1,0,-476368,126373524 17 - 1 7 - 9 2 4 -17 2 •s-'w3 17 0,0,0,-3179,-29478 17 -17 2 2 -17 2 4 5 3 0,1,0,-28,12 17 - 1 7 2 2 2 • 17 3 2 4 5 3 0,-1,0,-8188,107904 17 2- 17 17 2 2 5 3 13 3 0,1,0,-368,2596 17 2-17 2 1 7 3 2 2 5 3 13 3 0,-1,0,-106448,13392656 23 - 2 3 - 3 2 3 • 23 2 2-'3JS'i 23 0,0,0,2645,73002 23 2-23-3 - 2 3 3 2-3J 5 J 7 J 23 2 0,0,0,-18515,754354 31 31 2 3 - 31 2 22 23 3 31 0,1,0, 7368,-74780 31 - 2 - 3 1 - 3 1 3 2-97J 31 2 0,1,0,-31072,-643692 31 31 2 3 • 31 _ 2 2 7 3 0,1 ,0 , -72 , -380 31 - 3 1 2 2 3 • 31 3 _ 2 2 ? 3 0,-1,0,-69512,10626560 31 - 2 - 3 1 -31 2 • 1273 0,1,0,-1312,-18732 31 2 -31 2 - 3 1 3 2•127 3 0, - 1 , 0 , -1261152,545434592 Table C.4: Extraneous curves of conductor 24p2. Appendix C. Tables with small p 321 Conductor: N = 25p2 P a 4 j-invariant minimal model 7 7 2 • 7 2 7 0,1,0,82,-176 7 - 7 2 • 7 2 2b 5 3 7 0,-1,0,82,176 7 2-7 - 7 3 0,-1,0,-408,1940 7 - 2 - 7 - 7 3 2a 5 b 7 2 0,1,0,-408,-1940 7 7 2 • 7 - 2 6 0 , 1 , 0 , - 2 , - 8 7 - 7 2 • 7 - 2 6 0 , - 1 , 0 , - 2 , 8 7 7 2 2 • 7 3 - 2 6 0,1,0,-114,-2528 7 - 7 2 2 • 7 3 - 2 6 0,-1,0,-114,2528 7 2 • 7 - 7 2 3 31 3 0,-1 ,0 , -72,260 7 - 2 • 7 - 7 2 3 31 3 0,1,0, -72,-260 7 2 • 7 2 - 7 3 2 3 31 3 0,-1,0,-3544,82104 7 - 2 • 7 2 - 7 3 2 3 31 3 0,1,0,-3544,-82104 Table C.5: Extraneous curves of conductor 2 5 p 2 . Appendix C. Tables with small p 322 Conductor: N = 26p2 P 0-2 0.4 j-invariant minimal model 5 2 - 5 - 9 2 4 5 3 2 4 33 Ji 5 2 0,0,0,-700,-6000 5 - 2 - 5 - 9 2 4 5 3 24 3 3 7- : i 5 2 0,0,0,-700,6000 5 2 2 • 5 • 9 2 2 5 2 22 3 3 1 0 7 a 5 0,0,0,-10700,426000 5 - 2 2 - 5-9 2 2 5 2 22 3 3 1 0 7 3 5 0,0,0,-10700,-426000 5 2 2 • 5•11 2 2 5 5 - 2 " 1 0 93 58 0,1,0,-3633,-129137 5 - 2 2 - 5 • 11 2 2 5 5 - 24 1 0 9 3 5 6 0,-1,0,-3633,129137 5 2-5-11 - 2 2 5 2 2I 4 3 1 3 5 3 0,-1,0,-4133,103637 5 - 2 • 5 • 11. - 2 2 5 2 2i 4 3 1 3 5 3 0,1,0,-4133,-103637 5 2 2 • 5 • 3 2 2 5 4 22 3 3 1 3 3 0,0,0,1300,-34000 5 - 2 2 - 5 - 3 2 2 5 4 22 3 3 1 3 3 5 4 0,0,0,1300,34000 5 2 - 5 - 3 - 2 4 5 2 24 3 3 7 3 52 0,0,0,-700,6000 5 2 - 5 - 3 - 2 4 5 2 24 3 3 7 3 5 2 0,0,0,-700,-6000 7 2-7-13 2 9 7 2 53 4 3 3 2 e 7 3 0,-1,0,14047,-1080127 7 - 2 - 7 - 1 3 2 9 ? 2 53 4 3 3 2 6 7 3 0,1,0,14047,1080127 7 2 2 • 7 • 13 - 2 2 7 5 53 1 1 3 3 1 3 2 a 7 6 0,1,0,-111393,11692351 7 - 2 2 - 7 - 1 3 _ 2 2 ? 5 53 1 1 3 3 1 3 2 3 7 6 0,-1,0,-111393,-11692351 17 2 2 • 17 • 71 2 217 5 5b 7 3 3 1 3 2 1 7 6 0,1, 0, -2090433, -804591713 17 - 2 2 -17-71 2 2 17 5 5b 7 3 3 1 3 2-17 6 0, -1 ,0 , -2090433, 804591713 17 2-17-71 2 717 2 53 2 3 3 4 3 3 2 2 1 7 3 0, - I 7 O , -1905473,1012893665 17 - 2 - 1 7 - 7 1 2 717 2 53 2 3 3 4 3 3 2 2 - 1 7 3 0,1,0, -1905473, -1012893665 Table C.6: Extraneous curves of conductor 26p2. Appendix C. Tables with small p 323 Conductor: N = 2 7 p 2 P 0-2 0,4 j-invariant minimal model 13 2 -13-5 2 • 13 3 - 2 ' 1 1J 13 2 0,1,0,-1239,-28079 13 - 2 - 1 3 - 5 2 -13 3 - 2 ' 11J 13 2 0, -1 ,0 , -1239, 28079 13 2 213- 5 2 3 13 3 — 2' 11J 0,-1,0,-4957,-219675 13 - 2 2 1 3 - 5 2 3 13 3 - 2 ' 1 1J 132 0,1,0,-4957,219675 13 2-13-5 - 1 3 2 2b103- i 13 0,1,0,-5802,168130 13 - 2 - 1 3 - 5 - 1 3 2 21 U03 J 13 0,-1,0 , -5802,-168130 13 2 213 • 5 - 2 2 1 3 2 2b103 ; i 13 0,-1,0,-23209,1368249 13 - 2 2 1 3 - 5 - 2 2 1 3 2 2b 103 J 13 0,1,0,-23209,-1368249 13 2 • 13•239 2 • 13 6 -2 '28559J 13 s 0,1,0, -3217647, -2223146015 13 - 2 • 13•239 2 • 13 6 -2 '28559J 13« 0, - 1 , 0 , -3217647, 2223146015 13 2 213 • 239 2 3 13 6 -2 '28559J 13 8 0, - 1 , 0 , -12870589,-17772297531 13 -2 2 13-239 2 3 13 6 -2 '28559J 138 0, - 1 , 0 , -12870589,17772297531 13 2 • 13 • 239 - 1 3 2 2b7"4663 J 13-1 0,1,0, -12871434,17769846862 13 - 2 - 1 3 - 2 3 9 - 1 3 2 2t , 7 ° 4 6 6 3 J 13-' 0, - 1 , 0 , -12871434, -17769846862 13 2 213 • 239 - 2 2 1 3 2 267 04663' i 13 4 0, - 1 , 0 , -51485737,142210260633 13 -2 2 13-239 - 2 2 1 3 2 2b 7 ° 4 6 6 3 J 13" 0,1,0, -51485737, -142210260633 Table C.7: Extraneous curves of conductor 27p2. Appendix C. Tables with small p 324 Conductor: N = 2 V P 0-2 (24 j-invariant minimal model 23 2 3 -23-39 2 • 23 5 2b 3 3 4057 : i 23 s 0,0,0, -4292306, -3419024336 23 - 2 3 -23-39 2 • 23 5 2b 3 3 4057 J 23 6 0,0, 0. -4292306, 3419024336 23 2 4 • 23 • 39 2 3 23 5 2b 3 3 4057 3 23 6 0, 0,0, -17169224, -27352194688 23 - 2 4 -23-39 2 3 23 5 2b 3 a 4 0 5 7 3 23 B 0, 0,0, -17169224, 27352194688 23 2 3 • 23 • 39 2 • 23 2 2b 3 3 16223 3 23 3 0,0,0, -17163934, 27369909840 23 - 2 3 -23-39 2 • 23 2 2b 3 3 16223 J 23 3 0,0,0, -17163934, -27369909840 23 2 4 • 23 • 39 2 3 23 2 2b 3 J 16223 J 23 3 0,0,0, -68655736, 218959278720 23 - 2 4 -23-39 2 3 23 2 2b 3 a 16223 J 23 3 0,0, 0, -68655736, -218959278720 Table C.8: Extraneous curves of conductor 2 8 p 2 . 

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