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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 in  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 y = x ± 2 3 . This information is required in providing the classification mentioned above. We show how the commercial mathematical software package MAGMA can be used in solving this problem. As an application, we turn our attention to the ternary Diophantine equations x + y = 2 pz and x + y = ±p z , 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. 2  5  n  a  n  6  a  2  3  3  m  ii  n  Table of Contents Abstract  ii  Table of Contents  iii  List of Tables  vi  Acknowledgement  vii  Dedication  viii  Chapter 1. Introduction 1.1 1.2 1.3 1.4  1  Introduction to Diophantine Equations Generalized Fermat Equations Statement of Principal Results Overview of chapters  1 4 6 9  Chapter 2. The Conductor of an Elliptic Curve over Q with 2-torsion 2.1 2.2 2.3 2.4  2.5 2.6  Introduction Statement of Results The Proof of Theorem 2.1 The case when v^{a) = 1, v (b) = 0 2.4.1 Proof of Theorem 2.1 part (vii) when 2.4.2 Proof of Theorem 2.1 part (vii) when 2.4.3 Proof of Theorem 2.1 part (vii) when 2.4.4 Proof of Theorem 2.1 part (vii) when 2.4.5 Proof of Theorem 2.1 part (vii) when The Proof of Theorem 2.3 The Proof of Theorem 2.4 2  u ( A) u (A) v ( A) v (A) u ( A) 2  2  2  2  2  = = = >  8 10 ; 11 12 13  11 11 12 17 24 25 26 26 27 27 27 29  Chapter 3. Classification of Elliptic Curves over < Q > with 2-torsion and cona ductor 2 3*V 30 3.1  Curves of Conductor 2 p 3.1.1 Statement of Results ' 3.1.2 The Proof for Conductor 2 p 3.1.3 List of (^isomorphism classes a  2  a  iii  2  31 31 45 45  Table of Contents  3.2 3.3 3.4  3.1.4 The end of the proof Curves of Conductor 2 3^p Curves of Conductor 2 3 V Proofs of 2 3^p and 2 3 V a  a  5.3 5.4 5.5  a  Diophantine Lemmata  150  Useful Results Diophantine lemmata  Chapter 5.  5.1 5.2  46 55 90 146  a  Chapter 4.  4.1 4.2  iv  150 152  Rational points o n y  2  = a? ± 2 *3 5  C  /3  177  Introduction and Statement of Results 177 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 Data for the curves y = x ± 2 3^ 192 The family of curves y = x + A 202 Proof of Theorem 5.1 204 5.5.1 A = 2 3 204 5.5.2 A = 2 3 206 5.5.3 A = 2 207 5.5.4 Rank > 2 cases 207 2  5  2  6  2  6  3  Q  5  5  Chapter 6.  Classification of Elliptic Curves over Q with 2-torsion and conductor  6.1 6.2  2p a  Statement of Results The Proof  Chapter 7.  209 227  O n the Classification of Elliptic Curves over < Q > with 2-torsion and conductor  7.1  209  2  23p a  229  2  Statement of Results  229  iv  Table of Contents 7.2  v  The Proofs 7.2.1 Proof of Theorem 7.1 7.2.2 Proof of Theorem 7.2 7.2.3 Proof of Theorem 7.3 7.2.4 Proof of Corollary 7.4 7.2.5 Proof of Lemma 7.5  Chapter 8. O n the equation x  n  8.1 8.2 8.3 8.4 8.5 8.6 8.7 8.8  n  = 2 pz a  244  2  Introduction Elliptic Curves Outline of the Proof of the main theorems — Galois Representations and Modular Forms Useful Propositions Elliptic curves with rational 2-torsion Theorems 8.1 and 8.2 Concluding Remarks  Chapter 9. O n the equation x  3  9.1 9.2 9.3 9.4  + y  233 233 234 235 237 239  + y  3  = ±p  z  m  244 245 246 247 248 249 251 254 255  n  Introduction Frey Curve The Modular Galois Representation p% Proof of Theorem 9.1  255 256 259 261  b  Bibliography  263  Appendix A . O n the Q-Isomorphism Classes of Elliptic Curves with 2Torsion and Conductor 2 *& p <  3  270  s  A.l 6>0 A. 2 6<0  270 309  Appendix B. Tables of S-integral Points o n Elliptic Curves.  B. l 5-integral points on Elliptic Curves B.2 Computing S-integral points on Elliptic Curves — B.3 Tables of 5-integral points on the curves y = x ± 2°3 2  Appendix C .  z  311  6  311 312 314  Tables of Q-Isomorphism Classes of Curves of Conductor 2p a  2  with Small p.  317  List of Tables 2.1 2.2 2.3 2.4  Neron type Neron type Neron type Neron type  at at at at  2 of y 2 of y 3 of y p of y  =x =x =x =x  5.1  Theorem 5.1: A l l points onC :y = x ± 2 3  5.2  Data for y = x + 2 3^  5.3 5.4 5.5 5.6 5.7 5.8  Data for Data for Data for Data for Data for Data for  (con't) (con't)  195 196 197 198 199 200  5.9  Data for y = x - 2 3 (con't)  201  B.l B. 2  ^-integral points on y = x + 2 3 5-integral points on y = x - 2 ° 3  Cl C .2 C.3 C.4 C.5 C.6  Extraneous curves Extraneous curves Extraneous curves Extraneous curves Extraneous curves Extraneous curves  C.7 C.8  Extraneous curves of conductor 2 p Extraneous curves of conductor 2 p .  2  3  2  3  2  3  2  3  + ax + ax + ax + ax  + bx + bx (con't) + bx + bx  2  2  2  2  2  2  y y y y y y  2  2  2  2  2  2  2  5  =x =x =x =x =x - x  5  5  5  5  5  5  5  + 2<*3 + 2 3^ +2 3 - 2 3 - 2 3^ - 2 3^  /?  a  a  /?  a  /3  a  a  a  /3  178 194  a  5  a  15 16 16 16  (con't) (con't) (con't)  13  2  3  2  3  of conductor of conductor of conductor of conductor of conductor of conductor  a  315 316  b  6  2p 2p. 2 p 2  2  2  3  2  2p 4  2  2 p 2 p  2  5  6  2  7  2  8  2  vi  318 318 319 320 321 322 323 324  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  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  1.1  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. M u c h 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 equation 2 , 2  x + y• = z  2  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(u - v ),2uvd, d(u + v )) 2  2  2  2  where u, v. d e Z and gcd(u, v) = 1. O n the other hand, the equations x + y s  3  = z, 3  x + y = z, 4  4  4  and x + y 5  5  z  5  have only the trivial solutions; solutions where one of the values is 0. Fermat ^eoi-uses.  1  1  Chapter 1.  2  Introduction  wrote in the margin of his copy of Arithmetica that, in fact, the equation x  +y  n  = z  n  n  has no nontrivial solutions for any n > 3, and commented that he had a marvelous proof of this fact but the margin was too small to contain it. This became known as Fermat's "Last Theorem" . The quest to prove (or disprove, 2  for that matter) Fermat's Last Theorem became the driving force for modern number theory over the last three hundred years. Amateurs and professionals 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. After 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 treating 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 y = x 2  + 1.  3  The only integer solutions are (-1,0),(0,±1),(2,±3). These are, in fact, the only rational solutions. O n the other hand, the Diophantine equation y = x + 17 2  3  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 : y  2  + a\xy  + a y = x  3  3  + ay  = x  3  3  + a x 2  2  + a x + ae 4  with ai 6 Z is called an elliptic curve (provided it is nonsingular). For curves in Weierstrass form y  2  = x  3  + a2£  + 4£+  2  a  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 P  — Pi + P . Geometrically,  3  2  this is done by taking the (rational) line through Pi and P and letting P be 2  4  the third point of intersection of the line with E. Next, take the vertical line through P (i.e. the line through P and O) and let P3 be the other point of in4  4  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  where £ ' ( Q )  fors  x Z  r  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 possible groups (see for example [69], p. 223). However, there is no known algorithm 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 : y = x + 1, Ei(Q) ~ Z / 6 Z 2  E  2  A hyperelliptic curve  3  : y = x + 17, E (Q) ~ 2  3  2  I. 2  is a curve of the form y = /(*) 2  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 theorem 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 y = x ± 2 3 . Chapter 5 can be read independently of all other chapters. It provides an introduction to the theory and practice of computing all rational 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. 2  1.2  5  a  fe  Generalized Fermat Equations  In relation with Fermat's last theorem the equation x + y' = z p  !  r  (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> ) = j, + \ + r  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 rational 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 . We have already mentioned that 5(3,3, 3) was empty and 3  2  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 = 3 , 2 + 7 = 3 , 7 + 13 = 2 , 2 + 17 = 71 , 4  3 + ll 5  4  2  5  2  4  3  2  9  7  3  2  = 122 , 17 + 76271 = 21063928 , 1414 + 2213459 = 65 , 2  7  3  2  3  2  7  9262 + 15312283 = 113 , 43 + 96222 = 30042907 , 3  2  7  8  3  2  33 + 1549034 = 15613 . 8  2  3  Notice that an exponent of 2 appears in each solution. This leads to the following 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), Tijdeman, Granville and Beal. The first known result in the hyperbolic case is due to Darmon and Granville [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 number 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.  6  Introduction  (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) (5,5,n),(7,7,n) (2n,2n,5) (4,2n,3) 1.3  Chen for 7 < n < 1000, n ^ 31 Darmon and Kraus (partial results) Bennett Bennett, Chen  Statement of P r i n c i p a l Results  Modularity techniques have since been applied to generalized Fermat equations with coefficients: Ax  p  + By" = Cz . T  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 = N , a > 1, with a  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 = 2 , 13  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 nonnegative integer, then the Diophantine equation x +y = n  n  pz a  3  has no solutions in coprime integers x, y and z with \xy\ > 1 and n >  p . 4p2  7  Chapter 1. Introduction  Their proof of this proceeds as follows. Attach to a supposed solution  (a, b, c) an elliptic curve E = E b with a 3-torsion point, and to this a Galois a>  tC  representation pE, on the n-torsion points. To pE, there corresponds a cuspin  dal newform  n  N (E), where N (E) can be explicitly  of weight 2 and level  n  n  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  pip  2 o  r  that there exists an elliptic curve over Q with rational 3-torsion and conductor 3 p . Hence a classification of such curves is needed to finish the T  u  argument. In Chapter 8, we apply a similar argument to the equation x  n  +y  n  =  2 pz a  2  and prove the following  Let p ^ 7 be prime. Then the equation  Theorem 1.3 (Bennett, Mulholland) x + y n  n  =  2 pz a  2  has no solutions in coprime nonzero integers x and y, positive integers z and a, and prime n satisfying n > p . 27p2  A key ingredient in the proof is a classification of the elliptic curves with conductor 2 y M  2  and possessing a rational 2-torsion point. In Chapter 6, we  provide such a classification. In Chapter 9, we study the equation x + y = 3  3  ±p z , m  n  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 + y = 3  3  ±p z m  n  has no solutions in coprime nonzero integers x, y and z, and prime n satisfying n > p and n\m. 8p  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 2 3 p, M  1 < M < 3. Thus, in this  2  case we need a classification of the elliptic curves with conductor 2 3 p, M  M  2  1 <  < 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 2 p M  2  or 2 3 p M  L  and  possessing a rational point of order 2 we start by considering the following more general question. Problem  1 Determine all the ^-isomorphism  conductor 2 3 p M  L  classes for elliptic curves over Q of  and having at least one rational point of order 2.  N  A s is well-known, there do not exist any elliptic curves defined over Q with conductor divisible by 2 , 3 , or q for q > 5 prime (see e.g. Papadopou9  6  3  los [57]). Furthermore, as we show in Chapter 2, the existence of rational 2torsion implies the conductor is not divisible by 3 . Therefore, we can suppose 3  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 3 M  L  or 2 3 . Coghlan in his dissertation [17] also studied the M  curves of conductor 2 3 M  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 p  and 2 p ,  N  conductor  M  N  and Ivorra, in his dissertion [37], classifies those of  2 p. M  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 results analogous to those of Ivorra for conductor 2 p . N  2  In Sections 3.2 and 3.3  Chapter 1.  Introduction  9  we obtain results for conductor 2 S p N  L  and 2 3 p , N  L  2  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. A s 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 following more general problem, which we do not know how to attack in general. Problem 2 Determine all the ^-isomorphism conductor  classes for elliptic curves over Q of  2 3p. M  L  N  Let us note that Brumer and McGuinness have determined the elliptic curves of conductor p < 10 . The definitive web source for tables of all the 8  elliptic curves of conductor < 130000 is John-Cremona's home page . These 3  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 constructing his tables (and, indeed, a fine introduction to the arithmetic of elliptic 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 2 m k  1.4  2  with m < 23 prime and also m = 15 and 21.  O v e r v i e w of chapters  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  w w w . m a ths.nottingham.ac.uk/personal/jee/  Chapter 1. Introduction  10  i.e. curves of the form  —x  y  2  3  + ax  + bx.  2  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 2 3 p , M  L  N  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 ;  y = x± 2  2 3 ,  3  Q  /?  and the genus 2 curves = x  y  2  5  ±  23. a  p  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 2 p M  2  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 2  M  Z p, 1 < M < 3, with rational 2  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 x + y n  n  = 2 pz a  2  and x + y 3  3  =  ±p z , m  n  respectively. A modified version of Chapter 8 has appeared in print [4].  C h a p t e r2 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: y = x + ax 2  3  2  + 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 v denotes the p-adic valuation on Q. p  2.1  Introduction  Let E be the elliptic curve over Q defined by + a x + ag  E : y + a\xy + ayy = x + a x 2  3  2  4  2  with ai G Z . Let b , 64, b§, bg, C4, CQ, and A be the standard invariants associated 2  with E: 02  = a + 4a ,  6s =  2  2  a a6 2  64  = a\a$ + 2a ,  be = a | + 4ac  2  a\asa4 + 4a a$ + a a\ — a 2  2  c = b\ - 246 , c = -b\ + 366 6 - 21 66 4  A = -bjb  4  8  6  2  4  6  - 86^ - 276| + 96 6 6 . 2  4  6  The conductor of an elliptic curve over Q is defined to be  p  11  (2.1) (2.2)  2  (2.3) (2.4)  Chapter 2. The Conductor of an Elliptic Curve over Q with 2-torsion  where f  p  12  = v ( A) +1 — n . Here n is the number of irreducible components of p  p  p  the special fibre of the minimal Neron model at the prime p (see [69]). Essentially, 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 ) or additive (E has a cusp over ¥ ) depending on whether f p  p  respectively. It is well known that for p determined by the values of v (ci),  2,3, the value of f  p  v (c ) and v (A).  p  = 1 or > 2,  p  p  G  p  is completely  This is not always the  case when p = 2 or 3. Papadopolous [57] has determined when the triple (•u (c ),'U2(c ),f2(A)) (resp. (v {c ), v (c ), 2  4  6  3  mine the value of /  2  4  3  v (A)))  6  is not sufficient to deter-  3  (resp. / 3 ) and in these cases he has given supplementary  conditions involving the values of a\, a , a , a , ae, 6 , 6 , £>e and b%. In the 2  3  4  2  4  case of the prime 3 these supplementary conditions involve checking a single congruence involving c and CG modulo 9. However, for the prime 2 the 4  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(c ),^(cc), ^ ( A ) ) = (6, > 9,12) one is unable to decide from 4  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 isomorphic to a curve of the form y = x + ax 2  where a, b e Z are such that v (a) p  3  2  + bx,  > 2 and v (b) > 4 do not both hold for all p  p. The discriminant in this case is A = 2 b {a 4  2  2  -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 / is completely determined 2  by the values of u (a), i> (6) and the congruence classes of a and 6 modulo 4, 2  2  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 R e s u l t s .  Let p denote a prime > 5. We will prove the following theorems.  13  Chapter 2. The Conductor of an Elliptic Curve over <Q> with 2-torsion  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 y  = x  2  + ax  3  + bx is given by Tables  2  2.1 and 2.2 on pages 15 and 16. In the cases where /  = 0 or 1 the model y  2  2  =  x + ax + bx is non-minimal at 2, this is indicated in Table 2.1 by the appearance of s  2  "non-minimal" y = x + ax 3  2  2  in the corresponding column. In the cases where /  ^ 0,1 the model  2  + 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) : y  = x  2  + ax  3  + bx is non-  2  minimal at 2 we have the following: 1. Ifv2(a) = 0, v (b) > 4 and a = 1 (mod 4) then 2  (^p)  y + xy = x + 2  3  x + 2  (J£)  x  is a minimal model for E(a, b) at 2. 2. Ifv (a) 2  = 1, v (b) = 0, v (A) 2  2  > 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 y = x + ax 2  page 16. In all cases the model y  2  3  — x + ax 3  2  2  + bx is given by Table 2.3 on  + bx is minimal at 3.  Theorem 2.4 Let pbea prime > 5. If a, b G Z are such that not both v (a) > 2 and p  v (b) > 4 hold, then the Neron type at p of the elliptic curve y  2  p  = x + ax 3  given by Table 2.4 on page 16. ln all cases the model y = x + ax 2  3  2  2  + bx is  + 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 v (b) > 4 hold and N( ^ q  then:  a  such that not both v (a)  is the conductor of the elliptic curve y  q  2  > 2 and  = x + ax 3  2  + bx  Chapter 2. The Conductor of an Elliptic Curve over Q with 2-torsion  d) Q I (a,b) if and only ifq\A N  = 2 b (o? i  2  - 46),  (ii) if q || A^(„,6) then q does not divide a, (iii) q || N/ j 2  ab  if and only if q divides a and b.  14  0 1  0  "2(a) v (b)  0  2  a E E 1 (4) 6 E E -1(4)  a = - 1 (4)  b = 1 (4)  Kodaira symbol  II  IV  III  II  III  Case of Tate  3  5  4  3  4  Exponent v {N) of conductor  4  2  3  4  5  Supplementary conditions  a  1 (4)  EE  6  a  EE  a  -1 (4)  0  2  3 a  1 (4)  EE  EE  a  - 1 (4)  1 (4)  EE  a =-1(4)  6= -1 (4)  1 (4)  EE  0  13  7  15  III*  7  9  6  2  0 4  V2(a) v (b) 2  "2  1  1  > 5  0  0  a  Supplementary  1 (4)  EEE  non-minimal  Kodaira symbol  Io  a  1 (4)  EE  EE  a - b  -1 (4)  a - b  13 (16)  EE  5 (16)  ES  non-minimal II  13  7  7  3  6  1  4  7  4  Iv (A)-12  I4  2  7  1  Case of Tate  a  a EE-1(4)  conditions  Exponent  8  7  (A)  4  3  4  3  0  It  7  v (N) 2  4  0  of conductor  1 0  0  8 (con't)  9  10  2  "2(A) Supplementary  1  a - b = 1 (16)  a - b = 9 (16)  3 1 0  1  1 0  Vi{a) v (b)  f =  K4)  |  11 f  -1(4)  EE  EE  1 (4)  § = -1 (4)  conditions  13  Kodaira symbol Case of Tate  6  Exponent v (N) of conductor  IV*  IS  8  6  2  5  III* 9  7  13  II*  7  10  4  3  2  4  4  v (a)  1  1  v (b)  0  0  2  2  f  EE  1 (4)  conditions  f s - 1 (4) non-minimal  § = 1 (4)  1  2  > 3  > 2 0 6  f = - 1 (4) non-minimal  Kodaira symbol  1;  Io  Iv (A)-12  III  Case of Tate  7  1  7  2  4  4  0  4  1  7  Exponent  1  > 2 1  > 13  12  "2(A)  Supplementary  3 •1 1  2  13 6  !C (A)-10 7  6  6  EE  1 (4)  b  EE  -1(4)  II  III  III  3  4  4  6  5  8  2  v (N) 2  of conductor  Table 2.1: Neron type at 2 of y = x + ax + frc. 2  3  2  •"2(a) vi{b)  2  2  2  3  > 3 3  > 3 2  * (A) 2  Supplementary  ! = 1 (4)  H-l  conditions Kodaira symbol  I  i = M<)  (4)  f = -1(4)  v (A)-10  15  III*  7  7  9  la 7  7  9  6  7  7  5  6  8  2  Case of Tate  III*  Exponent V2(N) of conductor  Table 2.2: N e r o n type at 2 of y = x + ax + bx (con't) 2  3  V3(a)  0  0  > 1  > 1  1  v (b) Supplementary  0  > 1  0  1  > 2  3  bs  1(3)  1  > 2 2  > 2 3  6 = - 1 (3)  conditions Kodaira symbol  WA)  lo  I,,.., (A)  lo  III  '«.-l(A)-6  1  0  1  0  2  2  III*  Case of Tate Exponent V3(N) of conductor  2  2  Table 2.3: N e r o n type at 3 of y = x + ax + bx l  6  1  v {a)  0  0  > 1  > 1  1  1  0  > 1  0  1  2  > 3  > 2 2  > 2  v (b)  2t,„(6)-4  12  nr  2  2  2  p  p  Supplementary  a  2  £ 46 (p)  a  2  a  = 46 (p)  2  £ 46 (p ) 3  a  2  3  = 46(p ) 3  conditions Kodaira symbol  lo!  ^ptA)  l2u (6)  lo  III  13  *»p(A)-6  1  0  2  2  2  p  I  Case of Tate Exponent  v (N) P  of conductor  0  1  Table 2.4: N e r o n type at p of y = x + ax + bx. 2  3  2  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] except in cases 1  (ix) v (a) = 1,1*2(6) = 2 and (xiv) u (a) = 2, i>2(6) = 2 where we will need to 2  2  apply Tate's algorithm directly. The seventeen cases we consider are labeled as follows: (i)  (ii)  (iii)  (iv)  (v)  (vi)  (vii)  (viii)  (ix)  0 0  0 1  0 2  0 3  0 4  0 > 5  1 0  1 1  1 2  (x)  (xi)  (xii)  (xiii)  (xiv)  (xv)  (xvi)  (xvii)  1 3  1  > 2 0  > 2  2 2  2 3  > 3 2  > 3 3  V2{a)  v {a) 2  Mb)  > 4  1  The standard invariants for the curve y — x + ax 2  3  2  + bx  are (see (2.1)) ai = 0, a2 — a, b = 4a,  03  04  = 6,  = 0,  2b, b = 0, b =  64 =  2  = 0, 6  s  -b , 2  c = 2 ( a - 36), c = 2 a(96 - 2a ), A = 2 6 ( a - 46). 4  2  5  4  2  4  2  2  6  Some of the cases immediately follow from Table IV of [57] so we quickly deal with these first. We have the following table. Case of V2{a)  (viii) (ix) (xi)"  1 1 1  (XV)  > 2 2  (xvii)  > 3  (xiii)  V2(b)  1 2  ^2(04)  V2{CQ)  5  7  > 4  > 6 6  1 3 3  5 7 7  «2(A)  8 10  8 9  > 14  > 8 10  9 14  > 11  15  Tate  Kodaira  h  4  III  6 7  T* ^ (A)-10  7 6 6  III III* III*  • 8 7 8  2  4 9 9  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 v (b) = 0 we have 2  [5  /  if 6 E E 1 (mod 4),  V2 c ) = < 4  [>6  if b= - 1 (mod 4),  w ( c ) = 5, 2  6  v (A  = 4.  2  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),  [0(mod4)  if a = - 1 (mod 4),  a + a = 6+a = < 4  2  implies that if a = 1 (mod 4) we are in case 3 of Tate and f  2  = 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 6 + 3r& + 3r t> + r b + 3r = 2(1 + 2a) £ 0 (mod 8), 2  8  6  3  4  4  2  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  EE  1 (mod 4),  { 2 (mod 4)  if a  EE  -1  a + a =b+ a= < 4  2  (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 f — 2 . 2  (ii) When ^ 2 ( 0 ) = 0 and ^2(6) = 1 we have V2{c4) = 4, v {c ) > 7, v (A) = 6, 2  6  2  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 v (a) = 0 and v (b) = 2 we have 2  2  v (c ) = 4., v {c ) = 6, u (A) = 8. 2  4  2  6  2  19  Chapter 2. The Conductor of an Elliptic Curve over Q with 2-torsion  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 + 3r& + 3r o + r b + 3r = 0 (mod 32). 2  b  8  4  3  6  4  2  The integer t = 2 satisfies the congruence ae + ra + r a  + r - ta^ - t - rta\ = 0 (mod 8).  2  4  3  2  2  Moreover, for r = t = 2 we have the congruence a + ra + r a + r - i a - t 2  6  4  3  2  2  3  rtai  = 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 f — 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 2  0 = a + 3r - ta,] - t = 3 - t (mod 4) 2  2  2  has no solutions for t thus it follows that we are in case 7 of Tate and f = 3. (iv) When v (a) = 0 and v (b) = 3 we have 2  2  2  v {c ) = 4, v (c ) 2  A  2  = 6, v (A)  6  2  = 10,  and, from Table IV of [57], we are in case 7 or 9 of Tate. The integer r = 0 satisfies the congruence b + 8  3r6 + 3r b + r b + 3r = 0 (mod 32). 2  6  4  3  4  2  Moreover, we have the congruence 1 - t (mod 4)  if a = 1 (mod 4),  3 -t  if a = - 1 (mod 4),  2  2  0 = a + 3r — ta\ — t 2  _  2  (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 f = 4, whereas if a = 1 (mod 4), we are in case 9 of Tate and / = 3. 2  2  Chapter 2. The Conductor of an Elliptic Curve over Q with 2-torsion  20  = 0 and V2{b) = 4 we have  (v) When 1*2(0)  ^ (c ) = 4, v (c ) = 6, v {A) = 12, 2  4  2  G  2  and, from Table IV of [57], we are in case 7 of Tate or the model is nonminimal. The integer r — 0 satisfies the congruence b + Srb + 3r b + r b + 3r = 0 (mod 32). 2  8  3  6  4  4  2  Moreover, the congruence  (1 - t (mod 4) 0 = a + 3r - tai - t = < [3 -t (mod 4) 2  2  if a =  1 (mod 4),  2  2  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 f = 4, whereas 2  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' ,a ,a ,a ) 2  3  4  = (1, — ^ , 0 , ^ ° ) >  6  and such that v (c' ) = 0, v (c' ) = 0, and v (A') = 0. Hence we are in case 1 2  A  2  6  2  of Tate and f = 0. 2  (vi) When v (o) = 0 and v (b) > 5 we have 2  2  v {c ) = 4, v {c ) = 6, u (A) > 14, 2  4  2  G  2  and, from Table IV of [57], we are in case 7 of Tate or the model is nonminimal. The integer r = 0 satisfies the congruence b + 3rb + 3r b + r b + 3r = 0 (mod 32). 2  s  6  3  4  4  2  Moreover, the congruence  „ f 1 - t (mod 4) if a = 1 (mod 4), 0 = a + 3r - tai - t = < 1 3 - t (mod 4) if a = - 1 (mod 4), 2  2  2  Chapter 2.  21  The Conductor of an Elliptic Curve over Q with 2-torsion  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 f  2  = 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[,a ,a' ,a' ,a' ) 2  3  A  = (1,  6  Then v (c' ) = 0, i>2(c ) = 0, and v (A') 2  and f  4  6  2  ,0, j ^ O ) .  > 2, whence we are in case 2 of Tate  = 1.  2  (vii) When v (a) = 1 and v (b) — 0 we have 2  2  v (c ) 2  4  = 3, v (c ) = 6, u (A) > 7. 2  We consider the cases v (A)  6  2  = 7, 8, 9, 10, 11, 12, > 13 separately.  2  If v (A) 2  = 7 then from Table IV of [57] we are in case 3 of Tate and f  If v (A)  = 9 then from Table IV of [57] we are in case 6 of Tate and f  2  2  2  In the remaining cases; v (A) 2  = 7. = 5.  — 8, 10, 11, 12, > 13, some work is required  to determine f . We defer the proof for these cases until Section 2.4. 2  (x) When v (a) = 1 and v (b) = 3 we have 2  2  ^ (c ) = 6, 2  4  « ( C 6 ) > 9 , «2(A) 2  = 12,  and from Table IV of [57] we are in case 7 of Tate. There are, however, two possibilities for f . We need to apply Tate's algorithm directly in this case. 2  We will use the pseudocode for Tate's algorithm given in [26]. 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 notation of loc. cit. since xa3 = f- = 0 is even, xa6 = f | = 0 is even, and xa4 = ^ = | is odd (J  we exit the loop after line 54, with m = 2. Thus f  2  = v (A) 2  Kodaira symbol is Z* • (xii) When v (a) > 2 and v (b) = 0 we have 2  2  ^2(c ) = 4, v (c ) 4  2  6  > 7, v (A) 2  = 6,  — 6 = 6 and the  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 a +ra +r G  „  4  , [2 (mod 4) 2 a +r —ta —t —rtai = b+a+1 = < (0(mod4) 2  if b = 1 (mod 4),  3  if b = - 1 (mod 4),  that if 6 EE 1 (mod 4), we are in case 3 of Tate and f  = 6, whereas if 6 EE  2  -1 (mod 4), we are in case 4 of Tate and f — 5. 2  (xiv) When v (a) = 2 and v (b) = 2 we have 2  2  v (c ) = 6, v (c ) = 9, v (A) > 13, 2  4  2  6  2  so, from Table IV of [57], we are in case 7 of Tate. There are, however, two possibilities for f depending on whether v (A) = 13 or v (A) > 14. We claim 2  2  2  v {A) = 13 if and only if 6/4 EE - 1 (mod 16). Indeed, since A = 166 (a - 46), 2  2  2  the hypothesis on a and 6 imply - ( A ) = . 3 « * ( ( ! ) ' - ! ) = - . But (a/4) = 1 (mod 4), from which the claim follows. Thus, f 2  2  = 7 if 6/4 =  -1 (mod 4) and f == 6 if 6/4 = 1 (mod 4). 2  (xvi) When v (a) > 3 and 1*2(6) = 2 we have 2  v (c ) = 6, u (c )-> 10, v (-A) = 12, 2  4  2  6  2  and, from Table IV of [57], we are in case 7 of Tate. There are,-however, two possibilities for f . We need to apply Tate's algorithm directly in this case. 2  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 = a + 6, a = 0, a = 6 + 4a + 12, a = 26 + 4a + 8. 2  3  4  6  It follows that v (a ) = 1, v (a ) = 3, v (a ) > 5, 2  2  2  4  2  6  23  Chapter 2. The Conductor of an Elliptic Curve over Q with 2-torsion  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 f — v (A) - 6 = 6 and the Kodaira symbol is I\. 2  2  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, a = a+ 18, a = 8, a = 6 + 12a + 108, a = 6 6 + 36a + 200, 3  2  4  6  and find v {a ) = l, i* (a ) = 3, 2  2  2  3  1*2(04) > 4 ,  v (a ) > 2  6  5,  (here we've used the fact that 6 / 4 = 1 (mod 4)). Moreover, ^2(62)  = 3,  1*2(64) > 5 ,  i» (6 ) 2  6  = 3, v {b ) > 7. 2  8  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 notation 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' ,a' , a' , a' ) such that 2  3  A  c  o'i = 0, v {a' ) = 1, i* (o ) = 3, v (a' ) > 4 , v (a' ) > 5. 2  2  2  3  2  A  2  6  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 f = v (A) — 2  2  7 = 5 and the Kodaira symbol is 7 . 3  To finish the proof it remains to verify the cases when v (a) = 1,1*2(6) = 0, 2  and v {A) = 8,10,11,12, and > 13. We do this in the next section. 2  Chapter 2.  2.4  24  The Conductor of an Elliptic Curve over Q with 2-torsion  T h e c a s e w h e n v (a)  = 1, ^2(6) = 0.  2  We have already determined in part (iv) of the proof of Theorem 2.1 the values of /  2  when v (A) 2  = 7 or 9. In this section we determine the value of f  2  for the  remaining cases: ^ ( A ) = 8,10, 11, 12, > 13. First we make two observations.  I/a, 6 G Z such that v {a) = 1, v (b) = 0 and v (A) — v (l6b (a 46)) > 8 then 6 = 1 (mod 4). furthermore, if v (A) = 8 then 6 = 5 (mod 8). 2  Lemma 2.6  2  2  -  2  2  2  2  Proof. If v2(A) =  v (16b (a - 46)) > 8 then w ((§ ) - 6)) > 2. It follows that 2  2  2  2  2  6 = 1 (mod 4) since a/2 is odd. Moreover, if i; (A) = 8 then t> ((|) - 6) = 2 2  2  2  thus 6 ^ 1 (mod 8).  •  We will use the next lemma when applying Proposition 4 of [57]. Lemma 2.7  For a, 6  Z swcfo f/zaf i>2(a) = 1fl^d^2(6) = 0 f/ie congruence  G  - 6 + 6r 6 + 4r a + 3r = 0 (mod 32) 2  2  3  4  has no solutions for r ifb = —1 (mod 4), whereas for 6 = 1 (mod 4) it has solutions 3  if a = 2 (mod 8),  1  if a = 6 (mod 8).  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  - 6 + 6r 6 + 4r a + 3r = 16fc(fc + 1) = 0 (mod 32). 2  2  3  4  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  2.4.1  Proof of Theorem 2.1 part (vii) when v (A)  = 8  2  It follows from Lemma 2.6 that 6 = 5 (mod 8). Since v (c ) 2  v (A) 2  25  4  = 4, v (c§)  = 6 and  2  = 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 b + 3 r 6 + 3 r 6 + r 6 + 3 r = 0 (mod 32) 2  8  3  6  4  4  2  has solutions if a EE 2 (mod 8),  _ J3  if a EE 6 (mod 8).  \ l  In either case the integer t — 2 satisfies the congruence a + r a + r a + r - i a -t 2  6  3  4  2  3  2  - rta  = rb + r a + r - t = 4 - t = 0 (mod 8). 2  x  3  2  2  Fix t = 2 and r as above. Suppose a = 2 (mod 8). We have the congruence a + ra + r a  + r  2  6  4  2  - ta  3  - l - rta 2  3  x  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 f  2  = 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 + 3r - sai - s EE 3 - s = 0 (mod 4) 2  2  2  has no solution for s, whereby we are in case 7 of Tate and f  2  = 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 + ra + ra 2  6  4  + r - ta 3  2  - t - rta\ = b+ a - 3 = 0 (mod 16) 2  3  if and only if a — 6 EE 9 (mod 16). Thus, we are in case 6 of Tate (and f  2  = 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 a + 3r - sai - s = 1 - s EE 0 (mod 4) 2  2  2  Chapter 2. The Conductor of an Elliptic Curve over Q with 2-torsion  has solution s = 1, whereby we are in case 8 of Tate and f  2  26  = 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 v (A)  = 10  2  In this case we have = 4, v (c )  v (c ) 2  4  2  = 6, v {A)  6  = 10,  2  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  6 + 3r6 + 3r 6 + r b + 3r = 0 (mod 32) 2  8  6  4  3  4  2  has solutions  _ f3  if a =  \l  2 (mod 8)),  i f a = 6 (mod  8)).  Furthermore, the congruence  f 3 - t (mod 4)  if  a EE 2 (mod 8),  (mod 4)  if  a EE 6 (mod 8),  2  0 EE a  2  + 3r  2  t EE < [1-t  2  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 v (A) 2  = 11  In this case v {c ) = 4, v (c ) 2  4  2  6  = 6, v {A) 2  = 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.  27  Chapter 2. The Conductor of an Elliptic Curve over Q with 2-torsion  2.4.4  Proof of Theorem 2.1 part (vii) when v {A) 2  = 12  In this case v (c )=4, 2  v (c )  4  2  = 6, v (A)  6  2  = 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\  (  (ai, a , a a a ) = I I , 2  3 )  4 )  a? - 46 a(a  a,+ 2  2  — ,0,  6  ^—,  - 46) \ — I,  (2.6)  which are all integers (by assumptions on a and 6). Also, v (c' ) = 0, v {c' ) = 0, and v (A') = 0, whence we are in case 1 of Tate and f = 0. 2  2  2.4.5  4  2  (i  2  Proof of Theorem 2.1 part (vii) when v (A) 2  > 13  In this case v (c ) 2  4  = 4, v (c ) 2  6  = 6, v (A) 2  > 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 i n (2.6). Since v (c' ) = 0, v (c' ) = 0, and v (A') > 1, we are in case 2 of Tate and f = 1. This completes the proof of Theorem 2.1. 2  2  2.5  A  2  2  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].  6  Chapter 2.  28  The Conductor of an Elliptic Curve over Q with 2-torsion  Case of vj, (a)  v (b)  V3(C4)  V3{C )  0 >1 1  >1 0  0 >3 3  >2 >2 >3  1 2 3  0 1 2 2 3 4  3  >2  6  >5 >6 >8  « (A)' >2 0 >6 3 6 9 3  Tate  Kodaira  2 1  I (A)  1 0 2 2 2 2  U3  lo  6 or 7 4 6 9  There are only three remaining cases to check:  /3  ^3  (A)-6  in T* III*  (1) 1*2(0.) = 0, v (b) = 0; (2) 2  1, v {b) = 1; (3) v {a) = 2, v (b) = 3. (1) Suppose 1/3(0) = 0 and 1/3(6) = 0. Then v (c ) = 0, 1/3(00) = 0, and 3 divides  v {o) = 2  2  2  2  2  4  A if and only if b = 1 (mod 3). It follows that 1  if6=l(mod3),  0  if 6 E E - 1 (mod 3),  and the Neron type at 3 is I„ (A) if 6 EE 1 (mod 3) and In if b = —1 (mod 3). 3  (2) Suppose vs(a) = 1 and 1/3(6) = 1. Then v (c ) 2  A  > 2, VS{CQ) = 3, and i>3(A) =  3. We consider the intervening condition P in Table II of [57]. P is decided if 2  2  we have  Hi) H (D ))' - ((i) -i)^ '1  +2ss  4 2  a  9  or equivalently  (l)'(i) (i) (i)+  a+2  -  ls0(mod9)  Since ^ 3 ( 0 ) = v (b) = 1, this is certainly the case. Therefore / s  3  = 2 and the  Neron type at 3 is III.  (3) Suppose  1/3(0)  = 2 and w (6) = 3. Then v (c ) > 4, ^(co) = 6, and i> (A) = 3  2  4  3  9. We consider the intervening condition P 5 in Table II of [57]. P 5 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  2 and the  case. Therefore f$ —  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]. Case of v (a)  v (b)  v (c )  v {c )  v {A)  0 > 1 > 1 1  > 1  0 0 1 2 2 3  0 > 1 >2 3  >2 0 3 >8  >4  6  6  T* o  > 5  9  9  III*  p  p  p  0 1 >3 2 3  >2 >2  4  P  p  6  Tate  Kodaira  2  hv {b)  1  Io  4 7  III  There are only two remaining cases to check: v (a) = 1,1-2 (6)  f 1 P  p  0 2 2 2 2  T* 1  (1) 1*2(0) = 0, 1*2(6) = 0; (2)  = 2.  2  (1) Suppose i* (a) = 0, ^2(6) = 0. In this case, p can divide at most one of C4, 2  CG and A . If p does not divide A then f  p  or C6, whence /  (2)  2  = 0. If p | A then p does not divide C4  = 1 and the Neron type at p is I„ (A) •  Suppose 1*2(0)  =  1,1*2(6)  = 2. Then i* (c ) > 2, v (ce) p  4  >  p  Moreover, in this case, p  can divide at most one of a  a - 46. If v (A)  2  3  2  p  f  p  2, and  —  p  2  > 7, i.e. a — 46 = 0 (mod p ), then we are in case 7 of Tate, 3  the Neron type at p is I* ( A ) - 6 ^ -  n  t  n  e  o  t  n  e  r  i.e. a — 46 5= 0 (mod p ), then we are in case 6 of Tate, / 2  3, and w (A) > 6.  — 36, 96 — 2a and  2  hand,  3  type at pis IQ. This proves Theorem 2.4.  p  if"i* (A) = 6, p  = 2, and the Neron  Chapter 3 Classification of Elliptic C u r v e s over Q with 2-torsion and conductor 2 3 V a  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 2 3 p , M  L  having  N  a rational point of order 2 over Q. The first nine theorems list curves of conductor 2 p . M  The next nine list curves of conductor 2 3 p,  2  M  list those of conductor 2 3 p . M  2, M  L  L  and the last nine  Together, with the work of Ogg on conductor  2  Coghlan on conductor 2 3 , Setzer on prime conductor, and Ivorra on M  conductor 2 p, M  L  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 analogous 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 following properties of E: i. A minimal model of E of the form y  2  + a\xy  = x  3  + ci2X + a x 2  4  + 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 — a = 0, 3  6  so in the statements of these theorems we omit the columns corresponding to these coefficients.  30  Chapter 3. Elliptic Curves with 2-torsion and conductor 2 '3 p c  l3  31  5  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 A 4 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: = 1 mod 4  if n is odd  (3.1)  if n is even .  3.1  C u r v e s of C o n d u c t o r 2 p a  2  The tables presented here are an intermediate step in the classification problem for curves of conductor 2 p . In Chapter 6, we refine these tables by using a  2  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 2 p N  2  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 p , and having 2  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 2 ° 3 / V 5  32  1. p = 7 and E is Q-isomorphic to one of the elliptic curves: a-2  Al  7-3  T  A  [1,-1,0,-2,-1]  2  7  3  [1,-1,0,-107,552]  2  7  9  [1,-1,0,-37,-78] [ 1 , - 1 , 0 , - 1 8 2 2 , 30393]  2 2  7 7  2 -7  A2  -7 -3 2  BI B2  minimal model  0-4 4  4 .  2  3  ?  -7 -7  -2-7-3 2• 7 • 3  3  2  2  3  9  2. p = 17 and E is Q-isomorphic to one of the elliptic curves: 1  a Cl C2  17-33 -2-17-33  C3 C4  17-9 2-17-15  2 • 17 17 4  3  2  2 - 17 4  1  7  A  minimal model  T  [1,-1,1,-1644,-24922]  4  1  [1,-1,1,-26209,-1626560]  2  17  [1,-1,1,-199,510] [1,-1,1,-199,-68272]  4 2  17 io  C14  2  2  4  2  7  8 7  7  1 7  3. p — 64 is fl square and E is Q-isomorphic to one of the elliptic curves: ai Al  1  A2  1  a  \T \  A  0  2  v  p - V P - 64  2  -P  04  2  P\/p—64— 1 4 PVP~ 64— 1 4  4p  a  2  6  2  1  8  Theorem 3.2 The elliptic curves E defined over Q, of conductor 2p , and having 2  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 2 p Q-isomorphic to one of the elliptic curves: m  a  2  Al  epy/2 p  A2  -e2py/2 p  m  m  +1  n  n  +1  n  + 1 is fl square and E is  a.4  A  2m-2p7i+2  22mp2n+6  P  2  where e e {±1} is the residue ofp modulo 4. 2. there exist integers m > 7 and n > 0 such that 2 Q-isomorphic to one of the elliptic curves:  m  + p  n  is a square and E is  Chapter  3.  Elliptic Curves with 2-torsion and conductor  epsJ2  -e2p 2  + p  m  n  /  v  33  d  m-2 2  22m,pn+6  n+2  2"i-|-6p2n+6  p  2  +p  m  !3  A  4  2  B2  a  a  a  Bl  23 p  n  p  where e G {±1} is the residue ofp modulo 4. — p is a square and E is  3. there exist integers rn > 7 and n > 0 such that 2 Q-isomorphic to one of the elliptic curves:  rn  a epy/2  C2  -e2p /2  A  0,4  2  CI  n  -  m  p  n  -  m  x  p  2  n  m-2 2  22rripn+6  -pn+2  2"i+6p2n-h6  p  where e G {±1} is the residue ofp modulo 4. 4. there exist integers m > 7 and n > 0 SMCTI that p — 2 Q-isomorphic to one of the elliptic curves: n  DI  tV\lp  D2  -e2p^p  -  2  -2 -V  22m n+B  n+2  _2m+&p2n+  m  m  -  n  A  0,4  0,2 n  2  m  is a square and E is  rn  p  p  6  where e £ {±1} is the residue ofp modulo 4.  5. £h<?r<? exist integers m > 7 and £ G {0,1} such that *" 2  is a square and E is  +1 p  Q-isomorphic to one of the elliptic curves: 04  A  2m-2p2t+l  22rrip3+6i  2t+l  2?n-f 6^3+6£  a  2  El E2  p  w/zere e G {±1} is the residue ofp  w  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: a  A  FI  2m-2p2t+l  22rrip3+6t  F2  _p2t+l  2^+6^3+6t  0,2  4  Chapter 3. Elliptic Curves with 2-torsion and conductor 2 3 p' a  where e e {±1} is the residue ofp  34  l3  modulo 4.  +  Theorem 3.3 The elliptic curves E defined over Q, of conductor 4p , and having at least one rational point of order 2, are the ones such that one of the following conditions is satisfied: 2  1. there exists an integer n > 0 such that p — 4 is a square and E is Q-isomorphic to one of the elliptic curves: n  A  0,4  0-2 Al  epy/p — 4  A2  -e2pv> - 4  n  2p 4  -P  2  n+2  n  2  p  n+G  8 2n+6 p  where e e {±1} is the residue of p modulo 4. Theorem 3.4 The elliptic curves E defined over Q, of conductor 8p , and having at least one rational point of order 2, are the ones such that one of the following conditions is satisfied: 2  1. there exist integers m e {4, 5} and n > 0 such that 2 p E is Q-isomorphic to one of the elliptic curves: m  Al  A2  m  2  p  -e2p 2 > + 1 /  7  2 2 77). p2n-\-6  m-2 n+2  1  n  n  + 1 is a square and  A  CI4  a.2 tpy/2 p ^-\r  n  P  2  v  where e £ {±1} is the residue ofp modulo 4. 2. there exists an integer n > 0 such that 4+p is a square and E is Q-isomorphic to one of the elliptic curves: n  2  BI B2  A  0,4  a  -epJ4+p e2py/4 + p n  v  2  2p 4  n+6  n+2  n p  where e € {±1} is the residue ofp modulo 4. 3. there exist integers m 6 {4,5} and n > 0 such that 2 + p is a square and E is Q-isomorphic to one of the elliptic curves: rn  n  Chapter  3.  Elliptic Curves with 2-torsion and conductor  a  epy/2 + p -e2p^/2 + p m  n  m  m-2  2  p  2  35  s  2m n+&  2  n+2  n  l3  A  CL4  0-2  Cl C2  23 p  p  m+6 2n+6  p  2  p  where t € { ± 1 } is the residue ofp modulo 4. 4. there exist integers m e {4,5} and n > 0 such that 2 E is Q-isomorphic to one of the elliptic curves:  — p is a square and  rn  A  a  2  DI D2  ep^/2  -  m  -e2p 2 /  n  p  n  ™ - 2  2  n  v  2  __ 2m n+d 2  _ n+2  - P  m  p  p  2"i.+6p2n+6  p  where e € { ± 1 } is the residue ofp modulo 4. 5. there exists an integer n > 1 such that p — 4 is a square and E is Q-isomorphic to one of the elliptic curves: n  a  a  2  El E2  epy/p — 4 -t2py/p - 4 n  ' A  4  2  -P  2  p  _ 8 2n+C  n+2  n  4 n+6  p  2  p  where e e {±1} is the residue ofp modulo 4. 6. there exist integers rn e {4, 5} and n > 0 such that p — 2 E is Q-isomorphic to one of the elliptic curves: n  a Fl F2  epy/p - 2 -t2psjp  -  n  _  m  2  m-2  p  2  n+2  2  m  is a square and  A  0-4  2  n  rn  2m n+6  2  p  _2 + p2n+6 m  p  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: a  2t+l  GI G2  p  pt+\/±±l V  e2  y  A  a.4  2  2t+l  p  V  2p 4  3+6t  2V  +  6  T  Chapter 3. Elliptic Curves with 2-torsion and conductor  where e € {±1} is the residue ofp  23 p a  l3  36  s  modulo 4.  t+1  8. there exist integers m € {4,5} and t € {0,1} such that  2 m  is a square and  + 1 p  E is Q-isomorphic to one of the elliptic curves: A  0,4  a  2  HI  2m-2p2t+l  H2  22m^3+6t 2 m+6  2t+l  p  where e € { ± 1 } is the residue ofp  ^3+61  modulo 4.  t+1  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:  A  0,4  «2  11  2m-2p2t+l  22rrip3+6t  12  _p2t+l  2^71+6^3+6^  where e e { ± 1 } zs ffoe residue ofp  modulo 4.  l+l  Theorem 3.5 Tfte elliptic curves E defined over Q, of conductor 16p , and having 2  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 2 p Q-isomorphic to one of the elliptic curves: rn  Al A2  -tpy/2 p t2p /2 p  n  m  n  A  04  a-2 m  + 1 is a square and E is  n  +1 +1  x  2m.p2n+6  2m—2pn+2  2  m+6pn+6  P  2  2  where e e {±1} is the residue ofp modulo 4. 2. there exists an integer n > 0 such that 4+p to one of the elliptic curves:  is a square and E is Q-isomorphic  n  a  Bl B2  a  2  epy/4 + p  n  -e2py/4+p  n  P  2  n+2  p  A  4  2p 4  n+6  2Sp2n+6  Chapter 3. Elliptic Curves with 2-torsion and conductor  23 p a  f3  s  37  where e G { ± 1 } zs the residue ofp modulo 4. 3. there exist integers m > 4 and n > 0 such that 2 Q-isomorphic to one of the elliptic curves:  + p is a square and E is  m  a 2  2 ~p  n  m  m  2  2  pn+2  e2p 2 + p /  A  (14  2  -qV ™ +p  Cl C2  n  n  2"T-+6p2n+6  v  where e G { ± 1 } is the residue ofp modulo 4. 4. there exist integers m > 4 and n > 0 suc/z that 2 Q-isomorphic to one of the elliptic curves:  - p is a square and E is  rn  a DI D2  -ep^/2  e2p 2  A  0,4  2 m  /  n  - p  2 ~p  - P  _pn+2  m  n  m  v  n  2  2  2m+6p2n+6  where e G { ± 1 } is the residue ofp modulo 4. 5. there exists an integer n> I such that p -4 is a square and E is Q-isomorphic to one of the elliptic curves: n  El  a-2  a  epy/p — 4  -p  -e2p^/p  E2  2  pn+2  - 4  n  A  4 2  n  4 n+6 p  _2 p 8  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 — 2 Q-isomorphic to one of the elliptic curves: n  a  a-2  Fl  -epy/p  F2  e2p P™ - 2  n  /  -  2  m  zs a square and E is  A  4  -2 - p  m  m  m  2  2  pn+2  22m,pn+6 _2"i+6p2n+6  v  w/zere e G { ± 1 } zs £«e residue ofp modulo 4. 7. £«ere exz'sfs an integer t G {0,1} SWCH that isomorphic to one of the elliptic curves:  is a square and E is Q-  Chapter 3.  Elliptic Curves with 2-torsion and conductor  GI  '  V  2  t3  A  m 2t+l  2  2t+l  2  8 3+6t  p  38  s  4 3+6t  2  /4+1  a  ct4  a t+i  23 p  p  p  G2  p  p  where e £ {±1} is the residue ofp modulo 4. 1  <§. 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:  4  A  rrip2£+l  22?n.p3+6£  p2t+l  2m-f-6p3+6t  a  a  2  HI  2  H2 wfaere e € { ± 1 } is  the residue of p  modulo  t+1  9. there exist integers m >  4 ana* t  £  4.  {0,1} such that  2 m  ~  x  is a square and E is  Q-isomorphic to one of the elliptic curves: a  a  2  11  2  e  { ± 1 } is  m 2t+l  22m^3-|-6£  p  2m4-6p3+6£  _ -i+i  12 if^ere e  A  4  p  the residue of p  modulo  i+l  4.  Theorem 3.6 The elliptic curves E defined over Q, of conductor 2>2p , and having 2  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 p -lis to one of the elliptic curves:  a square and E is Q-isomorphic  n  a  a  2  Al A2 Al' A2'  2py/p - 1 n  -4py/p  n  -2p  v  /  P  n  - 1  -P 4p  - 1  -P  4py/p - 1 n  A  4  2  2  n+2  2  2  4 p  n+2  p  12 2«+6 p  2  2  6 n+6  6 7i+6 p  12 2n+6 p  Chapter 3.  2 3V Q  Elliptic Curves with 2-torsion and conductor  /  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 ( m o d 4); BI B2  (b) p =  A  C14  a-2  0 0  2<y  -p  2t+1  -  2t+l  4p  p  l 2 2  +6t  3 + 6 t  - 1 ( m o d 4);  a Cl C2  0,4  A  2t+l  -2V  + 6 t  ]  + 6 t  2  0 0  p  -4p  2 V  2 i + 1  3. there exists an integer n > 0 such that 8p + 1 'is a square and E is Qisomorphic to one of the elliptic curves: n  a DI D2  p^8p + 1 -2p 8p + 1  DI' D2'  -p^8p 2p^8p  /  n  v  +1 + 1  n  n  2p  2p + 2p  n+2  6  2  p  2p 9  2  A  0,4  2p  El E2  +P -2 /8 + p  El' E2'  -p^/8 + p  2p  2 /8 + p  n+2  n+2  p  n  5. f/tere exists an integer n > 1 SWCH  6  2  2  n  Px  2p  2  n  Py  n+6  is a square and E is Q-isomorphic  n  n  6  n+6  <2& 2n+Q  n+2  2  2n  9  P 2p p  4. there exists an integer n > 0 such that 8+p to one of the elliptic curves:  a  A  0,4  2  n  p  9 2n+6 p  2  2  n+li  6 n+6 p  9p 7i+6 2  8—p" fs a square and E is Q-isomorphic  to one of the elliptic curves:  a  a  Fl F2  - P" - 2 / 8 -p  2p  Fl' F2'  -P\/8 - p 2 ^ 8 - p"  2  A  4  n  2  _ n+2 p  -2 p 6  2  n + 6  9 2n+6 p  P v  n  2p  2  _ n+2 p  _ 6 n+6 2  2  p  9p2n+6  Chapter 3. Elliptic Curves with 2-torsion and conductor  2 3^p a  40  s  6. there exists an integer n > 1 such that p -Sis a square and E is Q-isomorphic to one of the elliptic curves: n  a  0-2  GI G2 GI' G2'  n  P  n n  6  n+2  n+&  -2 V c  p  ~2p  2  -8 -8  -pVP 2p^p  2p  2  - 8  -2 y/p  A  4  -2p  -8  Ps/p  n  2  n + 6  6 n+6 p  n+2  p  7. there exists an integer t € {0,1} such that  is a square and E is Q-  isomorphic to one of the elliptic curves:  a  a  2  A  4  2  6 3+6t  _p2t+l  2  9 3+6t  2t+l  2  6 3+6(  2  9p3+6t  HI  2p  H2  2t+1  HI'  2p  „p2t+l  H2'  p  p  p  T h e o r e m 3.7 The elliptic curves E defined over Q, of conductor 64p , and having 2  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 p -lis to one of the elliptic curves: n  a  a  2  Al A2  -4ps/p  Al' A2'  -2?Vp - 1 4pyjp - 1  W P "  A  4  pn+2  -1  - 1  n  a square and E is Q-isomorphic  - 2 V  -4p  2  2  n+2  n  ^4p  n  +  G  p  - 2 V  p  N  12 n+6 N  +  6  2 p  2  12  n+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 ( m o d 4); 02  BI B2  0 0  a  _4 2t+l p  A  4  p2t + X  _ 6 3+6t 2  2  p  12 3+6t p  Chapter 3. Elliptic Curves with 2-torsion and conductor 2 "3 p c  l3  5  41  (b) p = -1 ( m o d 4); a  Cl C2  a  2  A  4  _ 2t+l  0 0  p  2  6 3+6t p  _2l2 3+6£  4 2t+l  p  p  3. f/tere exist integers m > 3 and n > 0 swc/z i/zaf 2 j/ + 1 is a square and E is Q-isomorphic to one of the elliptic curves: TO  a  a  2  DI D2  2p 2 p + 1 ~4p^/2 p + 1  DI' D2'  -2p 2 P +1 4p /2 p + 1  /  m  n  2  1  A  4  m n+2  22m+6 2n+6  p  p  v  rn  /  4p  n  m  n  v  m  2 p m  x  22m+6 27i+6  n + 2  4p  n  2m+12p?i+6  2  p  2m+12pn+6  2  4. i/iere exz'st integers m > 2 and n > 0 swc/t f«af 2 Q-isomorphic to one of the elliptic curves: a  a  2  +  m  is a square and E is  A  4  22m+6 n+6  El E2  2py/2 + p ~ 4 p 2 +p  2p n+2  4p  2m+12p2n+6  El' E2'  -2pv/2 + p 4py/2 + p "  2 p 4p + .  2m+12p2?i+6  m  n  /  m  m  n  v  m  n  m  2  m  p  2  n  2  5. fHere exzsf integers m > 2 and n > 0 stzc/z Q-isomorphic to one of the elliptic curves: a  Fl  2py/2  F2  a  2  - p  m  -4p 2  - p  n  Fl'  -2p 2  m  - p  n  F2'  4p 2  v  /  v  /  - p  TO  v  -4p  n  A  -4p  22m+6 n+6  2  p  n + 2  2"'p  n  — p is a square and E is  m  4  m  m  /  2  2 p  n  22m+6p7i+6  2  m+12p2n+6 2 "T-+6pn+6 2  2  2m+12p2n+6  n + 2  6. fnere exz'st' integers m > 2 and n > 0 swc/z f«flf Q-isomorphic to one of the elliptic curves: a  GI  a  2  2p^3 p e  - 2  n  m  G2  -4p /3 p  - 2  m  GI'  -2p>/3V - 2  m  G2'  4PV/3V - 2  f  v  1  n  m  4p  A 2  m  n + 2  22m+6pn+6 _2m+12p2n+6  n + 2  -2 p 4p  is a square and E is  m  4  -2 p  m  —2  2  22m+6 n+6 p  _ m+12 2n+6 2  p  Chapter 3. Elliptic Curves with 2-torsion and conductor  42  23p 0c  7. there exist integers m > 2 and t £ {0,1} such that  2 m  l3  + 1 p  s  is a square and E is  Q-isomorphic to one of the elliptic curves: 0  HI H2  A  4  2mp2t+l  22m+6p3+6t  -v+V ^ 2  2m+12p3+6t  HI'  2  m 2t+l p  2  2m+12p3+6t  2t+l  H2'  2m+6p3+6t  4p  8. f/tere exz'st integers m > 2 and £ £ {0,1} swdz f^iaf — -^ is a square and E is 2  p  Q-isomorphic to one of the elliptic curves: a  «2  11 12  -4p  11' 12' T h e o r e m 3.8  2  v+V ^ 2  A  4  m 2t + l  2  p  2  2m+6p3+6t  _ m + 1 2 3+6t  2 t + 1  2  m 2t+l p  2  _4 2(+l  2m+6p3+6t  _ m+12p3+6(  p  2  Tfte elliptic curves E defined over Q, of conductor 128p , rmd having 2  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 Qisomorphic to one of the elliptic curves: a  a  2  A  4  Al A2  2py/2p - 1 -Apy/2p - 1  2p + -4p  Al' A2'  -2py/2p - 1 4p /2p - 1  2p"+ -4p  Bl  2p 2p™ - 1  -p  B2  -4p 2p" - 1  Bl' B2'  -2py/2p - 1 4p 2p" - 1  n  n  n  n  v /  v  /  v  n  /  v  n  2  2  2  2  2  2  2  8p 8p  m+12pn+6 2  2  2  n + 2  m+12pn+6  2m+6p2n+6 2  2  n + 2  -p  2m+6p2n+6  2m+12p2n+e 2  2  Tn+6pn+6  m+6pn+6  2m+12p2n+C  Chapter 3.  Elliptic Curves with 2-torsion and conductor  2. there exists an integer n > 0 such that 2+p to one of the elliptic curves:  2p  n  Py  Cl' C2' DI D2  2psJ2 + p -4pv/2 + p  DI' D2'  -2ps/2 + p 4py/2 + p  n  n  Py  n  i  n+2  p  n  2p 4p"+  p7i+6  <22m+6 2  2m+12p2n+6  n+2  2"i+6p2n+6  p  8p  22m+12pTi+6  2  pn+2  8p  n  43  2m+12p2n+6  2  n  5  22m+6pn+6  2  -4pv/2 + p -2p^/2 + p " 4 /2 + p  0  A  (I4  «2  2 /2 + p  a  is a square and E is Q-isomorphic  n  Cl C2  23p  2m+6p2?i+6 2m+12pn+6  2  2  3. there exists an integer n > 0 such that 2-p is a square and E is Q-isomorphic to one of the elliptic curves: n  2  El E2  2 /2 - p  n  Py  n  El' E2'  -2 /2  n  Fl F2  2 /2 - P -4pv/2 - P  Fl' F2'  -p  4pV2 - p  n  n  Pv  - 2 p V 2  -  p  4PV2 - p  2p -4p +  22m+6pn+6  2  -4pv/2 - p Ps  A  0,4  a  n  n  n  n  2p -4p  2  2m+\2p2n+G 22m+G  2  n + 2  -p"+ 8p  2  pn+G  12p2n+6  m+Gp2n+6  2  _22m.+12pn+6  2  —pn+2  8p  2m,+  2  m+6 2n+6 p  _ 2m+12 n+6  2  2  p  4. there exists an integer n > 0 such that p -2 is a square and E is Q-isomorphic to one of the elliptic curves: n  a  a.2  A  4  p"+  2  22m+6pn+6  -2  -8p  2  _2'Ti+12 2n+6  GI' G2'  -2p p" - 2  p"+  2  22777.+6pn+6  4p p - 2  -8p  2  _ 2 " i . + 12p2n+6  HI H2  2pv/p™ - 2  2  _ m+6p2n+6  -4pv/p - 2  -2p 4p  n + 2  22m+12pn+6  HI' H2'  -2pVP - 2 4pv/p - 2  -2p  2  GI G2  2 /p  -2  n  Py  -4p p /  n  v  /  v  /  n  v  n  n  n  p  2  _2m+6p2n+6 22m+12pn+6  Chapter 3. Elliptic Curves with 2-torsion and conductor  44  23 p a  l3  5  T h e o r e m 3.9 The elliptic curves E defined over Q , of conductor 256p , and having 2  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  £  is a square and E is Q-  l  isomorphic to one of the elliptic curves:  a-2  a  A  4  29 2n+6  Al  p  A2  8P\/ "2 P  +1  8 p  Al'  2 p 15  2p  2  n+2  A2'  8p  Bl  2p  8p  2  9 2n+6  2  15 n+6  -*PXJ^  n+2  8p  2  2. there exists an integer n > 0 SMC/J fTzaf  p  2  2  9 n+6 p  15p2n+6  2p  B2'  p  2  2  n+6  2  n+2  B2 Bl'  n+2  9 n+6 p  15p2n+6  is a square and E is Q-isomorphic  to one of the elliptic curves:  0.2  a  A  4  CI  2p +  C2  -8p  c r  n  2  2  2p"+  9  2  2  C2'  -8p  2  DI  -2p  2  D2  g n+2  DI'  -2p  D2'  g n+2  p  p  2 n + 6  15 n+6 p  _ 9 2n+6 2  p  2 p + 1 5  n  2 p™ 9  6  +6  _ 15 2n+6 2  2  3. E zs Q-isomorphic to one of the elliptic curves: 1  -2 p  p  2  9 n+6 p  _ 15 2n+6 2  p  Chapter 3. Elliptic Curves with 2-torsion and conductor  CZ4  El  0 0  2p -8p  -2 p  E2 FI  0 0  -2p  2 p  F2  2  2  2  8p  2  a  l3  s  45  A  2  a  23 p  9  2  1 5  p  9  -2  1 5  6  6  6  p  6  4. t/*zere exz'sts an integer t e {0,1} size/ ! f/zaf Z? zs Q-isomorphic to one of the elliptic curves: 1  A  04  02  2p  GI G2  0 0  2p -8p  2l5 3+6t  HI H2  0  -2p  2V  2t+l  2t+1  2t+l  8p  3.1.2  The Proof for Conductor 2 p  3.1.3  List of Q-isomorphism classes  a  3+6t  p  2t+1  0  9  +6T  2l5 3+6t p  2  Let E be an elliptic curve over Q of conductor 2 p M  2  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 y = x + ax + bx, 2  3  2  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 b (a 2  2  -46) = ± 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 Qisomorphic 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  46  5  y = x + a x + C14X 2  3  2  2  Diophantine Equation 1 2  6? - 2 p = ± 1 d - 2 = ±p pd - 2 = ± 1 d -p = ±2 d - 1= 2p m  2  n  m  2  5  10 11 14 19 20 23  n  m  2  n  m  2  m  n  pd - 1 = 2 2d -p = ± 1 2d - 1 = p 2pd - 1 = ± 1 2  2m+2r!-2 n+2  ri  2m+2ra-2p2  2 pd 2 pd 2p d 2 ^ pd 2 pd 2 p d ri  p  r3+1  r  +1  2  2r  2 l  r3+1  2  2  l p  2r  +l  3  p  2 pd  n  2r  22ri + l n + 2  2 i+ pd r  n+2  l p  2 V  ri+1  n  2  0,4  2  ri+1  m  2  a ri  ri+2  2  2r-  l 2  1 +  p  22ri+l 2r +l  2  p  3  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}. y = x + a x + C14X 2  3  2  2  Diophantine Equation 2  10 11 14 20 24  0-2  d +2 = p d +p = 2 d + 1= 2p 2  m  2  n  2  n  2 pd  m  2 i pd  m  m  2  n  2  _ 2r 2  2 pd ri+1  3+1  r i +  2  n + 2  p  r  l  - 2  2 '+V d 2 V r  p  2  r +1  n  pd + l = 2 2d + l= p 2pd + 1 = 1 2  _ m+2ri-2 2  ri  3  + i  d  _22r _ 2r 2  p  V  2  l p  1  n+2  2r +l 3  l 2 p  +  _22r!+l 2r +l p  3  3.1.4 The end of the proof In this section, we verify that the elliptic curves appearing in Theorems 3.13.9 are the only curves, up to Q isomorphism, having the stated properties.  Chapter 3. Elliptic Curves with 2-torsion and conductor  23 p a  /3  5  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 2 p , N  2  having at least one rational point of order 2. A c -  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 follows 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 2 p , N  2  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 Qisomorphic 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 isomorphic 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  2 3^p a  48  s  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 isomorphic to one of the curves B l or B2 in Theorem 3.4. If F' is isomorphic to one of the curves C I or C2 of Theorem 3.5; if m 6  {4,5}, then F is isomorphic to one of the curves C I 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 D I or D2; if m G {4, 5}, then F is isomorphic to the curve D I or D2 of Theorem 3.4; if m > 7, then F is isomorphic to one of the curves C I or C2 in Theorem 3.2. If F' is isomorphic to one of the curves E l or E2 then the curve F is isomorphic 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 = d + 64 and F is isomorphic to one of the 2  curves in 3.1; if m > 7, then F is isomorphic to one of the curves D I or D2 in Theorem 3.2. If F' is isomorphic to one of the curves G I or G2 then the curve F is isomorphic to one of the curves G I 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. A l 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  49  23 p a  l3  5  Assertion (**) holds for Lemma 3.10: Since assertion (**) is concerned only with the curves up to quadratic twist we may choose the sign of a which makes calculations most convenient. This 2  usually involves specifying the congruence class of pd, a factor of a , modulo 2  4. We will make extensive use of the tables in Chapter 2 for computing conductors. 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)' +  satisfy d  i  2  = 2p m  curve with coefficients a — 2 pd and a — 2 Tl  2  + 1, and E is the elliptic  n  m + 2 r i  4  ~ p 2  n + 2  . We may assume d  is such that pd = — 1 (mod 4). Thus, using the tables in Chapter 2, the conductor of E is 2^ p 2  2  where if r\  0, m = 3;  ifn  0, m > 4;  ifn  l,m  > 2.  (Observe how the assumption pd = -1 (mod 4) reduced the number of possibilities for the value of f  2  in the case when r\ = 0 and m > 4.)  Now we can easily see that E is curve D I , A l , or D I in Theorems 3.6, 3.5, 3.7, respectively. 1") Suppose that (p, d, m, n) curve with coefficients a  2  satisfy d = 2 p 2  = 2 pd and a = 2 T1  2  4  m  n  m + 2 r i  - 1, and E is the elliptic ~ p 2  n + 2  d is such that pd = — 1 (mod 4). The conductor of E is 2 p 7  2  . We may assume 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 2 p . In what 7  'Then m > 3. T h e n m 6 {0 1}.  2  ;  2  Chapter 3. Elliptic Curves with 2-torsion and conductor  50  23 p a  f3  6  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? = 2  + p , and E is the elliptic  m  +  curve with coefficients a  = 2 pd and a  = 2  T1  2  n  ~p.  m+2ri  4  2  We may assume d  2  is such that pd = — 1 (mod 4). Thus, from Theorem 2.1, the conductor of E is 2Ep  where  2  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)  satisfy d  3  curve with coefficients a — 2 pd 2  such that pd=-l  — 2  2  ri  — p , and E is the elliptic  m  and a = 2  n  ~p.  rn+2ri  4  2  We may assume d is  2  (mod 4). The conductor of E is 2^ p 2  {  5  if r\ = 0, rn, = 3;  4  if n = 0, m > 4;  6  if  n = l,m  2  where  > 2;  Thus £ is curve F l , DI, or F l , respectively. 5 ) Suppose that (p,d,m,n)  satisfy pd  4  +  2  — 2  rn  and o, = 2  + 1, and E is the elliptic p  '  . We may as-  curve with coefficients a  — 2 p' d  sume d is such that p ' d  = —1 (mod 4). The conductor of E is 2 ^ p where  Tl  2  T i+1  r i+i  m + 2 r i _ 2  4  2 r  3 + 1  2  {  3  i f n =0,TO = 2;  4  if  6  if n  n  2  = 0,rn> 4;  = 1, 777 > 2.  Thus, E is curve G I (if n = 0, m = 2), H I (if n = 0, m > 4) and H I or HI' (if n = 1, 777 > 2). 5~) Suppose that (p,d,m,n)  5  3  4  5  T h e n m > 3. T h e n m ^ 1,3. T h e n m > 3.  satisfy pd  2  = 2  m  - 1, and E is the elliptic  Chapter 3. Elliptic Curves with 2-torsion and conductor 2 3 p a  curve with coefficients a2 — 2 p  5  51  . We may assume d is such that p ' d = — 1 (mod 4). The conductor of E is 2^ p where n  r 3 + 1  d and a = 2  l3  ~~ p  7n+2ri  4  2  2r3+1  7 3+l  2  {  5  if n = 0, m = 3;  4  if n = 0,m > 4;  6  ifri = l , m > 2 .  2  Thus £ is curve H I or H I ' (if n = 0, m = 3), II (if n = 0, m > 4) and II or II'  (if r = l,rn> 2). 1  10 ) Suppose that (p, cf, 777, n) satisfy d +  6  curve with coefficients a2 = 2 pd Tl+1  2  = p" + 2 , and E is the elliptic m  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^ p where 2  /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  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 = p - 2 , and E is the elliptic 2  curve with coefficients a, — 2  n + 1  2  n  p d and a = 2 4  2 n  rn  p  n + 2  . We may assume d is  such that pd = 1 (mod 4). The conductor of E is 2^ p where 2  6  Then m ^ 0.  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\  5  if ri = 1, 777 = 0;  7  if r\ = 1, 777 = 1;  6  if r i = 1, 777 > 2.  777  > 4;  2  Chapter 3. Elliptic Curves with 2-torsion and conductor 2 3 p a  l3  52  5  Thus E is curve A l (or A l ' ) , GI (or GI'), E2, G2 (or G2'), F2, A2 (or A2'), G2  (G2'), and H2 (or H2'). 11) Suppose that (p, d, ?n, ra) satisfy d = 2 p + 1, and E is the elliptic curve with coefficients a = 2 pd and 0,4 = 2 p . We may assume d is such that pd = 1 (mod 4). The conductor of E is 2^ p where 7  2  m  n  Tl+l  2 n  2  2  2  2  Thus E is curve D2 (or D2'), A2, and D2 (or D2'), respectively. 14) Suppose that (p, d, m. n) curve with coefficients a2 = 2  d is such that p ' 7  3+1  satisfy pd  8  = 2  2  p'  + 1, and E is the elliptic  rn  d and a = 2 p . We may assume d = 1 (mod 4). The conductor of E is 2^ p where r i + 1  3 + 1  2 n  2 r 3 + 1  4  2  2  0,m = 2; 0, m > 4; 1, m > 2. Thus E is curve G2, H2, H2 (or H2'), respectively.  19 ) Suppose that (p,d,m,n) satisfy 2d = p + 1, and E is the elliptic curve with coefficients a = 2 p d and a = 2 p + . The conductor of 1? +  2  n + 2  n  2ri+1  2  n  2  4  is 2 V / thus £" is curve B l (or Bl') if n = 0, and A2 (or A2') if r = 1. a  19~) Suppose that (p,d,rn,ri) satisfy 2d = p — 1, and E is the elliptic curve with coefficients a = 2 pd and a = 2 p . The conductor of E 2  ri+2  n  2ri+l  2  n+2  4  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 2d = p" +1, and I? is the elliptic curve 2  with coefficients  02  =2  ri+2  p d and a = 2  2 r i + 1  4  p . The conductor of E is 2 p , 2  8  2  thus £ is curve A l (or A l ' ) if n = 0, and B2 (or B2') if n = 1.  23 ) There are no solutions to 2pd = 1 + 1 so we have no curves corre+  2  sponding to this case.  23~) Suppose that (p, d, m, n) satisfy 2pd = 1 — 1, then d = 0, and E is the elliptic curve with coefficients a = 0 and a = 2 p . The conductor of E is 2 p and E is the curve Gl if r\ = 0 or the curve H2 if r\ = 1. 2  2 r i  2  8  2  Then m > 3.  7  8  Thenm ^ 0,1,3.  4  2 r 3 + 1  Chapter 3. Elliptic Curves with 2-torsion and conductor  23 p a  l3  5  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  = p  — 2 , and E is the elliptic curve  n  with coefficients a = 2 pd and a = —2 Tl  m  ~p.  m+2ri  2  A  2  We may assume d is such  2  that pd = — 1 (mod 4). Thus, using the tables in chapter 2, the conductor of E is 2^ p 2  2  where f  4 5  if  r i = 0, m = 2;  if r\  = 0, m = 3;  4  if r\  = 0,77i >  7  if  6  if n = l , m  7  if  4;  r i = 1, m = 1; ri  > 2;  = 2,777 =  1.  Thus E is curve E l , G I , F l , H I (or HI'), G I (or GI') and G I (or GI'), respectively. 10) Suppose that (p, d, rn, rif satisfy d  2  curve with coefficients a — 2  such that pd =  2  ri+1  = 2  rn  p d and a = - 2 4  — p , and E is the elliptic  2 r i  n  p  n + 2  . We may assume d is  1 (mod 4). The conductor of E is 2^ p where 2  2  5  if  ri = 0, 777 = 0 (i.e. 77 = 0)  7  if 7*i  = 0,777 =  5  if  7*1  = 0,777 = 3;  4  if  7*1 =  6  if  ?*i = l , m = 0 (i.e. 77 = 0)  7  if  ri = 1, 777 = 1;  6  if  7*1  0,777 >  1;  4;  = 1, 777 > 2.  Thus E is curve A l (or A l ' ) , F l (or Fl'), F2 (or F2'), D2, A2 (or A2'), and F2 (or F2'), E2 (or E2') respectively. 9  Then m ^ 2.  Chapter 3. Elliptic Curves with 2-torsion and conductor  11) Suppose that (p, d, m, n)  2  curve with coefficients a = 2 pd  m  l3  5  - 1, and E is the elliptic  n  and 0,4 = —2 p . We may assume d is  ri+1  2ri  2  such that pd  a  satisfy d = 2 p  w  54  23 p  2  = 1 (mod 4). The conductor of E is 2^ p where 2  5  if r\  7  if r i  2  — 0, m- 0;  6  — 0, m= i ; if r*i = 1, m = 0;  7  if r\ = l , m = i ;  Thus E is curve A l (or A l ' ) , BI (or BI'), A2 (or A2') and A2 (or A2'), respectively. 14) Suppose that (p, d, m, n)  satisfy pd  n  tic curve with coefficients a  2  assume d is such that p ' d T i+1  = 2  2  2  p  m  r 3 + 1  d and  — 1, and E is the ellip-  . We may = 1 (mod 4). The conductor of E is 2-^ p where —  n + 1  0,4  = -2  2 r i  2 r 3 + 1  2  6  if r\  = 0, rn=  5  if  = 0, rn = 0,p = 1 (mod 4);  5  if r\  4  if r\ = 0,777  5  if r\  =  6  = 0,p = - 1 (mod 4) if r\ = 1, 777 = 0,p = 1 (mod 4);  6  if r ]  = 1, 777 > 2.  v.  p  7*1  0,77 =  2  - 1 (mod 4)  — 0, m= 3; > 4;  1,777  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 2d = p — 1, and E is the elliptic curve 2  with coefficients a = 2 2  r i + 2  pdand  0.4  = -2  n  2 r i + 1  p - The conductor of E is 2 p , 2  8  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 2pd = 1 - 1 , then d = 0, and E is the elliptic curve with coefficients a = 0 and 0 4 = — 2 p ' . The conductor of E is 2 p and E is the curve HI if r\ = 0 or the curve G2 if 7*1 = 1. 2  2ri  27  3+1  2  8  2  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  3.2  Curves of Conductor  55  23 p a  l3  s  23p a  p  A s we mentioned in the introduction to this chapter, the models presented in the following table are minimal except i n 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 3 p, and having b  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 3 p E is Q-isomorphic to one of the elliptic curves: &  a Al  e • 3 "V 3 V + 1 -e • 2 • 3 " V 3 V + 1 ft  A2  a  2  2 6  b  2  b  1  + 1 is a square and  n  A  4  2l232f+6(b-l) 27  43^+2(fc-l) n  p  p  32(6-1)  2 < 5  where e G { ± 1 } is the residue of3  e  2  123f+6(6-l)  J  n  modulo 4.  2. there exist integers £ > 2 — b and n > 1 such that 2 3 + p is a square and E is Q-isomorphic to one of the elliptic curves: 6  0.2  Bl B2  e-3 -V fa  6  n  A  1I4  263f  -e-2-3 -V  e  2 6  +P  n  43<?+2(b-l)  2  ^+P"  where e G { ± 1 } is the residue of 3  b  3  1  2  1232«?+6(b-l) n  2123^+6(6-l)p2n  2(b-iy  modulo 4.  3. there exist integers £ > 2 - b and n > 1 such that 2 3 — p • is Q-isomorphic to one of the elliptic curves: 6  CI C2  e ^ - V ^ -c-2-3 - y/2 3 h  1  6  ~P  n  2  -p  i  where e G { ± 1 } is the residue of3  n  b  1  n  is a square and E  04  A  43<M-2(fe-l)  _ 1232<*+6(6-l)pn  0.2 2  e  _ 2(b-l) n 3  modulo 4.  p  2  2  123«+6(b-l)p27i  Chapter 3.  Elliptic Curves with 2-torsion and conductor  a  l5  4. there exist integers £>2 — b and n > 1 such that 2 p is Q-isomorphic to one of the elliptic curves: 6  a  5  + 3 is a square and E  n  l  A  04  2  DI D2  56  23 p  e • 3 - V 2 V + 3* -e-2-3 -V V+3^ b  b  2  4  b  1  2(6-iy  3^+2 ( b - l )  2  where e e { ± 1 } is the residue of3  3  2  123^+6(b-l) 2n  2  12 2'H-6(b-l)  p  3  6  a 6  6  !,  1  e  is a square and E  n  A  a.4  2  £-3 -V2 + 3 V -e-2-3 - v 2 + 3 V /  6  where e e { ± 1 } is the residue of3  b  1  2  3  4 2(6-l) 3  2  <'+2(6-])pn  2  123«?+6(b-l)pn  1232€+6(b-l)p2n  modulo 4.  6. there exist integers £ > 2 — b and n > 1 such that 2 — 3 p is Q-isomorphic to one of the elliptic curves: 6  a  Fl F2  e-3 -V2 -3V 6  -t • 2 • 3 ~' yj2 h  i  e  n  -  fi  2  3p e  where e € { ± 1 } is the residue of3  b  n  1  _2l2g-!+6(6-l)  4 2(6-l) 3  _^t+2(b-l)pn  2  modulo 4. e  is a square and E  n  A  _ 4 2(6-l) „ 2  b  6  0,4  0,2  (  where e € {±1} is the residue of3  b  3  2  p  3^+2(6-1)  -e-2-3 -V3 -2y 1  n  1232^+6(ft-I)p2n  7. there exist integers £ > 2 — b and n > 1 such that 3 — 2 p is Q-isomorphic to one of the elliptic curves:  GI G2  is a square and E  A  0,4  2  6  n  modulo 4.  5. there exist integers £>2-b and n > 1 such that 2 + 3 p is Q-isomorphic to one of the elliptic curves:  El E2  ? : )  123(;+6(b-l)p2n  _ 1232/;+6(b-l)pn 2  modulo 4.  8. there exist integers £ > 2 — b and n > 1 such that p is Q-isomorphic to one of the elliptic curves:  n  —23 6  e  is a square and E  Chapter 3.  Elliptic Curves with 2-torsion and conductor  a  «2  HI  H2  a  6  FT  4  £  3  b  6  A  £+2(B_1)  2  1232r?+6(6-l) n p  _2l23t»+6(b-l) 2r  263  w/zere e G {±1} zs r/ze residue of3  f3  4  -2 3 e • 3 " VP™ " 2 3 2(i.-l) n -e • 2 • 3 " V P " - ^ 6  57  23 p  ?J  p  t  modulo 4.  1  In fne rase iTzaf b = 2, i.e. N = 2 • 3 p , rx>e furthermore could have one of the 2  2  following conditions satisfied: 9. there exist integers n > 1 and s £ {0,1} such that Q-isomorphic to one of the elliptic curves: a  0-2  11  where  A  4  4 2.s+l  2 3 >  3 +V  2 3 >  2  12  3  s+1  a  2  4  - .2.3^V^ E  s+1  zs a square and E is  p  _ 12 3+6. n  2 5+1  2  -3 p 2s+1  .3^V^  _ . 2.3-+^^ e  where e G {±1} is the residue of 3  S + 1  3  5p  12 3+6 n  n  2  3  Sp  modulo 4.  a  02  2N  A  11. there exist integers n > 1 and s 6 {0,1} szicn tttflf Q-isomorphic to one of the elliptic curves:  £  ~  (I4  2  where e £ {±1} is the residue of3  K2  3+6  N  4.  modulo  23'  KI  3+6  12  20. f/zere exz'sf integers n > 1 and s G {0,1} sizc/z zTzaf Q-isomorphic to one of the elliptic curves:  n J2  12  2 s  e £ {±1} is the residue o / 3  is a square and E is  2  p  ~  zsflsquare and E is  2  A  4  -2 3  2S+1  2s+1  n  4  3 p  modulo 4.  23 p 12  3+6s  n  _ 12 3+6 2n 2  3  S p  Chapter 3. Elliptic Curves with 2-torsion and conductor 2 3 V a  /  58  5  T h e o r e m 3.13 The elliptic curves E defined over Q, of conductor 2 • 3 p, and having b  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 2 3 p square and E is Q-isomorphic to one of the elliptic curves: m  Al A2  €-3 -V  2rn-2^e+2{b-l)pn  V +l  2 m 3  -e-2-3 - v 2 3y + l b  /  1  m  i  3  where e G { ± 1 } zs the residue of3  b  2(6-l)  + 1 is a  n  A  0,4  (12 b  e  22m 2€+6(6-l)p2n 3  2m+6^+6(t-l)pn  modulo 4.  1  2. there exist integers m > 7, £ > 2 — b and n > 1 such that 2 3 square and E is Q-isomorphic to one of the elliptic curves: m  0,4  A 22m 2/?+6(i)-l)pn  2(6-l) n  2m+6 <?+6(6-l)p2n  2  BI B2  t • 3 - /2 3 +p -e - 2• 3 ~ ^2 3 +p 1  m  e  n  y  h  1  m  e  n  3  where e G { ± 1 } is the residue of3  b  n  2m-2<^t+2{b-l)  a h  + p is a  e  p  3  3  modulo 4.  1  3. there exist integers m > 7, £ > 2 — b and n > 1 such that 2 3 square and E is Q-isomorphic to one of the elliptic curves: m  0,4  A _22m 2/!+6(b-l)pn  2  Cl C2  e • 3 ~ y/2 3 l  m  e  -e-2-3 - y/2 3 b  1  m  -p  n  3  -p  e  _ 2(6-iy  n  3  where e G {±1} zs the residue of3  b  1  3  2^1+6^1+6(6-\)p2n  modulo 4.  4. there exist integers m > 7, £ > 2 — b and n > 1 such that 2 p square and E is Q-isomorphic to one of the elliptic curves: m  DI D2  e • 3 ~ ^2 p -t-2-3 - ^/2 p h  1  1  m  m  n  n  +3 +3 e  A 22m ^+6(6-l) 2n  3<+2(6-l)  2m+6 2<;+6(b-l)pn  i  b  e  0,4 3  where e G { ± 1 } is the residue of3  + 3 is a  n  2m-2 2(6-l)p7i  a.2 h  n  2?n-2 ^+2(')-l)  a b  - p is a  i  1  modulo 4.  3  3  p  Chapter 3.  Elliptic Curves with 2-torsion and conductor  23p a  0  59  s  5. there exist integers m > 7 , £ > 2 - b and n > 1 such that 2 square and E is Q-isomorphic to one of the elliptic curves:  + 3p  m  a El  e- 3 - V  E2  -e • 2 - 3 -  b  6  <z  A  2117,-232(6-1)  22m3£+6(b-l) n  3<?+2(6-l) n  2777+632^6(6-1)^277  4  2  +3 V  2 m  1  / v  2  m  + 3 V  p  p  where t e {±1} zs zTze residue of3  b  1  is a  n  modulo 4.  6. t/zere exz'sf integers m > 7, £ > 2 — b and n > 1 swc« z7zaf 2 — 3^> zs fl m  n  square and E is Q-isomorphic to one of the elliptic curves: a FI F2  «-3 b  1  a  2 /  v  2 -3V m  -e-2-3 -V b  2 m  -3V  where e € {±1} is the residue of3  b  1  A  4  2771-232(6-1)  _ 2 " 7 . 3^+ 6 ( 6 - l ) p 7 l  _3«+2(6-l)p77  277i+632»?+6(b-l) 2n  2  p  modulo 4.  7. f/zere exz'sf integers m > 7, £ > 2 — b and n > 1 swdz t/taf 3 p — 2 e  n  m  zs a  square and E zs Q-isomorphic to one of the elliptic curves: a GI G2  f  a  2  •^ V ^ "- 2  A  4  _2777-232(6-l)  m  3«+2(6-l)  -e • 2 • 3 - V 3 V - 2 6  m  where e £ {±1} is the residue of3  b  1  2277737? + 6 ( 6 - l ) p 7 1  n  _2777.+632<?+6(6-l)p2n  modulo 4.  8. there exist integers m > 7, £ > 2 — b and n > 1 such that 3 - 2 p square and E is Q-isomorphic to one of the elliptic curves: e  a  «2  HI H2  e -3 -^3 b  -  e  -e • 2 • 3 - ^/3 h  1  e  _ 2  2p m  -  n  2p m  m  3  3<+2(6-l)  n  where e e {±1} is the residue of3  b  1  n  is a  A  4  ~ 2 2 ( 6 - 1)  rn  n  22m3<?+6(6-l)p2n •  —2 + 3 + ( - )p m  6  2e  6  b  1  n  modulo 4.  9. there exist integers m > 7, £ > 2 — b and n > 1 such that p square and E is Q-isomorphic to one of the elliptic curves:  n  —2 3 m  e  is a  Chapter 3. Elliptic Curves with 2-torsion and conductor 2 3 p a  A  A  e• 3~VP" b  "  2  —2  ^  M  1  T O  - 3^+ ( - ) 2  2  h  1  2m 2£+6(b-l) n  2  2(b-l)n  - e • 2 • 3 - yV - 2 3* 6  60  s  a  0,2  11 12  l3  m  3  wnere e e {±1} is the residue of3  b  1  p  3  p  _2*n+63<+6(b-l) 2n p  modulo 4.  In fne case i«af b — 2, i.e. N = 2 • 3 -p, we furthermore could have one of the following conditions satisfied: 2  10. there exist integers m > 7, n > 1 and s € {0,1} such that and E is Q-isomorphic to one of the elliptic curves:  JI  2  J2  e - 2 . 3 + y ^ ± ^  m-2 2.s+l  s+l  2 2 ?n  3+6  3  ln  2,+ p  a  2  A  4  ^2"»-232s+l e  .  2  .  3  -  5  +  y ^  _2»+ln 3  wner-? e G {±1} fs fne residue o / 3  s + 1  22TnQ3+6spn 2 rn,+6 2 3-(-6-f p 2 77.  p  modulo 4.  22. inere em'sf integers m > 7, n > 1 and s G {0,1} swcn fnaf -""g and E is Q-isomorphic to one of the elliptic curves:  _2m-2 2.s+l 3  L2 wnere e € {±1} is the residue of3  3  s+1  is a square  2 2m 2 3+6  p  n  modulo 4.  n  2 +^33+6- p2n rn  2 s + 1  2 1  A  04  02 LI  fs a square  modulo 4.  a  _  ?/l  2"?-+6^3+6sp2n  2r  K2  is a square  SpTi  11. there exist integers m > 7, n > 1 and s G {0,1} swctt fnaf " ~ and E is Q-isomorphic to one of the elliptic curves:  KI  P  3  s  wnere e G {±1} is the residue of3  g  A  04  02  2  5  Chapter 3. Elliptic Curves with 2-torsion and conductor  23 p a  l3  61  s  Theorem 3.14 The elliptic curves E defined over Q, of conductor 2 3 p, and having 2  b  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^ + p is a square, 3 = - 1 (mod 4) and E is Q-isomorphic to one of the elliptic curves: n  e  04  A  3^+2(fa-l)  2432/J+6(6-l) n  2(6-l) n  283»M-6(b-l) 2n  0-2  Al A2  -e-2-3 -V -3^+p b  4  where e G { ± 1 } is the residue of3  b  n  3  p  p  p  modulo 4.  1  2. there exist integers £ > 2 — b and n > 1 such that 4 • 3 - p is a square, 3 = — 1 (mod 4) and E is Q-isomorphic to one of the elliptic curves: e  n  e  a  a  2  Bl B2  A  4  e-3 -V -3^-p _ 2(6-l) -e • 2 • 3 " V - 3' -p" b  3^+2(6-1)  n  4  b  4  3  where e £ {±1} is the residue of3  b  p  n  -2 3 4  M+6(ft  ~ p 1)  n  283<'+6(6-l)p2n  modulo 4.  1  3. there exist integers £ > 2 — b and n > 1 such that 4p — 3 is a square, p = — 1 (mod 4), and E is Q-isomorphic to one of the elliptic curves: n  e  n  a  «2  CI C2  e• 3 "V P ~ ^ -e • 2 - 3 " V P " ~ b  4  n  3  4  6  where e G {±1} is the residue of3  b  3  p  _3^+2(b-l)  '  3  A  4  2(6-l) n  _ 4 «+6(fc-l) 2n 2  2  p  3  832f+6(b-1)  n  modulo 4.  1  4. there exist integers £ > 2 — b and n > 1 such that p — 4 • 3 is a square, 3 = 1 (mod 4), and E is Q-isomorphic to one of the elliptic curves: n  e  e  a DI D2  e-3 ~ y/p b  1  -e-2-3 - ^p  n  b  1  n  A  04  2  - 4-3  '  e  -  4-3  e  _ «+2(6-l) 3  3  2(b-l) n p  2  4 3 2 ^ + 6 ( 6 - 1) n  _28 £+6(6-l) 2n 3  p  Chapter 3.  Elliptic Curves with 2-torsion and conductor 2 3 p ct  where e £ { ± 1 } is the residue of3  b  l3  62  5  modulo 4.  1  In the case that 6 = 2, i.e. N = 2 3 p , we furthermore could have one of the 2  2  following conditions satisfied: 5. there exist integers n > 1 and s £ {0,1} such that ~ is a square, p 1 ( m o d 4), and E is Q-isomorphic to one of the elliptic curves: A p n  a  a-2  El  3  E2  e-2-3'  5 + 1  / 4 v  ^i  where e £ { ± 1 } is the residue of3  =  p  n  2 s + 1  -2 3 4  2 3 8  V  3 + 6  3 + 6 s  p"  modulo 4.  s+1  6. there exists an integers n > 1 and s £ {0,1} swcft f/zaf is Q-isomorphic to one of the elliptic curves:  p  FI  3  F2  2s+l  3 p 2s+1  s + 1  n  2f is a square and E 4  A  0,4  a-2  where e £ { ± 1 } is the residue o / 3  n  A  4  2 s + 1  -3  l  2 3 4  2 3 8  p  n  V  n  3 + 6 s  3 + 6  modulo 4.  Theorem 3.15 The elliptic curves E defined over Q, of conductor 2 3 p, and having z  b  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 2 3 p a square and E is Q-isomorphic to one of the elliptic curves: m  4  2  Al A2  e-3 - /2 3 p 1  m  e  2m~2^e+2{b-l)pn  +1  n  y  n  + 1 is  A  a  a h  e  22m 2(!+6(b-l) p2n 3  -e • 2 • 3 - V 3 V + 1 f c  2 m  where e £ { ± 1 } is the residue of3  b  1  3  2(6-l)  2m+6^(+6(b-l)pTi  modulo 4.  2. there exist integers £ > 2 — b and n > 1 such that 4 • 3^ + p is a square, 3 = 1 ( m o d 4), and E is Q-isomorphic to one of the elliptic curves: n  e  Chapter 3. Elliptic Curves with 2-torsion and conductor 2 3@p a  a -e-3 - /4-3 +p e - 2 - 3 - V ' 3 * +p  a.4  A  3  £+2(b-l)  24 2/?+6(b-l) n  3  2(b-l) n  28 ^+6(b-l) 2n  2  BI B2  h  1  e  n  63  s  3  p  y  b  4  n  where e e {±1} is the residue of3  b  1  3  p  p  modulo 4.  3. there exist integers m e { 4 , 5 } , ! > 2 - [ i and n > 1 swc/t tTzat 2 3 + -p™ zs a square and E is Q-isomorphic to one of the elliptic curves: m  a  0,2  Cl C2  l  m  t  A  4  2m-2^+2(t-l)  e • 3 ~ • /2 3 + p -e-2-3' - v " '3*+p" h  n  <?  22m 2^+6(b-l) n . 3  p  s  ,  1  /2  32'b-l-yi  1  where e £ {±1} is the residue of3  b  1  2m+6 ;+6(6-l)p2n ,  3  modulo 4.  4. i/zere exzsi integers £ > 2 — 6 and n > 1 swc« rTzaf 4 • 3 — p is a square, 3 = 1 ( m o d 4), and JE zs Q-isomorphic to one of the elliptic curves: e  n  e  a DI D2  1  04  A  f+2(b-l)  __2 3 +6(b-l)pn  2  -e-3 - v -3 --p e-2-3 -V -3 -p ( ,  / 4  <  f c  4  f  n  3  _  n  where e 6 {±1} zs the residue of3  b  1  3  4  2 ( 6 - i y  2  2 f  8 ^+6(b-l) 2n 3  p  modulo 4.  5. there exist integers m £ {4,5}, £ > 2 — b and n > 1 SMC/? r7za£ 2 3 square and E is Q-isomorphic to one of the elliptic curves: m  a El E2  £-3 -V b  2 m  n  2m.-2 '!+2(b-l)  3^-p"  -e-2-3 -V b  p is a  —  A  114  2  2 m  <?  3  3 -p f  „  n  where e 6 {±1} zs the residue of3  b  1  3  2  (  2m+63< +6(6-l) 2n ,  b - i y  p  modulo 4.  6. frtere exz'st integers m £ {4,5}, £ > 2 — b and n > 1 swdz tTzflt 2 p + 3 zs a square and E is Q-isomorphic to one of the elliptic curves: m  a Fl F2  e-3 b  1 v  A  04  2  / 2 m  P +3 n  £  n  2m-232(b-l) n  22m3<?+6(b-l)p2n  3^+2(6-1)  2Tn+632£+6(b-l)p«  p  £  Chapter 3.  Elliptic Curves with 2-torsion and conductor  where e £ { ± 1 } is the residue o/3  b  1  23 p a  l3  64  s  modulo 4.  7. there exist integers £ > 2 — b and n > 1 such that 4p - 3 is a square, p = 1 (mod 4), and E is Q-isomorphic to one of the elliptic curves: n  e  n  -e • 3 - v^p™ - 3 e • 2 • 3 " V P - 3^ b  A  04  «2  GI G2  J  £  b  4  3  2(6-l) n p  _3^+2(6-])  n  2  zy/zere £ £ { ± 1 } is the residue o/3  h  1  832f+6(6-l) n p  modulo 4.  8. frtere exist integers £ > 2 — b and n > 1 such that 4 + 3*p is a square and E is Q-isotnorphic to one of the elliptic curves: n  a-2 HI H2  -e • 3 " 6  1  o  32(6-1)  + 3V  / 4 v  £-2-3 -V + 3 V (,  4  w/tere E G { ± 1 } is r«e residue o / 3  b  A  4  2  3«+2(b-l) n 1  2  4 3^+6(6-1) n  832«+6(6-l)p2n  modulo 4.  9. frtere exzsf integers m G {4,5}, ^ > 2 — b and n>\ such that 2 square and E is Q-isomorphic to one of the elliptic curves:  + 3p  rn  a-2 11 12  2 m  b  2  m-232(b-l)  2 m  where c G { ± 1 } is the residue o/3  b  3  1  M-2(fc-l) «  2  2  2m ?+6())-l)pTi 3<  modulo 4. — 3p  rn  A  m-232(fc-l)  _22m f+6(b-l)pn  _3f+2(fe-l) n  2?n+632(!+6(6-l) 2n  2  6  -e • 2 • 3  1 b _ 1  - 3  y/2 yj2 m  m  -  V  2  3V  p  where t G { ± 1 } is the residue ofZ  h  1  n  is a  m  is a  e  04  a  £• 3 -  is a  m+632f+6(6-l)p2n  10. there exist integers m G {4,5}, £ > 2 — b and n > 1 such that 2 square and E is Q-isomorphic to one of the elliptic curves:  Jl J2  n  A  04  £-3 -V +3 V -£-2-3 "V +3 V b  e  3  p  modulo 4.  11. there exist integers rn G {4,5}, £ > 2 — b and n > 1 such that 3 p square and E is Q-isomorphic to one of the elliptic curves: e  n  —2  Chapter 3. Elliptic Curves with 2-torsion and conductor 2 3 p a  a  a-2  e • 3 " y/3 p  KI  1  K2  1  -e - 2 • 3  e  - 2  n  m  3  where e G {±1} zs the residue of3  b  22.  A 22m^e+6(b-l)pn  _2m+6g2e+6(b-l) p2n  3<?+2(b-l)pn  m  65  5  4  _2m-2g2(6-l)  V V -2  b _  l3  modulo 4.  1  integers m G {4,5}, £ > 2 — b and n > 1 swc« square and E is Q-isomorphic to one of the elliptic curves: fnere exfsf  LI L2  w/zer.? e  e• 3 "V * -e-2-3 - /3 b  1  m  e  {±1} zs  tTze  m  n  m  s  G  -2 ~ 3 ( )p  2p -2 p  3  2  2  6_1  n  residue o/3  b  m  is a  n  22m^e+6(b- 1) p2n  %e+2(b-i)  n  3^ - 2 p  A  a.4  a-2 fa  t«ar  _2m+632t'+6(fe-l)pn  modulo 4.  1  23. fnere exz'sf integers £ > 2 — b and n > 1 swc/z (7zaf p — 4 • 3 is a square, n  e  3 = — 1 (mod 4), and £ zs Q-isomorphic to one of the elliptic curves: e  a Ml M2  A  a.4  2  _3«+2(6-l)  e-2-3 - /p h  1  - 4-3  n  (  y  3  2(6-l) n p  432*!+6(f>-l)pn  2  _ 83<f+6(6-l)p2n 2  zw/zere e G {±1} is the residue of3  b  1  modulo 4.  14. there exist integers m G {4, 5}, £ > 2 — b and n > 1 szzrfz that p — 2 3 is a square and E is Q-isomorphic to one of the elliptic curves: n  NI N2  e-3 ~ y/p 1  n  - 23  £ • 2 • 3 "VP" b  m  2  M  _ m-23«"+2(()-l)  l  3  2  ^  w/zere e G {±1} is the residue of3  3  b  1  e  A  CJ4  a-2 h  rn  2(6-l) n p  2  2m 2£+6(fc-l)pn 3  _ m+63<?+6(6-l)p2n 2  modulo 4.  In the case that 6 = 2, i.e. N = 2 3 p, we furthermore could have one of the 3  2  following conditions satisfied: 15. there exist integers m G {4,5}, n > 1, and s G {0,1} such that "+P" is a square and E is Q-isomorphic to one of the elliptic curves: 2  66  Chapter 3. Elliptic Curves with 2-torsion and conductor 2"2>^p  5  0,4  A  2m-2g2s+l  22m-^3+6Spn  0.2  Ol e-2-3  02  a + 1  ^±2l  where e 6 {±1} is the residue o / 3  s+1  3  2 s +  V  2m  + 6 2 3+6 s p 2 n  modulo 4.  26. f/zere exz'sf integers m G {4, 5}, n > 1, and s € {0,1} swcn tTztzi "32  0  is a  square and E is Q-isomorphic to one of the elliptic curves: a  a  2  PI P2  2m-232s+l  _22m-^3+6,Spn  _ 25+l n  2^71+6 ^3+6.Sp2n  3  i^nere e € {±1} is the residue o / 3  s+1  A  4  p  modulo 4.  27. f«ere exz'sf integers n > 1 and s G {0,1} such that " ~ 4p  is a square and E is  1  3  Q-isomorphic to one of the elliptic curves: a  A  0,4  2  Qi  3 p  Q2  -3  2s+1  where e € {±1} is the residue o / 3  s+1  n  2 s + 1  -2 3 4  2 3 8  3 + 6 s  3 + 6 s  p p  2 n  n  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: a Rl Pv2 where e € {±1} zs  —3  t-2-3 yJ^  2 s + 1  S+1  residue o / 3  A  0,4  2  3  s + 1  2 s + 1  p™  2 3 4  3 + 6  -2 3 8  ' p 5  3 + 6  n  V  n  modulo 4.  29. f/zere exz'sr; integers m G {4, 5}, n > 1, and s G {0,1} swcn z7za£ square and E is Q-isomorphic to one of the elliptic curves:  p  3  2  zs a  Chapter  3.  Elliptic Curves with 2-torsion and conductor  67  23 p a  l3  s  A  a  a-2  4  _2m-2g2s+l  SI  2 +6^3-f-6Sp2n  2s+l n  S2  3  where e e {±1} is the residue of 3  m  p  modulo 4.  S + 1  T h e o r e m 3.16 The elliptic curves E defined over Q, of conductor 2 3 p, and having 4  b  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 2 3 p square and E is Q-isomorphic to one of the elliptic curves: rn  4  2  Al A2  - e • 3 - y/2 3 p 1  m  £ •2 • 3 b  1  e  /2 3 p m  y  22Tn 2<?+6(6-l)p2n  + 1  n  e  3  + 1  n  + 1 is a  n  A  a  a b  i  3  where e e. {±1} is the residue of3  b  2(b-l)  2m+6^i+6(b-l)  n  modulo 4.  ]  2. there exist integers £ > 2 — b and n > 1 such that 4 -3 + p is a square and E is Q-isomorphic to one of the elliptic curves: e  0,4  A  *+2(b-l)  2  4 2<!+6(b~l)pn  . 2(6-l) n  2  8 (!+6(b-l)p2n  02  Bl  e-3 - y/4-3 b  B2  1  +  e  -£-2-3 6  1  / v  p  n  3  4-3 +p f  n  where £ 6 { ± 1 } is the residue of3  e+b  3  1  n  p  3  3  modulo 4.  3. there exist integers rn > 4, £ > 2 — b, and n > 1 such that 2 3 square and E is Q-isomorphic to one of the elliptic curves: m  C2  -t-3 h  sj2 3  x  m  +p  l  n  £ • 2 • 3 "V2 3 + p b  m  £  A  2m-23«f+2(6-l)  22?n 2i?+6(b-l)pn  2(b-l) n  2m+6 ?+6(6-l)p2ri  n 3  w/iere e € {±1} is rTze residue of3  b  + p is a  04  02  CI  e  1  p  n  3  3<  modulo 4.  4. there exist integers £ > 2 — b and n > 1 such that 4 • 3 — p is a square and E is Q-isomorphic to one of the elliptic curves: e  n  Chapter 3. Elliptic Curves with 2-torsion and conductor  DI  e-3 ~ ^4-3  D2  l  a  l3  68  s  A  a.4  0-2 b  23 p  -2 3  -p  l  4  n  -e-2-3 -V - ^-P 6  4  3  _ 2(b-l) n  n  3  where e € {±1} is the residue of3  e+b  M + 6 ( f t  " p 1 )  2S^e+6(b-l)p2n  p  modulo 4.  1  5. there exist integers m > 4, £ > 2 — b, and n > 1 such that 2 3 square and E is Q-isomorphic to one of the elliptic curves: rn  a El E2  -e-3 ~ ^2 3 1  m  -p  e  _ 2(6-l) n  e • 2 • 3 - y/2 3* - p 6  1  —2 3 ^+ ( 2m  2m-2g-?+2(b-l)  n  m  n  3  —p  e  is a  n  A  0,4  2  h  n  2  6  fe_1  )p  n  2m+6 <?+6(b-l)p2n  p  3  where e <E {±1} is the residue of3  b  modulo 4.  1  6. f«ere ex/sf integers m > 4, £ > 2 — b, and n > 1 such that 2 p m  + 3 is a  n  e  square and E is Q-isomorphic to one of the elliptic curves: 04  A  2m~2 2(b-l)pn  22m '+6(b-l)p2Ti  ^+2(b-l)  2m+6 2^+6(b-])p7i  a  2  Fl  -e-3 - v ' P + ^ e-2-3 -V P + b  F2  1  / 2  b  T l  n  2 m  n  ,  3  3  3  3 ( ?  3  where eg {±1} is the residue of3  b  3  modulo 4.  1  7. there exist integers £ > 2 — b and n > 1 such that 4p is Q-isomorphic to one of the elliptic curves:  — 3 is a square and E  n  e-3 -V P - * b  4  n  4  n  2  3  3  -e-2-3 -V P b  A  0,4  a.2  GI G2  e  b  b - i y  _ f+2(b-l)  3 < ?  • where e G {±1} is the residue of 3  (  3  p  1  _24 ^+6(b-l)p2n 3  28 2^+6(b-l) n 3  p  modulo 4.  n  8. there exist integers £ > 2 — b and n > 1 such that 4 + 3 p is Q-isomorphic to one of the elliptic curves: e  a HI H2  e-3 -V + V 4  3  -e • 2 • 3 - s/4 + 3 V 6  3  1  3  24^e+6(b-l)pn  2(b-l)  i+2(b-l)pn  is a square and E  A  (34  2  b  n  2  8 2'!+6(6-l)p2r 3  l  Chapter 3.  Elliptic Curves with 2-torsion and conductor  where e £ {±1} is the residue of3  b  1  2°3 p l3  69  s  modulo 4.  9. there exist integers rn > 4, £ > 2 - b, and n > 1 such that 2 square and E is Q-isomorphic to one of the elliptic curves:  rn  -E • 3 - V b  e-2-3 6  1  /  A  v  2m.-2g2(t.-l)  22m -?+6(6-l) n 2 m + 6 2 £ + 6 ( b - 1) 2n  A  2 m  2  m  + 3V  where e G {±1} is the residue of3  b  modulo 4.  1  m  a b  1  h  m  1  m  s  2m-232(b-l) „3^+2(b-l) n  - 3 V - 3p e  n  —3p  n  is a  rn  is a  e  A  C14  2  -e • 3 - ^2 t • 2 • 3 ~ /2  is a  3  10. there exist integers m > 4, £ > 2 — b, and n > 1 such that 2 square and E is Q-isomorphic to one of the elliptic curves:  JI J2  n  3  3<+2(b-l) n  + 3V  e  a  a.2  11 12  + 3p  _22m3^+6(b-l)pn  p  where e G {±1} z's the residue of3  b  modulo 4.  1  11. there exist integers m > 4, £ > 2 — b, and n > 1 such that 3 p' — 2 square and E is Q-isomorphic to one of the elliptic curves: e  «2  KI K2  b  A  0,4  _2m-2g2(b-l)  - £ - 3 - V V -2"' e • 2 • 3 -V3V - 2 3  b  l  22m3^+6(b-l)p7i  —2 + 3 ^+ C'- )p2n m  m  n  £+2(b-l)pn  6  2  6  1  3  where e G {±1} is the residue of3  b  1  modulo 4.  12. there exist integers m > 4, £ > 2 — b, and n > 1 swc«  tTzai  3^ — 2 p m  n  is a  square and E is Q-isomorphic to one of the elliptic curves: 4  2  LI L2  -e-3 - /3 1  —2 ~~ 3 ( ~ )p  -2 p  e  m  s  m  n  e-2-3 - v 3 -2 p" b  1  /  A  a  a ' b  €  m  2  2  b  1  n  22m.3^+6(b-l)p2n  _2 + 3^+6(b-l)pn  where e G {±1} is the residue of3  3<M-2(6-l) b  1  TO  6  2  modulo 4.  13. there exist integers £ > 2 — b and n > 1 SMC/Z that p E is Q-isomorphic to one of the elliptic curves:  n  - 4• 3  £  zs a  square and  Chapter 3.  Elliptic Curves with 2-torsion and conductor  a £'3 -VP - 4-3'  Ml M2  -e •  a  6  s  A  N  2• 3 "  l3  a.4  2  B  70  23 p  VP" - • * 4  3  where e g { ± 1 } zs (Tie residue ofS  3  2432*?+6(6-l)pn  _ 83f+6((>-l) 27i  2(6-l) n p  2  p  modulo 4.  e+b  24. f/iere exz'sf integers m > A, t > 2 — b, and n > 1 swc/z f/zaf p " — 2 3 ^ is a square and E is Q-isomorphic to one of the elliptic curves: m  NI N2  -e •  3 -Vp  A  a.4  0,2 h  n  _2 - 3<'+2(b-l) m  ~ 2 3 m  f  e • 2 • 3 -VP™ - 2 3^ b  m  where e G { ± 1 } is the residue o / 3  2  3  6  22m32t?+6(b-l) n p  _2m+63f+6(6-l) 2n  2(6-l) n  p  p  modulo A.  1  In the case that 6 = 2, i.e. N = 2 3 p , we furthermore could have one of the 4  following  2  conditions satisfied:  15. there exist integers n > 1 and s G {0,1} such that ' ~ Q-isomorphic to one of the elliptic curves: Ap  is a square and E is  l  i  a  3  Ol 02  A  04  2  e•2•3 + / s  1  4 p ,  v  ;-  where e € { ± 1 } zs the residue ofZ p' s  1  2 s +  -3  1  V  -2 3 4  2  2 s + 1  8  3 3  6s  modulo A.  a  a  2  PI  3 £-2-3  i+1 v  2  + p™  16. there exist integers n > 1 and s G {0,1} such that Q-isomorphic to one of the elliptic curves:  P2  p "  3 + 6 s  /Z£4  3  is a square and E is  A  4  2 3 4  2 a + 1  2 s + 1  p  3 + 6 s  p"  n  where e G { ± 1 } is the residue of 3 modulo A. s  17. there exist integers m > A, n > 1 and s G {0,1} such that ' " P " is a square and E is Q-isomorphic to one of the elliptic curves: 2  +  Chapter 3.  71  Elliptic Curves with 2-torsion and conductor 2 3@p a  a  a  2  6  A  4  2m-2g2s+l  Ql Q2  £  .  2  .  3  3  +  1  ^  2  ^  22m ^3+6spTi  3 p 2s+1  n  where e G { ± 1 } Z'S f/ze residue of 3 modulo A. s  28. z7zere exz'sf integers m > 4, n > 1 azzd s G {0,1} szzc/z that " ~ ' zs a square and E is Q-isomorphic to one of the elliptic curves: 2  a-2  a  A  4  Rl  2 ^ . - 2 3 2 ^ + 1  R2  __ 2 +l n 3  p  2m+Gg3+65p2n  p  S  where e G { ± 1 } z's f/ze residue of 3 modulo 4. s  29. fTzere exz'sf integer n > 1 a n d s G {0,1} stzcTz f/zaf Q-isomorphic to one of the elliptic curves:  a-2  A  04  _ 2 +l  SI S2  3  e-2-3" y/ '+1  p7  z's a square and E is  4  3  5  2 5 + 1  p™  2 3 4  3 + 6 s  p  n  _ 8 3+65 2n 2  3  p  if/zere e G { ± 1 } is the residue of 3 modulo 4. s  20. tTzere exz'sf integers rn > A, n > 1 a n d s G {0,1} szzc/z that ' g and E is Q-isomorphic to one of the elliptic curves: p  2 ~2^2s+l m  TI T2  -e-2.3'+V^  3  2 s +  z's a square  A  a4  02  2  V  22m^3+6Spn 2"i+6 ^3+6s p 2 n  w/Vre e G {±1} z's f/ie residue of 3 modulo A. s  T h e o r e m 3.17 The elliptic curves E defined over Q, of conductor 2 3 p, and having b  b  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 2 3@p a  1. there exist integers £ > 2 — b and n > 1 such that 3 p is Q-isomorphic to one of the elliptic curves: e  a  A  A2 Al'  -2 • 3 " v / 3 V + 1  A2'  4-3 -V3V + l  ft  2  4 . 32(b-l)  b  6  + 1 is a square and E  n  04  2  2-3 -V3V + l -4-3 -V3V + l  Al  72  s  ^e+2(b-l) n  1  2  p  4.  b  632f+6(b-l) 2n p  2  12 M-6(b-l) n 3  p  632-?+6(6-l) 2n p  2  123<;+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  BI B2  2 • ^-V^'+p" -4-3 -V3 b  2(b-l)n  3  +p  f  B2'  -2-3 "V3 b  2  . 3<?+2(6-l)  2(b-l)n  +P  f  n  3  4 • 3 - V ^ +P b  p  n 4  BI'  A  0,4  2  3  n  p  123 f+6(b-l)pn  2  2  26^+G{b~l) 2n p  p  . ^+2(b-i)  4  63^+6(b-l) 2n  3  2  12 2«+6(b-l) n 3  p  (b) £ is odd; a Cl -C2 Cl' C2'  2•  3 ~' y/3 +p [  e  n  -4-3 "V3'+P t  b  4-3 -  3  2  2  3  n  3  v^+p™  1  (?+ (b-l)  . 2(b-l) r  4  n  -2-3 ~V3^+P 6  A  04  2  b  4  p  l  «+2(6-l) p  3  p  123<!+6(b-l) 2n p  2  -. 2(b-l) n 3  2  6 2<!+6(b-l) n  632'!+6(6-l) n p  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; 04  A  _ 2(6-l) n  63<;+6(6-l) 2n  02  DI  2-3 - v 3 '-p" b  1  /  (  3  D2  - 4 - 3 - - <f3  -p  DI'  -2-3 ~ sj3  -p  D2' (b) £ is odd;  b  b  e  l  l  4-3^V3'-P  4  3  _ 2(fc-l) n  n  n  p  . <?+2(6-l)  n  3  4  .  p  3^+2(6-1)  2  p  _ 1232£+6(b-l) n 2  2  p  63/?+6(b-l) 2n p  _ 1232<!+6(b-l) n 2  p  Chapter 3. Elliptic Curves with 2-torsion and conductor  2  El' E2'  2-3 - 3 -p fe  1  /  <!  v  - 4 - 3 ~ y/3 h  a  l3  s  A  0,4  a  El E2  73  23 p  1  n  —4.3 ( - )p 3^+2(6-1) 2 fc 1  -2-3 -V3^ -P fe  n  -4 •  4-3 -V3 -p" f  b  -2 3  3«+2(6-l)  n  -p  t  6  n  2 < , + 6 ( b  ~ ^™ 1  2l2 ''+6(b-l) 2n 3  JD  _632*;+6(b-l)pn 2  Z^-Vp"  123£+6(b-l)p2n  2  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; a  Fl F2 Fl' F2'  2-3 "VP b  - 3  n  e  -4-3 -Vp - * -2-3 -Vp - 3 b  n  b  n  4•3 " b  632(!+6(b-l)pn  _3<?+2(6-l) 4• S ^- ^" 2  3  £  y/p" - 3 -  1  A  a.4  2  _3'!+2(b-l) 4 • S ^" 2  f  2  -2 3'  1  ? + 6 ( b _ 1  ^p  2 n  62":+6(b-l)pn  2 3  V  1  1 2  _ 123"J+6(b-l) 2n 2  p  (b) lis odd; a  a  2  GI G2  -4-3 "VP  GI' G2'  -2-3 ~VP - 3 4-3 P -3  2-3 ~ /p h  l  -3  n  f  s  b  b  n  b  1  v  /  3  " 3^  n  n  £  A  4  2(6-l) n p  _ 4 . 3-2+2(6-1)  _ 63^+6(6-l) 2n 2  2  f  _  3 4  2(6-l) n . 3^+2(6-1) p  2  HI' H2'  p  2  p  1232£+6(6-l) n p  8 • 3 p + 1 is a square and n  A  0,4  a  HI H2  1232«+6(6-l) n  _ 63«+6(6-l) 2n 2  5. f«f?re exist integers I > 2 — b and n > 1 swcn E is Q-isomorphic to one of the elliptic curves:  p  3 -V - V + l 2 • 3«+2(b-l) n 6 2«+6(b-l) 2?i -2-3 -V -3V + l 2(b-l) 93<!+6(b-l) n -3 -V8-3V + l 2 • 3«+2(6-l) n 6 3 2 f + 6 ( 6 - l ) 2 n 2-3 -V8-3V + l 2(b-l) 93<!+6(6-l) n b  8  3  p  b  8  2  3  p  l  3  2  p  b  p  2  p  b  3  2  6. there exist integers £ > 2 — b and n > 1 such that 8 • 3 + p E is Q-isomorphic to one of the elliptic curves:  p  e  n  is a square and  Chapter 3. Elliptic Curves with 2-torsion and conductor  cc  l3  4  11 12 11' 12'  2•  3 ~W8 • 3 + P e  -2-3 b  -3 b  1  n  8-3^+p  / v  e  n  8  3  2632^+6(6-1)^1 293^+6(6-l)p2n  p  . 3<+2(6-D  2  2632£+6(ft-l) n p  2 - 3 - V - * +P b  b  3  y/8 • 3 + p  1  3 ( -V e+2  2(b-l)n  T l  s  A  a h  74  23p  n  32(6-l) n p  2  93t"+6(6-l)p2n  7. there exist integers £ > 2 — b and n > 1 swcfr f/zaf 8 • 3 — p is a square and E is Q-isomorphic to one of the elliptic curves: e  a  A  04  2  Jl J2  3 ~ yJ% -3 -p -2-3 -V8-3 -p  Jl' J2'  - 3 - V 8 - 3 - p "  b  x  e  n  €  6  2-3 "V8-3^-p b  3  _2632£+6(b-l) n p  _ 2(b-l) n 3  €  b  . /+2(b-l)  2  n  2  2  p  p  p  3  3  93(?+6(b-l) 2n  _2632f+6(b-l) n  . /+2(6-l) _ 2(6-l) n  n  n  2  p  93<?+6(6-l)p2n  8. there exist integers £ > 2 — b and n > 1 such that 8p + 3 is a square and E is Q-isomorphic to one of the elliptic curves: n  ai KI K2 KI' K2'  l  4  +3  n  t  x  -2-3 -V8p b  - 3  b  A  a  3 ~ /8p b  e  - V % >  +  n  3  n  p  3<?+2(6-l) 2  3 < r  2-3 -V8p + 3 b  2 6 3 ^ + 6 ( 6 - l)p2n  . 2(6-l) n  2  + 3^  n  932^+6(b-l)pn  . 2(b^l) n  263f+6(b-l) 2n  «?+2(6-l)-  2932<"+6(b-l)pn  3  £  2  3  p  p  9. there exist integers £ > 2 — b and n > 1 such that 3 p is Q-isomorphic to one of the elliptic curves: e  ai LI L2 LI' L2'  a  3 "V3V -2-3 -V V - 8 -3 ~V V - 8 2-3 -V V - 8 b  8  b  b  3  3  b  3  - 8 zs a square and E  n  A  •  4  -2 • 3 ( - )  26 <M-6(b-l) n  3^+2(b-l) n  _ 9 2^+6(b-l)p2n  2  b  x  p  _  2  .  2  32(6-1)  e+i{b-i)pn  3  3  p  3  2  63»'+6(b-l)pn  _ 932«+6(b-l)p2n 2  10. there exist integers £ > 2 — b and n > 1 such that 3 - 8p is a square and E is Q-isomorphic to one of the elliptic curves: e  n  Chapter 3.  Elliptic Curves with 2-torsion and conductor 2 3@p a  a  0.2 _  n  Ml M2  -2-3 -V ^- P  Ml'  - 3 - V 3 * - 8p  M2'  2-3 -  Z  b  8 p  3  1  -2 • 3 ( " 2  8  h  b  A  4  b-\^J 1  Z  75  s  / 3 f v  - P 8  b  263f+6(b-l)p2n  V  3^+2(6-1)  n  n  -2 • 3  <  2  - V  b  2  3«+2(b-l)  n  6 £+6(b-l)p2n 3  _ 9 2»M-6(6-l) n 2  3  p  11. there exist integers I > 2 — b and n > 1 such that p - 8 • 3 fs a square and E is Q-isomorphic to one of the elliptic curves: n  a  a  A  4  2  NI  3 -Vp -8-3^  -2 • 3 « + ( - D  N2  -2-3 -VP -8-3«  2(b-l) n  _ 93^+6(6-l) 2n  NI' N2'  -3 "VP" -8-3^  - 2 • ^+2(b-l)  2632«+6(b-l) n  f c  2  n  b  n  3  b  2-3 -Vp - 8 - 3 ' b  n  3  2&^2l+6(b-\) n  b  p  p  2  p  p  -2 3 9  2(b-l) n p  € + e  (  b _ 1  )p  2 n  In the case that b — 2, i.e. N — 2 3 p, we furthermore could have one of the 5  2  following conditions satisfied: 12. there exist integers n > 1 and s e {0,1} such that Q-isomorphic to one of the elliptic curves:  is a square and E is  a  A  4  Ol  2• 3  2s+l  S + 1  02 or  4• 3 2-3  2 3 6  3  / "  s + 1  p  v  3  + 1 3  2 s +  2  2 s +  3  2  V  p  n  12 a+63 2n  2 s + 1  4• 3  02'  V  3 + 6 s  p  6 3+6. n 3  5 p  2l2 3+6s 2n 3  p  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: a  a PI  2• 3  S + 1  3  P2 PI' P2'  A  4  2  25 + l n p  -4 • 3 _ 2 .  3  4-3'  ' ' + y ^ 5 + 1 v  /S  l 1  3  2 s + 1  2s + l n  -4-3  p  2 s + 1  _ 6 3+6.s 2n 2  2  3  p  12 3+6s n 3  _ 6 3+6 2  2  3  p  S p  2n  12 3+6.s n 3  p  Chapter 3.  Elliptic Curves with 2-torsion and conductor  2°3 p l3  76  5  14. 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: p  a  A  0,4  2  2•3  Ql  2 5 + 1  3 p 2s+1  Q2  2  Ql' Q2'  2  - ' 3  5 + 1  - 3  S  -3  s + 1  +  V "3 P  1 V  P V  "  + 8  2•3  ^  3  + 8  23 p  n 2  2s+1  2.s + l n p  3+6s  6  9 3+6. 3  6 3+6.  2  2  5 p  3  n  2n  5 p  n  9 3+6s 2n 3  p  3  15. there exist integers n > 1 and s e {0,1} such that — g - is a square and E is Q-isomorphic to one of the elliptic curves: 2  a4  A  2 • 3 +-  _ 6 3+6s 7i  0  2  Rl R2 Rl' R2'  2s  -2'3 V^F s+  -3  3  v/V  S + 1  2• 3  -3 V T s+  2  _ 2.,+y,.  _  8  3  2 5 + 1  2 , + 1 n p  2  2  3  p  9 3+6.s 2n 3  p  -2 3 p 3+6s  6  9 3+6.  2  3  S p  n  2n  16. there exists an 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  3  o,  o  2  SI  3'  S2  2 • 3-  SI'  -3' V^  S2'  2•3  5+1  -2 • 3  \/^? 5 + i  3  S+  S + 1  i/^V  A  4  2  6 3+6s n 3  p  2.s+l ,i p  -2 • 3 8  2 s + 1  2 s + 1  3 p" 2s+1  23 p 6  3+6s  _ 9 3+6 2  3  S p  n  2n  Theorem 3.18 The elliptic curves E defined over Q, of conductor 2 3 p, cmd having &  b  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 3 p is Q-isomorphic to one of the elliptic curves: e  n  + 1 is a square and E  Chapter 3. Elliptic Curves with 2-torsion and conductor 2 3 p oc  a-2 l  b  Al'  b  A2'  3  p  2(b-l)  2  4 . «+2(fe-l) n  3  3  63''+6(6-l)pn  2l232/?+6(b-l) 2n  p  3  b  2  . 3<-+2(6-l) n  4  77  A  4  32(6-1)  2-3 -V3V + l -4-3 -V3V + l -2-3 -V V +l 4-3 -V V + l  A2  s  a  b  Al  l3  2  p  63f+6(b-l)pn  1232<+6(b-l)p2n  2. there exist integers £ > 1 and n > 1 such that 3 + p is a square and E is Q-isomorphic to one of the elliptic curves: n  (a) £ is even; a BI  •fW  2-3 b  1  / v  3- +P ?  n  3  B2  -4-3 - v 3' +P"  BI'  -2-3 -V3 +P  B2'  4-3 -V3^+P  b  1  /  £  2  7 1  b  «+2(b-l)  4 • 3 ( "V  ?  b  3  b  2  2(b-l)  12 «+6(6-l)p2n 3  2  4 . C+2(b-l) n  n  3  p  2  C32'?+0(b-l) r7, p  123(!+6(b-l) 27i p  £ is odd; a  02  3.  A  a.4  2  Cl  2-3 -V3^+P  C2  -4•3 -V3'+P'  Cl'  -2-3 -V3'+P  C2'  4 • 3 " V 3 ^ +P  b  3  n  I ,  b  b  ,  3  2l2 2-;+6(6-l) n 3  2(b-iy  2  4 . f+2(6-l) 3  integers £ > 1 and n > 1 swdi Q-isomorphic to one of the elliptic curves:  63f+6(b-l)p2n  2  4 . 3^2(6-1)  n  n  A  4  2(b-iy  6 ':+6(b-l)p27i 3  21232^+6(6-1)^71  3 — p™ is fl square and E is e  t/iere exisf  (a) £ is even; a  «2  DI  2-3 f c  1  / v  3*-p  n  D2  -4-3 -V3 -p  n  DI'  -2-3 -V3 -P  n  D2'  (b) £ is odd;  b  b  4-3 ~W^ b  A  4  _26 2/?+6(b-l) n  3<?+2(b-l)  f  f  ~P  n  —4 • 3 ( - ) p 2  6  1  3  n  3^+2(6-1) -4  • S ' - ^" 2  6  1  p  2l23f+6(b-l) 2n p  -2 3 ( ~ )p 2l23-?+6(6-l) 2n 6  2 ( ! + 6  b  1  n  p  Chapter 3. Elliptic Curves with 2-torsion and conductor 2 3 p a  a  a-2  El E2  2•  e  n  b  El' E2'  b  1  l  i  3  i  n  3  4  3  M  12  6  b  1  63£+6(b-l) 2n  2  p  . <?+2(b-i)  p  _2 3 + ( - )p™  _ 2(6-l) n  n  63«+6(b-l) 2n  2  p  4 . €+2(6-l)  n  b  A  3  - 4 - 3 - V ^ ~P -2 • 3 ~ ^3 -p A-3 ~ y/3 -p  78  s  4  _ 2(6-l) n  3 ' y/3 ~p h 1  l3  p  _ 1232t'+6(6-l) n 2  p  4. f/ter<? exist integers £ > 2 — b and n > 1 swc/z zTzzzf p™ — 3 is a square and E is Q-isomorphic to one of the elliptic curves: e  (a) £ is even; 0  <22  FI F2  -4-3 -Vp -3  £  FI' F2'  -2-3 -Vp"-3 4-3 -Vp -3  f  2 - 3  -  b  V - 3 *  1 v  b  n  b  b  n  3  _  A  4  _ 63«+0(b-l) 2n  2(6-l) n  2  p  . 3^+2(6-1)  4  3  2  2(6-l) n  1232f+6(6-l) n p  _ 63«+6(fc-l) 2n  p  2  _ 4 . 3^+2(6-1)  £  p  p  126 2^+6(b-l) n 3  2  p  (b) £ is odd; a  a  2  2-3 "Vp -3^ - 4 • a - s/p - 3  GI' G2'  - 2 • 3 " V P " - 3*  n  6  n  1  4- 3 - V p  n  -3  2  3  e  n  3  _ 12 ('+6(b-l)p2Ti  _3«+2(fc-l)  632/!+6(6-l) n  2  6  3  2  2  p  _ 1 2 « + 6 ( b - l ) 2n  4 • S ' - )^  £  6 2£+6(b-l)  4 . 3 (b-i) « 2  p  b  b  A  4  _ ^+2(b-l)  GI G2  b  1  2  3  5. there exist integers m > 3, £ > 2 — b and n > 1 such that 2 3 p™ + 1 zs square and E is Q-isomorphic to one of the elliptic curves: m  a  a  2  HI H2  -4-3 -V  HI' H2'  -2-3 - v / S ^ V+ l 4-3 -V 3V +l  2- 3  b _ 1  y/2 3 p m  b  b  b  2  1  2  m  e  m  n  A  4  +1  2  4 . 2(b-l)  3V +l  m 3^+2(6-1) TI 4 . 2(b-l) 3  p  2  3  2  2m+632<?+6(b-l) 2n  2  m+123*+6(b-l) n p  2777.+632i"+6(b-1 ) 2 n p  2  m+123«+6(b-l) n p  6. there exist integers m > 2, £ > 2 - b and n > 1 such that 2 "3 + p" is a square and E is Q-isomorphic to one of the elliptic curves: T  Chapter 3.  Elliptic Curves with 2-torsion and conductor  2  11 12  - 4 • 3 ~ ^2 3  11'  -2  12'  2-3 -V2 3*+p m  +p  • 3 ~ y/2 3  +p  1  b  m  1  m  4 •3~ b  5  A  n  e  +  m  4. 3 ( f c - i ) p « 2m 3 ^ + 2 ( 6 - 1 )  2TO+122<?+6(b-l) 27i  4 : 3 ( -V  2m+123£+6(b-l)p2n  2  n  yf2 3t  1  ,3  22m+6^2e+6(b-l)pn  n  e  b  a  0,4  a 6  79  23 p  2  p  n  b  p  22rn+632i?+6(t)- l) n p  7. £/W<? exzsf integers m > 2, £ > 2 — b and n > 1 t^af 2 3 square and E is Q-isomorphic to one of the elliptic curves:  — p  m  04  A  2m3t?+2(b-l)  _22m+632£+6(b-l)p7i  0-2 2 •3~  Jl J2  b  y/2 3  1  m  -4 •3~ h  -p  e  y/2 3  1  m  e  -2-3 ~ y/2 3  Jl' J2'  h  1  m  4 •3~ h  l  -  p  -  p  y/2 3  l  m  n  n  -p  e  -4 • 3 ( " 2  b  V  27713/2+2(6-1)  n  n  277i+123 ?+6(b-l)p2n t  _22m+632/?+6(b-l)p7i  n  _ . 2(b-l) n 4  3  p  2777.+ 123(?+6(b-l)p27i  8. there exist integers m > 3, £ > 2 — b and n > 1 such that 2 p square and E is Q-isomorphic to one of the elliptic curves: rn  KI  4  2 •3 - ^ 2 p 1  m  277132(6—1)^77  + 3  n  e  K2  - 4 • 3 ~ y/2 p  + 3  KI'  - 2 - 3 - y/2 p  + 3  K2'  4 • 3 - y/2 p  b  h  1  m  l  m  n  4 . 3^+2(6-1)  e  n  e  2 b  1  m  m 2(b-l) n p  3  4 .3^+2(6-1)  + 3  n  e  + 3 is a  n  A  a  0.2 b  is a  22m+63>?+6(6-l)p2n 2m+1232<?+6(6-l)p77 22m+63<?+6(b-l)p2n 2777+1232/^+6(6-1)^77  9. there exist integers £ > 2 - b and'n > 1 such that 4p — 3 is a square and E n  is Q-isomorphic to one of the elliptic curves:  4  2  LI L2  2 • 3 - V P - 3* 4 . 2(6-l) n - 4 • 3 - V P - 3* _ . t+2(b-l) -2 • 3 - V P " - 3 . 2(6-l) r7 6  4  n  L2'  p  3  b  4  b  4  4  4  3  £  4 •3 -V P" - ^ b  _2l03«"+6(6-l) 27i p  n  4  LI'  A  a  a  3  p  3  _ . /:+2(6-i) 4  3  2  1432/?+6(6-l) n p  _2i03f+6(6-i) 2n p  2  1432^+6(6-l) n  10. there exist integers m > 2, £ > 2 — b and n > 1 such that 2 square and E is Q-isomorphic to one of the elliptic curves:  p  m  + 3p e  n  is a  Chapter  3.  Elliptic Curves with 2-torsion and conductor  Ml  2-3 6  1  -4 • 3  Ml'  -2 • 3  b _ 1  M2'  b  A  2  22m+63*!+6(6-l)pn  +3 V  2  + 3V  m  ^2  +  m  1  s  a.4  m  - V  4-3 -  l3  2mg2(6-l)  v  b  a  a-2 /  M2  80  23 p  2  / v  1  4 . 3«+2(b-l) n  2m+1232^+6(b-l)p2n  2m.32(fc-l)  22771+63^+6(6-1)^71  p  3 V  4 . 3^+2(6-l) n  +3 V  m  p  2m+1232^+6(b-l)p2n  11. there exist integers m > 2, £ > 2 — b and n > 1 such that 2 square and E is Q-isomorphic to one of the elliptic curves:  —3p  m  2 • 3 ~ yj2  N2  - 4 • 3 ~ y/2  NI'  -2 • 3 ~^2  N2'  h  1  h  m  h  4 •3  - 3V  m  1  m  b _ 1  yj2  -  3p  -  3 V  -  m  l  e  04  A _227n+63*?+6(6-l)p7i  _ 4 . 3«+2(6-l) n  n  p  n  4  277i+1232£+6(b-l)p27i  . 3«+2(6-l) 7i p  2m+1232("+6(6-l)p2n  12. there exist integers m > 2, £ > 2 — b and n > 1 such that 3 p square and E is Q-isomorphic to one of the elliptic curves: e  01  2 •3 - V 3 V - 2  m  02  - 4 • 3 " y/3 p  -  2  or  -2 • 3 ~ yj3 p  -  2  02'  4 • 3 - ^/3 p  6  h  h  1  e  l  t  x  i  n  n  -  n  A  —2  7 7 l  3 ( 2  b _ 1  m  _ 2771+1232^+6(6-1)^271  p  _2  T O  +l 32^+6(6-l)p2n 2  13. there exist integers rn > 2, £ > 2 — b and n > 1 such that 3 - 2 p square and E is Q-isomorphic to one of the elliptic curves: m  PI  2 •3  P2  -4 •3  PI'  -2-3 ~ y/3  P2'  h  1  y/3  -  e  y/3  b _ 1  4• S ^ V  3  _2  n  -  2 p m  n  -  2 p  n  * -  2p  e  i  2 p m  m  m  n  4  r r a  3 C'-l)pn 2  . 3^+2(6-1)  -2 3 m  2 ( b - 1 )  is a  n  A  a.4  a-2 b _ 1  is a  22777+63^+6(6-1) n  )  4 . 3^+2(6-l) n  2  m  2277i,+63i'+6(6-l)p7i p  m  - 2  0,4  4 . 3«+2(6-l) n  m  n  _277i32(6-l)  a-2 b  is a  _2277i+63i°+6(6-l)p7i  277132(6-1) _  3p e  n  277132(6-1)  a-2  NI  e  p  4 . 3^+2(6-1)  22771+63^+6(6-1)^271 _277i+1232(!+6(b-l)pn  n  22m+63f+6(6-l) 2n p  _ 2771+1232^+6(6-l)p71.  14. there exist integers rn > 2, £ > 2 — b and n > 1 such that p square and E is Q-isomorphic to one of the elliptic curves:  n  —23 m  e  is a  Chapter 3.  Elliptic Curves with 2-torsion and conductor 2 3@p a  a  a2 Qi Q2 Ql' Q2'  n  2 m 3 f  —2 3^+ ( - )  22m+622^+6(b-l) n  m  6  -2 • 3  vV -  1  1  p  _2m+123<?+6(fc-l)p2n 22m+632i?+6(6-l)pn  4 . 32(b-l)pi  _27n+123«?+6(6-l)p2n  p  ^  4-3 - V-2 3* 6  fc  4 . 32(ft-i) n  m  2 m 3  2  _2TO3«?+2(b-l)  -4-3 -VP" -2 3* b _ 1  A  4  2- 3 - V p 6  81  s  m  v  In f/te case that b = 2, i.e. N = 2 3 p, we furthermore could have one of the 6  2  following conditions satisfied: 15. there exist integers n > 1 and s £ {0,1} such that ' Q-isomorphic to one of the elliptic curves: p  is a square and E is  +1 3  a  a  2  3  Rl  2'3  R2  -4-3  Rl'  -2-3^+V^  S + 1  / " p  X  s + 1 v  +1 3  ^P  4-3 V^l s+  R2'  A  4  V  2 s +  4• 3  3  2 s +  4•3  2 3 6  2 s + 1  V  n  2  a SI  2-3 + />+  1  v  S2  S2'  -3 4 • 3  2-3  SI' 4  .  ^  s + 1  3  p  2 3  3 + 6  2  3  3 + c  1  fs a square and E is  1 2  p  3  V  ^  ^  2 s + 1  2 s + 1  2 3 6  p  n  2  3 + G s  2 n  V  l  Sp  6  p  n  17. there exist integers n > 1 and s G {0,1} swdz f/taf Q-isojnorphic to one of the elliptic curves:  p™  3  2 3  3  2 s + 1  5  _ 12 3+6. 2n  _ 2»+l 4• 3  ' p  A  0.4  2  1  2  3  6  2 5 + 1  p "  12 3+6.s 7i  16. there exist integers n > 1 and s € {0,1} swc« that Q-isomorphic to one of the elliptic curves:  s  3 + 6 s  -2  4  1 2  p  3  1 3  3 + 6 s  p  3 + 6 s  n  p  2 n  fs a square and E is  Chapter 3. Elliptic Curves with 2-torsion and conductor 2° 3 p c  2-3 ['K=± s+1  4 •3  x  4• 3  T2  S + 1  y ^ r i  4  .  2s+1)  3  4• 3  TI' T2'  _  4.3»+i /4 ^-j. v  p  n  2s+l)  V  2 s + 1  —2 2  32S+1)  -4.  £  82  &  A  a.4  0-2  TI  l}  1 4  10  3  3  3+6s  3 + 6 s  p  p  n  _ io 3+6.yn 2  3  2 3 p 14  3+6s  n  18. there exist integers m > 2, n > 1 and s e {0,1} such that and E is Q-isomorphic to one of the elliptic curves: a UI  2  . ,+i y !l±2^ 3  <  E  2  U2 UI'  m 2s+l  2 •3  S + 1  xj^  U2'  2 2 m + 6 ^ 3 + 6 S p n  1  2  s  V  +  2m+12^3+6.Sp2Ti  19. there exist integers m > 2, n > 1 and s 6 {0,1} swc/z that and E is Q-isomorphic to one of the elliptic curves:  VI' V2'  -4 •3 ' 2  s +  S +  5 +  V  n  22"'+ 6 ^ 3 + 6 5 ^ ,7 1  2^225+1 - 4 •3 2  9+1  p  n  2m+12g3+6.Sp2n  WI'  2 32s+i  22m+623+6Spn  W2'  4• 3 -2-  4 •3*  + 1  2  s  +  V  2 32^+i  3 5 + 1  m  sJv^F-  2 3  A  m  W2  p  0,4  02  4• 3 ' 2  zs a square  2 ^ + 1 2 ^ 3 + 6 5 ^ 2 7 1  20. there exist integers m > 2, n > 1 and s e {0,1} such that and E is Q-isomorphic to one of the elliptic curves:  WI  P 3  22m+6^ |3 + 6 s pn  2nJ'32.S+l  -2-3 V^ 4"3 V^  2  A  04  02  V2  2  2m+12^3-f 6 s p 2 n  +V  2 s  4• 3  VI  * " is a square  2 2 m + 6 ^ 3 + 6 s ^ n  3  4•3  p  A  04  2  2n  5 +  V  _2"i+12^3+6Sp2n 22m+6^3+6.Spn _2 + 3 + - p m  12  3  6  5  2n  z's a square  Chapter 3. Elliptic Curves with 2-torsion and conductor 2 3 a  83  p  f3  s  Theorem 3.19 The elliptic curves E defined over Q, of conductor 2 3 p, and having 7  b  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 • 3 + p is a square and E is Q-isomorphic to one of the elliptic curves: e  2  Al  2-3 -V -3 b  2  n  Al' A2' BI B2 BI' B2'  3 -V  -2 •  -3 +p  1  b  2 . 3^+2(6-1)  +p  f  A2 . - 4 -S " A/2 6  e  2  4-3 ( "V  n  2  e  n  2  b  3  2  2  b  2  b  £  n  2  3  2  2(6-l) n 3  2  p  13 <;+6(6-l)p2n 3  7 M-6(6-l)p2n 3  14 2«+6(6-l)pn 3  2  p  . «+2(b-l)  8  n  3  2  pn  . <+2(k-i) 3  n  2  b  2(b-l)  8  n  f  2(  13 »2+6(b-l)p2n 28 2/J+6(b-l) n 3  3 - -Vp  3  n  b  2  3  4 •  n  b  . <+2(6-l)  2  -3 +P  4-3 -V - *+P 2-3 -V -3*+P -4-3 -V -3 +P - 2 - 3 - V - 3 +P 4-3 -V -3*+P b  A  a.4  a  11  7 «+6(6-l) 2n 3  p  14 2^+G(b-l) n 3  p  2. there exist integers £ > 2 — b and n > 1 such that 2 • 3 — p is a square and E is Q-isomorphic to one of the elliptic curves: n  0,4  A  2 . 3<+2(b-i)  _ 8 2£+6(b-l) n  a  2  Cl C2 Cl' -C2' DI  2 • 3 ~ y/2 - 3 - p h  l  e  n  -4-3 -V -3^-p -2-3 ~V -3^-p 4-3 -V -3 -p b  2  b  2  b  2  £  f  2  DI'  -2-3 -V -3^-p  D2'  b  l  i  n  -P  p  2 . 3<+2(6-l) 2  -p  2  4-3 ~ ^2-3 b  3  3  -4-3 ~' y/2-3  e  .2(6-l) n  4  - 4_ •2 ( 3b - <l ) "V n  n  D2  1  n  n  2-3^V -3 -77 b  n  _  8  n  n  p  . <+2(b-l) 3  3  p  . /+2(6-l) 3  3  p  13 »«+6(b-l) 2n 3  2  p  _ 8 2£+6(6-l) n 2  b  _ 2(b-l) n 8  2  2  3  p  13 i+6(b-l) 2n 3  2  p  7 ^+6(b-l) 2n 3  p  _ 14 2(!+6(b-l) n 2  2  3  p  7 /?+6(b-l) 2n 3  p  _ 14 2M-6(b-l) n 2  3  p  3. there exist integers £> 2 - b and n > 1 such that 2p + 3 is a square and E is Q-isomorphic to one of the elliptic curves: n  e  Chapter 3.  Elliptic Curves with 2-torsion and conductor 2 ° 3 / V 5  a  a  2  El E2  2-3^  1  2p  / v  + 3«  n  A  4  V  2 •3 ( " . 3^+2(6-1) 2  84  b  28 «+6(b-l)p2n 3  4  - 4 - 3 - V P + 3* 2l3 2/i+6(b-l)pTi 2 b - 2 - 3 - V P " + 3 2 • 3 ( -V 83«+6(b-l) 2n . 3^+2(6-1) 4 - 3 - V P " + 3 .. 2l332f+6(6-l)pn 3^+2(6-1) 7 2»?+6(6-l)pn 2 - 3 - V P + 3* 8 • 3 ( - >p -4-3 -V P + 2l43^+6(b-l)p2n -2-3 ~V P + <M-2(b-l) 732«+6(b-l)p7i b  2  n  3  El' E2' Fl F2 Fl' F2'  b  2  f  2  b  2  b  2  2  n  2  b  2  b  n  2  b  a  3<!  2  3  J  + 3  n  3  n  3<;  n  4- 3 - y/2p b  p  4  £  8•  e  3 ^ -^p 2  h  n  2l43f2+6(b-l) 2ra p  4. there exist integers t > 2 — b and n > 1 such that 2p - 3 is a square and E is Q-isomorphic to one of the elliptic curves: n  (12 GI  a  2-3 - vV -3 b  1  1  < ! 2  G2  - 4 • 3 " s/2p  - 3  GI' G2'  -2-3 - ^2p  - 3  b  1  n  e  . 2(b-l) 77 3  HI H2 HI' H2'  1  n  e  _ . <?+2(b-i)  4 • 3 " sj2p 6  1  n  b  l  n  - 3-  _3«+2(6-l)  l  2  b  n  2  b  n  2  8  £  n  3  2  p  p  2l332f+6(b-l) n 3  p  p  _2l4 3«+6(b-l) 2n 27 2e+6(b-l) n p  3  . 2(b-l) n 3  1332f+6(6-l) n _ 83t"+6(6-l) 2n  27 2^+6(6-l) n  . 2(6-l) n 3  8  3  p  _ <?+2(6-l)  f  2  2  p  _4 . 3^+2(6-1)  -4-3 -V P -3 -2 -3 -V P - 3 4- 3 - V P - 3* b  3  - 3  e  2 • 3 ~ sj2p  3  . 2(6-l) n  2  _ 8 *+6(&-i) 2 n  p  4  b  A  4  p  p  _ l 3f+6(b-l) 2n 4  p  2  5. there exist integers I > 2 — b and n > 1 such that 2 + 3 p is Q-isomorphic to one of the elliptic curves:  n  (12 11 12 11' 12' JI J2 JI' J ' 2  2-3 -V -4-3 -V -2-3 -V 4-3 -V 2-3 -V -4-3 -V -2-3 -V b  2  b  b  b  2  2  b  2  b  2  b  2  4-3 6  2  1  /  v  +3V +3 V +3 V +3V +3V +3 V +3 V  2 + 3y  a . 32(6-1)  A  4  l  i  2  4 . £+2(6-l) n 2 • S ^" ) 3  is a square and E  2  283^+6(6-1) n 1332t"+6(6-l) 277 p  p  2  1  4 . 3^+2(6-l) n  • 83^+6(b-l) n 2  p  2l332£+6(6-l) 2n p  p  3«?+2(b-l)n p  8  2732£+6(b-l) 2n p  1437?+6(b-l) n  . 2(b-l) 3  3^+2(b-l)n  2 2  p  7 2£+6(6-l) 2n 3  p  p  8 • S ^- ) 2  1  143*?+6(b-l) n  2  p  Chapter 3. Elliptic Curves with 2-torsion and conductor 2 3^p a  85  5  6. there exist integers £ > 2 — b and n > 1 such that 3 p — 2 is a square and E is Q-isomorphic to one of the elliptic curves: n  a KI  2 • 3  K2  - 4 - 3  KI'  -2 • 3  K2'  4 - 3  a  2  " V 3 V  6  - V 3 V  b  - V 3 V  b  -2 •  - 2  283^+6(6-1)^  3^ 2  4 . 3«+2(6-l) n  " V 3 V - 2  b  A  4  p  _  - 2  2  2  . 32(6-1)  2  4 . ^+2(b-l) n  -2  _ 1332^+6(b-l)p2n  3  83^+6(b-l) n p  —2 3 ^+ ( - .)p 1 3  p  2  6  b  1  2 r l  3^+2(b-l) n  _ 2732^+6(6-l) 2n  L2  2-3 -V3V-2 -4 • 3 - /3 p -2  _ . 2(6-D  2143*3+6(6-l)pn  LI'  -2-3 -V V-2  3«+2(b-l) n  _273 ^+6(6-l) 2n  LI  L2'  f c  b  1  e  p  n  8  y  b  4 • 3  3  " V 3 V  b  p  3  2  p  p  _ . 2(b-l)  - 2  8  2  3  14 f:+6(6-l)pn 3  7. there exist integers £ > 2 — b and n > 1 swc/z tTzaf 3^ - 2p is a square and E is Q-isomorphic to one of the elliptic curves: n  a Ml  2 •3  ~ V 3  b  - 2p  £  V3  €  M2  -4 • 3 -  Ml'  -2 • 3 " V 3 ^  b  b  M2'  4 - 3  NI  2 • 3  b  4 . 3«+2(b-l) -2 • S ^ - ) ? "  P  N2  -4 • 3 " \/3« -  NI'  -2 •  N2'  1  3~ V 3 * h  3  n  2  2p  n  - 2p"  -8 • 3 < " 2  b  1  2 3  p  283^+6(6-1)^272  _2l332«+6(6-l) n p  -2 3 7  V  3*3+2(6-1) _8-  83<'+6(6-l) 2n  2  3^+2(6-1)  n  2  _ 1332*3+6(b-l)pn  1  4 . 3^+2(6-1)  n  4-3 ~V ^-2p" b  1  - 2p  " V 3 ^ - 2p 6  -2 • S ' - ) ^ " 6  _ 2 «  - V 3 ^ - 2 p  b  A  4  2  n  (6-l)p"  2 £ + 6 ( b _ 1 )  p  n  2l43f+6(b-l)p2n -2 3 7  2 £ + 6 ( b  ~  1 )  p  n  2l43f+6(6-l) 2n p  8. there exist integers £ > 2 — b and n > 1 such that p — 2 • 3 is a square and E is Q-isomorphic to one of the elliptic curves: n  e  Chapter 3.  Elliptic Curves with 2-torsion and conductor 2 3 p a  a Ol 02  2-3 -VP" -2-3* -A-3 ~ y/p - 2- 3  01' 02'  -2 • 3 ~ V P "  PI P2 PI' P2'  6  b  1  n  2  A 2  2  p  2  3  _2l3 f+6(b-l) 2n 3  2  2  2  3  p  2  3  _ 7 «+6(b-l) 2n 2  1  2  f  N  3  _ 13 /:+6(6-l)p2n  p  £  6  p  8 2f+6(b-l) n  B  3  1  b  3  e+2  2  1  8 2f+6(6-l)pn  3  4 . 32(6-l) n  - • 3^ - 2 • 3 V>-V 4 • 3 < -V 4 • 3 " VP' - ' * 2(6-l) n 2-3^VP"-2-3' -4 • 3 - V P " - 2 • 3 - 8 • S^ ^- ) -2 • 3 - V P " - • 3 2(b-l) n 4-3 -Vp -2-3 _ . /+2(b-l) ft  86  s  4  _ . £+2(b-l) e  h  l3  3  p  14 2/J+6(6-l)pTi 3  _ 7 ^+6(b-l)p2n  p  2  <  8 3  2  3  14 2«+6(b-l) n 3  p  In the case that 6 = 2, i.e. N = 2 3 p, we furthermore could have one of the 7  2  following conditions satisfied: 9. there exist integers n > 1 and s € {0,1} such that Q-isomorphic to one of the elliptic curves: a  0.2 Ql  2-3  s + 1  - / " 2 p  v  Q2  4-3  Ql'  2-3 / +  Q2'  4-3 yJ +  Rl  2-3 yJ  s + 1  / =  2pn  ± i 3  s+1  2pn  s+1  l  Rl'  2-3 yJ s+1  / 2 p r v  ;  + 1  2pn +1 3  4-3 v ; s+i  /2p  +i  +  is a  1  square and E is  A 2  4 • 3  s  +  V 2 3 p  2 s + 1  13  3+6s  n  2 • 3 *+V  2  8 3+6s 2n  -,- 2*+l  2  13 3+6. n  3  3  3  s + i  3  4  2 • 3  4  2pn +1  4-3  p  2  1  ]  R2  R2'  3  2  A  s+1  + 1  2  5 +  l  2  2  5 + 1  Sp  3 + 6 s  p  n  I4 3+6, 2n 3  p  23 p  2 s + 1  8•3 -  3  7  8•3 ' V 3  p  2 3  2 s + 1  2  3  7  p  n  2  3+6s  n  14 3+6 2n 3  Sp  10. 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: P  L  Chapter 3.  Elliptic Curves with 2-torsion and conductor  a  a-2  SI  2-3  s+1  y" p  A  +2 3  2-3 yz^3±2 s +  n  23 p  2l3 34-6s 2n 3  2 s + 1  2  . ,+iy »+2 3  P  2s+1  23 p™  2l3 3+65 2n  8•3  n  2-3-  5+1  / " p  x  +2 3  4.3'+V^  3  Ul'  23 p 14  2 s + 1  VI V2  V2'  2 s +  3  n  4-3-^V ^ 2  2 s + 1  3 p 2s+1  -8 • 3  -2  n  2 5 + 1  p  1 3  n  3  V  3 + 6  23 'p 8  2 s + 1  V  -8 • 3  VI'  2  28 3+6s 7i  4•3 p 3  is a square and E is  ^-g- -  2 s + 1  2s+1  4•3 p s+  n  A  2s+1  2-3 V^  3+6s  4  -2 • 3  U2'  p  7 3+6s 2  -2 • 3  3  2n  2l4 3+6* n  2 s + 1  8•3  a  -4. *+y^  3+6 5  3p™ 23 p™  a2 Ul  p  7  11. there exist integers n > 1 and s £ {0,1} swcn that Q-isomorphic to one of the elliptic curves:  U2  p  4•3 p 3 p 23 -p n  +2s1  T2'  n  3  T2 TI'  3+6s  8 3+6s  2s+1  S2' TI  2 5 + 1  2• 3  5  A 2s+1  SI'  l3  8  4•3 p  S2  a  4  2•3  87  23 p  3+6 5  n  _ 13 3+6. 2n 2  3  5 p  -2 3 23 -2 3 23  p" p p p  7  3+6s  14  3+6s  n  7  3+6s  2n  14  3+6s  n  Theorem 3.20 The elliptic curves E defined over Q, of conductor  2  23p, and having 8 6  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™ Q-isomorphic to one of the elliptic curves:  -1  is a square and E is  Chapter 3.  Elliptic Curves with 2-torsion and conductor  a-2  a  2  8 . 3 6 - 1 ^ 3 ^ 1  3  _2 3 + ( 9  p  3 . 3 6 - 1 ^ 3 ^ 1  -8 • 3  BI  4.36-1^/3^1  _ .32(«>-i)  6  3 f p  ;'-  8•  BI'  - 4 . 3 - v ^ i  B2'  3 . 3 6 - 1 ^ 3 ^  293«"+6(6-l)pn 71  _ 1532«+6(6-l) 2n 2  1  2  . <+2(6-l) n p  3  a  a  2  4-3-V^  C2  cr C2' DI  8• 3" 6  1  y j  3  2  ^  -4-3»-V^F 8  .  3 6 - 1 ^ 3 ^  4.3"-  1  / ' /" 3  v  +  D2'  2932/J+6(6-l)pn  3  3 -- p 2l  h  1)  n  2l5 <M-6(6-l)p2n 3  . «+2(6-l)  2932£+6(b-l) n  3  8-3 ( -V 2  6  2• 3 ( -V 2  6  p  2  153t"+6(6-l)p2n  2  93t"+6(6-l)p2n  2l532/:+6(6-l)p7i  4-3 - y '+ " p  2-3 ( - V  293f+6(6-l)p2rc.  .  -  g . 3^+2(6-1)  2l532("+6(b-1)  b  8  A  4  g . 3^+2(6-1)  D2 DI'  2  is a square and E is  V 2  . £+2(6-l)  8•  93(?+6(6-l)pn  _2l532t*+6(6-l)p2n  2. there exist integers £ > 2 — b and n > 1 such that Q-isomorphic to one of the elliptic curves:  Cl  p  -2-3 ( - ) 8  3  i  3  6 - i y 3 ' +  P  2  2 n  p  tf+^-Vp FE  )p  2l53^+6(6-l) n  2 ( B _ 1 )  2  b _ 1  p  2  )  6  _2932^+6(6-l) 2n  p  A2'  M  2l53*"+6(6-l)pn  )  2 ( B _ 1  2 • 3«+2(ft-l) "  8-3 -y  88  s  A  . «+2(b-l) n  -8 •3  Al'  B2  f3  04  Al A2  23 p  f c  1  0/ _  n  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  a.2  a  El  El' E2' FI  8  4  p  2(b_1)  /  1 v  3  8  4 .  3  " -  1 A  /  3  ^  2£+6  ( - )p b  1  rl  p  3  _2l5 2€+6(b-l) n 3  3  _2. 2(6-l) n  9 «+6(b-l) 2n  2  p  g . f+2(6-l)  F2'  p  9 £+6(b-l)p2n  2  . ^+2((>-i) 3  3  3  p  n  ^  15 £+6(b-l) 2n  2l5 «+6(6-l) 2n  - 2 • &V>-Vp  -  p  9  . 3  3  2  3  _2 3  _ . 2(6-l) n  b  n  3  .36-1^/31^1  F2 FI'  2  . *+2(b-i) 8  s  _ 9 2^+6(b-l) n  3  2  l3  A  -8 • 3  E2  a  4  . <?+2(h-i)  2  89  23 p  3  p  _2 53 ^+6(b-l) n 1  2  p  3  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 GI G2 GI'  a  2  4 . 3 6 - 1 ^ 2 ^ - 8 - 3  b  A  4  2 • 3 ' p" 3  4 . 3 6 - 1 ^ ^  3  4.3b-V^ s-s"- ^"- '  _  H I ' H2'  4.  . /+2(6-l)  _  2(fc_1)  p  . £+2(b-l)  2  3  g. 2(b-l) n  8 . 3 - V ^  3  p  p  1532f:+6(b-l) n p  2  3  8•3  36-1 ^ £ ^ 1  2  3  2  1532«+6(6-l) n  p  H I  8  p  _293«+6(b-l) 2n  p  _ . «+2(6-l)  3  2  . 2(6-l) n  2  8.36-1^22^  1  2  _ g . «+2(6-l)  - y ^  G2'  H 2  _ 93«+6(6-l) 2n  2( ,_1)  n  9 2f+6(b-l) n 3  p  _2l5 <?+6(6-l) 2n 3  p  29 2t3+6(b-l) n 3  p  _ 15 «+6(b-l) 2n 2  3  p  In the case that 6 = 2 , i.e. N = 2 3 p, we furthermore could have one of the 8  2  following conditions satisfied: 5. there exists an integer n > 1 such that to one of the elliptic curves:  is a square and E is Q-isomorphic  Chapter 3.  90  Elliptic Curves with 2-torsion and conductor 2 3 V Q  a  02  2•3  11 12 11'  E  1  3  A  4  V  2 s +  8•3  _4."+V^  2 • 3  p  2 s + 1  8•3  ^^±1  8•3  2 s + 1  Jl  4-3»+V^  2•3  2 s + 1  J2 Jl'  g  3 S + 1  y "+i P  -4-3'+V^  J2'  8•3  8 • 3  2 s  + 1  p  3  Sp  3  2 3 9  n  2  Sp  3 + G s  3  1 5  2 3 9  2 s + 1  '  9 3+6. 2n  2l5 3+6. n  V  2 s +  2•3  2  2 s + 1  12'  S + 1  /  2  2 n  ' p s  3 + 6 s  p  n  n  15 3+6.s 2n 3  p  2 3 9  n  3 + 6  p  3 + 6  - p" 5  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: a  0.2  KI  4 - 3  ?  +  2 • 3 *'+V 2  y ^  K2 KI'  2  3  p  2s+1  -4-3 V^ s +  4-3  s +  V^  2 • 3  2 s + 1  p"  -8 • 3  2 s + 1  -2 • 3  2 s + 1  L2  -8-3- +y^  8 • 3  LI'  -4-3'+V^  -2 • 3  Curves of Conductor  8 • 3  2 s + 1  p  -2 3 9  2  p  3  9  n  2 3 9  n  3 + G s  3 + G s  3 + 6 s  p  2 n  p"  p"  2l5^3-f-6.s^2n  2 s + 1  2 s + 1  1 5  2 3  5  L2'  3.3  _ 9 3+6.s 2n  -8 • 3  K2' LI  A  4  3 + 6 s  p  n  _ 15 3+6, 2n 2  3  p  23 p a  (3  2  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  Elliptic Curves with 2-torsion and conductor  3.  91  23 p a  l3  s  Theorem 3.21 The elliptic curves E defined over Q, of conductor 3 p , and having b  2  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 3 p E is Q-isomorphic to one of the elliptic curves: 6  Al  e-3''-V\/2 3V + l  A2  l  -f-2-3 -V b  2  6  0,4  A 2l2 2f:+6(b-l)p2n+6  2(b-l) 2  2l2 f+6(6-l)pn+6  3y +l  where e G {±1} is the residue of3  + 1 is a square and  n  24g/J+2(6-l)pn+2  (12 6  e  3  3  3  p  p modulo 4.  b  l  2. there exist integers £ > 2 — b and n > 0 such that 2 3 + p" is a square and E is Q-isomorphic to one of the elliptic curves: n  4  2  e • 3 -V\/2 3<? + n b  6  p  -e-2-3 - p^/2 b  A  a  a  BI B2  e  1  ( i  3 +p e  2  n  2l2 2/?+6(b-l)pn+6  4 £+20-iy  3  3  3  2l2 <?+6(b-l)p2n+6  2(b-l)pn+2  3  where e G {±1} is the residue of3 - p modulo 4. J  b  3. there exist integers £ > 2 - b and n > 0 such that 2 3 - p is a square and E is Q-isomorphic to one of the elliptic curves: G  a  a-2  Cl C2  e-3 - p^/2 3 h  1  G  -p  e  n  -e-2-3 ~^psf2 3 h  G  i  n  where e € {±1} is the residue of3  b  A  4  _2l2 2£+6(b-l) n+6  _ 2(6-l) n+2  2l2 «+6(6-I)p2n+6  3  p  3  3  p  p modulo 4.  1  6  a  0.2  DI D2  e-3 - psj2 p l  G  n  +3*  2  - £ - 2 - 3 - V \ / 2 V + 3^ b  where e £ { 1 1 } is the residue of3  b  p  3  4. there exist integers £> 2 - band n > 0 such that 2 p is Q-isomorphic to one of the elliptic curves:  b  n  4 «+2(b-l) 2  2  -p  e  + 3 is a square and E e  A  4  4 2((,-l) n+2  2l2 j°+G(6-l) Ti+6  3«+2(b-l) 2  2l2 2«?+6(fc-l) n+6  3  j9  p  p modulo 4.  l  n  3  3  p 2  p  Chapter 3. Elliptic Curves with 2-torsion and conductor  92  23 p a  (3  s  5. there exist integers £ > 2 — b and n > 0 such that 2 + 3 p is Q-isomorphic to one of the elliptic curves: 6  a  a  2  E2  -e-2-3  p\/2 + 3 V  where e £ {±1} is the residue of3  b  A  4  2l23«+6(b-l) n+6  3<?+2(t.-l) n+2  2l232£+6(b-l) 2n+6  El 6  is a square and E  n  4 2(6-l) 2  2  6 _ 1  e  3  p  p  p  p modulo 4.  l  6. there exist integers £ > 2 — b and n > 0 such that 2 — 3 p is Q-isomorphic to one of the elliptic curves: 6  a  «2  Fl F2  e-3 -V\/2 -3V -t • 2- 3 ~ p^2 - 3p b  6  h  l  l  6  l  2  A _2l23<°+6(b-l) 7 +6  4 2(6-l)p2  p  3  2  p  where e £ {±1} is the residue of3  b  is a square and E  n  4  _3«+2(6-l) n+2  n  i  3  p modulo 4.  l  7. there exist integers £ > 2 — b and n > 0 such that 3 — 2 p is Q-isomorphic to one of the elliptic curves: e  a ~2 3 ( - V+  a t • 3 ^V\/3^ - 2 p -e-2-3 - p\/3' -2V  GI G2  6  b  1  6  is a square and E  n  A  4  2  b  4  n  2  b  1  2l23C+6(b-l) 2n+6  2  p  3«+2(b-l) 2  !  _2l232f+6(b-l) n+6  p  where e G { ± 1 } is the residue of3  b  1  12 2£+6(b-l)p2n+6  p  p modulo 4.  l  8. there exist integers £>2 — b and n > 0 such that p - 2 3 is a square and E is Q-isomorphic to one of the elliptic curves: n  a  0,2 HI  e - 3 - W P - 23<»  H2  - e - 2 - 3 " p \ / p - 2 3^  32(b-l) n+2  n  b  1  6  n  2  6  where e £ {±1} is the residue of3  b  e  A  4  _ 4 «+2(b-l) 2  b  6  3  2  p  p  1232£+6(b-l) n+6 p  _2l23*?+6(b-l) 2n+6 p  p modulo 4.  l  9. there exist integers £ > 2 - b and t £ {0,1} such that E is Q-isomorphic to one of the elliptic curves:  is a square and  Chapter 3. Elliptic Curves with 2-torsion and conductor  6-3 -y+y 6  12  2 6 3  .2.3 -y+y 6  £  a  l3  s  A  (24  0-2  11  93  23 p  ;  + i  2 0 3  ;  24 ^+2(6-l) 2t+l  21232*3+6(6-1)^3+64  32(6-1) 24+1  2l23*3+6(6-l)p3+6t  3  + i  where e € { ± 1 } is the residue o / 3  b  p  l  p  modulo 4.  t + 1  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:  e  J2  £  4  -3 -y+y ° ;-' b  Jl  .  2  .  3  A  a  a-2 2  3  2  43f+2(6-l) 2(+l p  _2l2 2*3+6(6-l)p3+6t 3  _32(i>-l)p2t+l  6 - y + y 2 « ' 3 « - i  where e <E { ± 1 } is ffte residue o / 3  b  p  1  t + 1  2  123*3+6(b-l)p3+6t  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  KI K2  .2-3 -y 6  £  04  A  2  4 2(6-l)p2t+l  2l23*3+6(6-l)p3+6t  3  «+2(6-l)2t+l  21232*3+6(6-1)^3+64  2  + l N  /  2 8 + 3  '  where e & {±1} is the residue of3  y  3  b  +  1  modulo 4.  12. there exist integers £>2 — b and £ G {0,1} such that —^ is a square and E 2  is Q-isomorphic to one of the elliptic curves: (24  A  4 2(6-iyt+i  _ 123*?+6(6-l)p3+6i  a  2  LI L2  2  -2.3 -y+y b  £  2 6  _ 3*3+2(6- i y t + i  ; ' 3  where e 6 { ± 1 } is the residue o/3  3  b  p  1  t+1  2  2  1232*3+6(6-iy+6i  modulo 4.  13. there exist integers £>2 — b and t e {0,1} such that ~ 3<?  is Q-isomorphic to one of the elliptic curves:  26  is a square and E  Chapter 3. Elliptic Curves with 2-torsion and conductor 2 3 p a  M2  - 3  £  e  .  2  6  - y + y  . 3  6  - y  94  3  A  0,4  «2  Ml  l3  +  3  _ 432(b-l)p2t+l  y  3  2l23^+6(6-l)p3+6t  2  ' ;  2  8  ' -  2  3«?+2(b-l)  6  zu/jere e 6 {±1} is the residue of3  p  b  1  _2l2 32«+6(b-l)p3+6«  2t + l  modulo 4.  t+1  In the case that 6 = 2, i.e. N = 2 • 3 p , we furthermore could have one of the 2  2  following conditions satisfied: 14. there exist integers n > 0 and s £ {0,1} such that ~^ Q-isomorphic to one of the elliptic curves: 2b  NI  e • 3-  N2  t • 2•  A  a.4  0,2  Vx/^  2 3 4  2 s + 1  p  2  32s+l n+2  y+i J*&i  is a square and E is  p  p  21233+6.5^71+6 2l233+6s 2n+C p  py  where e e {±1} is the residue of3 p  modulo 4.  s+1  15. there exist integers n > 0 and s e {0,1} such that g " is a square and E is Q-isomorphic to one of the elliptic curves: 2 &  A  a  02  4  Ol 02  P  2  -e • 2 • 3 ^  P  y  4 2 +I 2 3  S  p  _ 2s+l  J ^  n+2  3  where e G {±1} is the residue of3 p  _ 12 3+6.Spn+6 2  2  3  12334-6Sp?i-f 6  modulo 4.  s+1  16. there exist integers n > 0 and s e {0,1} such that ' Q-isomorphic to one of the elliptic curves: p  4  2  -2 3 p  PI P2  A  a  a  4  -e • 2 -  3-^pJ^  where e G {±1} is the residue of3 p s+1  2s+1  is a square and E is  2 3  2  32s+l n+2 p  modulo 4.  2  12 3+6s n+6  — 2  3  1 2  p  3  3 + 6 s  p  2 n + 6  Chapter 3. Elliptic Curves with 2-torsion and conductor  23p cc  t3  95  s  Theorem 3.22 The elliptic curves E defined over Q, of conductor 2-3 p , and having b  2  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 2 3 p square and E is Q-isomorphic to one of the elliptic curves: rn  a  Al  e • 3 - p^2 3 p  A2  - e • 2 • 3 - p /2 3 p  1  m  b  e  1  m  2m-2 e+2(b-l)pji+2  +1  n  e  n  y  3  + 1  where e e {±1} is the residue of3  b  3  + 1 is a  n  A  (I4  2  b  e  2(6-iy  22m 2<°+6(6-l)p27i+6 3  2m+6 e+6(b-l) n+6 3  p  p modulo 4.  1  2. there exist integers m > 7, £ > 2 — b and n > 0 such that 2 3 square and E is Q-isomorphic to one of the elliptic curves: rn  a  0-2  BI  e • 3 - p /2 3  B2  -e-2-3 - p^2 3  b  1  m  +p  e  n  y  b  1  m  2  +p  e  n  b  is a  n  A  4  m-23«+2(h-l)p2  22m 2t?+6(6-l)p7i+6  2(b-l) 7i+2  277i+6 <?+6(b-l)p2n+6  3  where e G {±1} is the residue of3  +p  e  p  3  3  p modulo 4.  Y  3. there exist integers rn > 7, £ > 2 — b and n > 0 such that 2 "3 - p square and E is Q-isomorphic to one of the elliptic curves: r  a  Cl  e-3 ~ p^2 3  C2  -e- 2- 3 - p /2 3  1  m  b  -p  e  1  n  m  y  e  2  4  A __2 TO 2^+6(b-l)p77+6  _32(b-l)p7i+2  -p  n  where e G { ± 1 } is the residue of3  b  2  3  2m+6 <?+6(b-l) 2n+6 3  p  p modulo 4.  1  4. there exist integers m > 7, £ > 2 — b and n > 0 such that 2 p square and E is Q-isomorphic to one of the elliptic curves: m  DI D2  4  e • 3 ~ p /2 p 1  m  n  y  -e-2-3 - p^2 p b  1  m  n  +3  £  e  2277i3<?+6(b-1 )p2n+G  £+2(b-l) 2  277i+632 ?+6(b-l) Ti+6  +3  e  3  b  + 3 is a  2m-2 2(b-l) n+2 3  where e G { ± 1 } is the residue of3  n  A  a  02 b  is a  n  m-23«+2(6-l) 2  02 b  e  p  p  p modulo 4.  1  t  p  Chapter 3. Elliptic Curves with 2-torsion and conductor  23 p a  l3  96  6  5. there exist integers m > 7, £ > 2 — b and n > 0 such that 2 square and E is Q-isomorphic to one of the elliptic curves:  + 3p  rn  a  El E2  l  -e • 2 • Z ~ pyj2 h  1  3«?+2(b-l) n+2  +Z*p  m  n  p  where e G { ± 1 } is the residue of3  2m+6g2e+6(b- l)p2n+6  p modulo 4.  b  l  6. there exist integers m > 7, £ > 2 — b and n > 0 such that 2 square and E is Q-isomorphic to one of the elliptic curves:  —3p  rn  1  _22m^e+6(b-l)  pn+6  _3f3+2(b-l)pn+2  2m+6 2(?+6(b-1  )p2n+6  - 3 V  3  1  m  p  v  where e € {±1} is the residue of3  3  p modulo 4.  b  1  7. there exist integers m>7,£>2-b and n > 0 szzc/z tTzaf 3 p square and E is Q-isomorphic to one of the elliptic curves: e  a.2  GI G2  1  / 3  - e • 2 • 3 ~ py/3 p b  1  e  _ m-2^2(b-l)p2  m  2  3^+2(b-l)^n+2  - 2  n  m  zf/zere e e {±1} z's zTze residue of3  b  n  —2  m  2  277i3*?+6(6-l)pn+6  _2"i+632«+6(b-l)p2n+6  p modulo 4.  l  8. t/zere exist integers m > 7, £ > 2 — b and n > 0 sizc/z that 3 — 2 p square and E is Q-isomorphic to one of the elliptic curves: e  a  HI H2  a  4  2  _2"J-2 2(b-l)pn+2  e3 "Vv 3 - 2 p -e2 • tf-^py/tf - 2 p 6  /  £  m  n  m  3  3«+2(b-l)p2  n  where e € {±1} z's f/ze residue of 3  b  is a  A  0,4  e• 3 - pv V - 2 f c  is a  A  04 2m-2 2(6-l) 2  -t • 2 • 3 - p / 2 - 3 V b  n  2  e • 3 ~ py/2 h  e  m  a  FI F2  is a  2m-232(b-l)p2  + 3V  m  n  A  a.4  2  e • 3 - p\/2 b  e  rn  n  is a  A 22m^e+6(b-l)p2n+6 _2m+632f?+6(6-l)pn+6  p modulo 4.  1  9. ffoere exist integers m > 7, £ > 2 — b and n > 0 swc/z f/zaf p" - 2 3^ z's a square and E is Q-isomorphic to one of the elliptic curves: m  Chapter  3.  Elliptic Curves with 2-torsion and conductor  11 12  e • 3 ~ py/p l  -  n  l  l3  5  A  -  n  22m 2e+6(b-l)pn+6  _2m-2 e+2{b-l) 2  2 3 m  -e • 2 • 3 - p^/p b  ct  0.4  a-2 h  97  23p  3  e  2{b-l) n+2  2 3 m  3  p  e  3  where e G {±1} is the residue of3  _2m+6 e+6(b-l)p2n+6  p  3  p modulo 4.  b  1  10. there exist integers m>7,£>2-b  and t  {0,1} such that  G  2 T  "^+i  j  s  a  square and E is Q-isomorphic to one of the elliptic curves: a  JI  e-3 -Y yJ '  J2  e-2-3 -'p  b  +1  2m-2 l.+2(b-\)p2t+l 3  2m3 +1  h  A  0,4  2  t + 1  ^  32(6-1)  2 i n 3 e + l  where e G {±1} is the residue of 3  b  p  1  t+1  2t+l  2277132^+6(6-1)^3+6* 2771+63^+6(6-l)p3+6t  modulo 4.  11. there exist integers m > 7, £ > 2 - b and t G {0,1} such that '"^~ 2  is a  1  square and E is Q-isomorphic to one of the elliptic curves: a  KI K2  e 3  £  2  A  04  2  2771-23^+2(6-l) 2t+l  6-y+y2".3<-i  •s -y+y b  p  s"' ;3  _32(6-l)  1  where e G {±1} is the residue of3  b  p  l  i + l  2t+l  _22m32)?+6(6-l)p3+6i 2 7 7 7 + 6 3 ^ + 6 ( 6 - l)p3+6t  modulo 4. and t G {0,1} such that "+  12. there exist integers m>7,£>2-b  2T  is a  3<!  square and E is Q-isomorphic to one of the elliptic curves: a  LI L2  e  -3  f t  _ . 2• e  -y+y 7  3  2  f c  04  A  2777.-232(6-1)^24+1  2277i3t +6(6-l)p3+6t  2  -y y  3 f  3^+2(6-1)  +  2 = ± £  where e e {±1} is the residue of3  b  p  1  t+1  2t+l  s  2TO+632»?+6(6- 1)^3+6*  modulo 4.  13. there exist integers m > 7, £ > 2 — b and t e {0,1} such that ' " ~ square and E is Q-isomorphic to one of the elliptic curves: 2  3 £  is a  Chapter 3. Elliptic Curves with 2-torsion and conductor  £  M2  .3  6-y+y  2™^3£  2m  6  A  2m-2 2(b-l)p2t+l  _ 2m «?+6(b-l)p3+6t 2  3  _3*i+2(b-l)p2t+l  e-2-Z»-V y/ ; ' +1  l3  0,4  0.2  Ml  98  2°3 p  3  w/zere e G {±1} zs fTze residue ofS  b  l  p  2  3  m+6 2»?+6(b-l)p3+6t 3  modulo 4.  t + l  24. there exist integers m > 7 , £ > 2 - b and t G {0,1} swdz that ^~ " z's a 3  2T  square and E is Q-isomorphic to one of the elliptic curves: a  NI N2  e  3  b - y  a  2  y 3 £ - _ 2 ^  +  - e 2 • 3 " - V+y  - 2 "  l  _  2  3  2  (  A  4  b  _  ' p  1  3«?+2(b-l)  3£f=-  where e G {±1} is the residue o/3  b  1  p  t + 1  2  t  +  1  2t+l  22m *?+6(b- l)p3+6t 3  _2 + 3 *+ (' - )jr, + m  6  2  6  ,  1  3  6 i  modulo 4.  7M zTze case zTzaf 6 = 2, z'.e. N = 2 • 3 p , we furthermore could have one of the 2  2  following conditions satisfied: 15. there exist integers m, > 7, n > 0 and s G {0,1} such that ™ * " z's a square and E is Q-isomorphic to one of the elliptic curves: 2  a  0.2  Ol 02  2  - .  2 . 3 - + ^ ^  e  A  4  m-2 23+y2  22777,^3 + 65^71 + 6  2.s+l n+2  2^+6^3+65^271+6  3  3  where e G {±1} is the residue of3 p  p  modulo 4.  s+1  16. there exist integers m > 7, n > 0 anti s G {0,1} such that and E is Q-isomorphic to one of the elliptic curves: a  PI P2  e  . * 3  a  2  2  +  -e . 2 • 3 - +  J P v  /3^  where e G {±1} is the residue of3 p s+l  p  ^  2  p  is a square  A  4  ,„-2 2 +l 2 3  s  p  _ 2s+l n+2 3  p  modulo 4.  2*71+6^3+65^271+6  Chapter  3.  Elliptic Curves with 2-torsion and conductor  2 3p oc  l3  99  s  17. there exist integers m > 7, n > 0 and s G {0,1} such that '-— — is a square and E is Q-isomorphic to one of the elliptic curves: 3  Ql Q2  _  e • 3 - V \ / " a '" 5 +  A  04  0.2 P  2  2 m  -2 2 +l 2 s  3  p  32s+l n+2  - e - 2 - ^ p s J ^  p  where e G {±1} is the residue of3 p  modulo 4.  s+l  18. there exist integers m > 7 and s, t e {0,1} such that -^T" 2  square and E  l s a  is Q-isomorphic to one of the elliptic curves:  Rl  t  1  2  22m.^3-\-6Sp3-\-6t  2m-2^2s+lp2t+l  c-3 'p + ^/ '"7 '" s+  A  04  02 1  i  R2  e-2-3 +y s  + 1 x  /  2 m  3  P  2  2s+lp2t+l  "  3  where e G {±1} is the residue of3 p s+1  +6 ^ 3-f-6 s p 3 + 6 £  modulo 4.  t+1  19. there exist integers rn > 7 and s,t e {0,1} such that " ~ 2  is a square and E  1  3  is Q-isomorphic to one of the elliptic curves: a SI  e  S2  tf«ere e  .  3  ,  +  £-2-3 G  s +  A  04  2  y + y 2 ^  - 2m-2 2s-+-lp2t+l  —2  3  _ 2s+lp2t+l  y+IN/2",3P"  3  {±1} is the residue of3 p s+l  modulo  t+1  2  2 m  m+6  3 + 3  3  6 s  p + '3  6  3+6sp3+6i  4.  Theorem 3.23 The elliptic curves E defined over Q, of conductor 2 3 p , 2  b  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 £ > 2 — b and n > 0 such that 4 • 3 + p is a square, 3 = — 1 ( m o d 4) and E is Q-isomorphic to one of the elliptic curves: e  n  e  2  Al A2  (.•3 ~ p^A-3 l  l  -e-2-3 - py/A-3 b  1  A  04  a h  f  + p  e+2(b-i) 2  n  +p  3  n  3  p  2(b-l)pn+2  2 4 g 2 £ + 6 ( 6 - l ) n+6 2  8 M-6(b-l)p2n+6 3  Chapter 3.  Elliptic Curves with 2-torsion and conductor 2 3@p a  where e G {±1} is the residue of3  100  5  p modulo 4.  b  l  2. there exist integers £ > 2 — b and n > 0 such that 4 • 3 — p is a square, 3 = — 1 (mod 4) and E is Q-isomorphic to one of the elliptic curves: e  n  e  a  Bl B2  e • 3 -V\/ -3 - P 4  e  -t-2-3 ~ py/4-3 h  A  OL4  2  b  x  n n  where e G {±1} is the residue of3  p  _ 2(6-l) n+2  -p  l  «+2(fc-l) 2  3  3  _2 3 ^+ ( - )p + 4  2  6 fe 1  n  6  283^+6(b-l) 2n+6  p  p  p modulo 4.  b  l  3. there exist integers £ > 2 — b and n > 0 such that 4p — 3 is a square, p = -1 (mod 4), and E is Q-isomorphic to one of the elliptic curves: n  e  n  4  A  32(6-1)^+2  _ 43^+6(b-l) 2n+6  _ <?+2(6-l) 2  2832^+6(6-1)^+6  a  a-2  CI C2  e • 3 ~ p /4p b  1  - 3  n  e  x  - e • 2 • 3 - p^4p h  x  n  - 3l  where e G {±1} z's the residue of3  3  2  p  p  p modulo 4.  b  l  4. there exist integers £ > 2 — b and n > 0 such that p — 4 • 3 is a square, 3 =. 1 (mod 4), and E is Q-isomorphic to one of the elliptic curves: n  e  e  2  DI D2  £-3 - py/p -4-3 n  e  -e-2-3 - p^p b  1  _ ^+2(6-l) 2  2432^+6(6-l) n+6  32(6-iy+2  _ 83f+6(6-l)p2n+6  3  -4-3 -  n  A  0,4  a b 1  e  where e G {±1} z's the residue of3  p  p  2  p modulo 4.  b  l  5. there exist integers £ > 2 - b and t G {0,1} such that ' 4  3  z's a square,  + 1 p  3 p = — 1 (mod 4), and E is Q-isomorphic to one of the elliptic curves: e  0,4  A  3«+2(6-l)p2t+l  432«?+6(6-l)p3+6t  0.2 El  E2  (  .  3 6 - y + y 4-3^+1  e • 2 • 3-y+y ; b  43  where e G {±1} z's the residue of3  b  +i  1  p  3  2(6-iyt+i  t + 1  modulo 4.  2  283^+6(6-l) 3+6t p  Chapter 3.  Elliptic Curves with 2-torsion and conductor  101  23 p a  l3  s  6. there exist integers £ > 2 — b and t <E {0,1} such that  4  3  is a square,  1 p  3 p = — 1 (mod A), and E is Q-isomorphic to one of the elliptic curves: e  d  a-2  A  4  FI  3<"+2(b-l) 2t+l  _2432f+6(b-l) 3+6t  F2  _ 2(b-l) 2t+l  2832«?+6(6-l)p3+6t  p  p  3  where e 6 { ± 1 } is the residue of3  b  p  l  p  modulo 4.  t + 1  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: a  «2  GI G2  £  . 6-y+1^4+^1 3  3  - . 2 •S^V+V^1  3  A  4  243/?+6(b-l)p3+6i  2(b-l) 2t+l p  «+2(b-l)  2t+l  28 2«+6(b-l)p3+6t 3  e  <x>/zere e € { ± 1 } is tTze residue of3  b  1  p  modulo 4.  i + 1  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: a  a  2  A  4  HI  _ 2(b-l) 2t+l  H2  3^+2(b-l) 2t+l  3  2  p  _2832f+6(6-i) 3+6t p  p  where e e { ± 1 } is f/ze residue of3  V  b  t  +  1  43<'+6(b-l)p3+6t  modulo A.  In the case that 6 = 2, i.e. AT = 2 3 p , we furthermore could have one of the 2  2  2  following conditions satisfied: 9. there exist integers n > 0 and s 6 {0,1} such that " "" z's a square, p = 1 (mod 4), and £ is Q-isomorphic to one of the elliptic curves: 4p  1  n  3  «2  a  11 12  A  4  32s+l n+2 p  t-2-y p\l " +l  Ap  x z  -3  2  s  +  1  p  2  -2 3 4  3  +  6  s  p  2  n  +  2833+68^71+6  6  102  Chapter 3. Elliptic Curves with 2-torsion and conductor 2 3 p a  l3  5  where e € {±1} is the residue of 3 p modulo 4. s  l  10. there exists an integers n > 0 and s G {0,1} such that -^- is a square and E is Q-isomorphic to one of the elliptic curves: p  (14  A  3 +y  2433+6.5^71+6  g2.s+l n+2  28g3+6.Sp2n+6  0.2  2s  Jl e • 2•  J2  3  J +  s+1  pn  4  p  Px  where c. G {±1} is the residue of 3 p modulo 4. s  l  Theorem 3.24 The elliptic curves E defined over Q, of conductor 2 3 p and having at least one rational point of order 2, are the ones such that one of the following conditions is satisfied: 3  6  2  /  1. there exist integers m G {4, 5}, £ > 2 - 6 and n > 0 such that 2 3 p a square and E is Q-isomorphic to one of the elliptic curves: m  a  a  2  Al A2  e • 3 ~ py/2 3 p b  l  -e • 2• 3  m  6_1  i  +1  n  m  n  + 1 is  A  4  2m-23^+2(6-l)pn+2  22m.32£+6(f)-l) 2n+6  2(fc-l)p2  2^+63^+6(6- l) n+6  pv 2 3V + 1 /  e  l  3  p  p  where e G {±1} is the residue of 3 p modulo 4. b  1  2. there exist integers £ > 2 — b and n > 0 swc« tTzflf 4 • 3 + p is a square, 3 = 1 ( m o d 4), and E is Q-isomorphic to one of the elliptic curves: e  n  e  Bl B2  -e-3 ~ py/4-3 e-2-3 - py/4-3 b  A  a.4  02 b  1  e  i  e  +p +p  n  fJ+2(6-l) 2  24 3 2 ^ + 6 ( 6 - l ) n + 6  32(6-l) n+2  283<"+6(6~l) 2Ti+6  3  n  p  p  p  p  where e G {±1} is the residue of3  p modulo 4.  b  1  3. there exist integers m G {4,5}, £ > 2 - b and n > 0 such that 2 3 + p is a square and E is Q-isomorphic to one of the elliptic curves: m  a  CI C2  /  b  1  A .  0,4  2  e • 3 p v 2 3 +p - e • 2 • 3 - py/2 3 +p 6_1  m  €  m  n  e  2  n  e  m-23«+2(6-l) 2  22m32£+6(6-l)pn+6  32(b-l) n+2  2m+63<2+6(6-l) 2n+6  p  p  p  n  Chapter 3.  Elliptic Curves with 2-torsion and conductor  where e G {±1} is the residue of3  103  23 p a  f3  s  p modulo 4.  b  1  4. there exist integers £ > 2 — b and n > 0 such that 4 • 3 — p is a square, 3 = 1 (mod 4), and E is Q-isomorphic to one of the elliptic curves: e  n  e  G2  DI  -e  D2  A  (I4  • 3 - ps/4 h  •3  1  -p  e  e^^^ps/A^  1  -p  1  3^+2(6-V _ 2(b-l) n+2  n  n  where c G { ± 1 } is the residue of3  b  3  -2 3 ^ ^ ^p 4  2  + 6  _ 1  83t"+6(c>-l)p2n+6  p  2  p modulo 4.  1  5. there exist integers m 6 {4, 5}, £ > 2 - b and n > 0 such that 2 3 square and E is Q-isomorphic to one of the elliptic curves: m  El E2  4  e-3 -ipy/2 3 m  -p  e  -c-2-3 - pv 2 3^-p b  1  /  277i-23<M-2(6-l)p2  n  m  _ 2(fc-l)  n  3  where e G {±1} is the residue of3  b  e  - p" is a  A  a  «2 h  n + 6  —2 3 + ( 2m  n+2  2f  6  b_  p modulo 4.  Fl F2  e • 3 - pyj2™p  -e • 2 • 3  b -  +  2m-2 2(b~l)pn+2  3  e  3  where e G {±1} is the residue of3  b  + 3 is a e  22m.3<'+6(6-l)p27i+6  3  V\/2 p" + 3 m  n  A  a.4  0.2 e  G  1  m  n  n  2m+63<?+6(b-l) p2n+6  6. there exist integers m G {4,5}, £ > 2 - b and n > 0 such that 2 p square and E is Q-isomorphic to one of the elliptic curves:  b Y  )p +  1  C+2(6-l) 2  2m+632<!+6(b-l)p?i+6  p  p modulo 4.  1  7. there exist integers £ > 2 — b and n > 0 such that 4p — 3 is a square, p = 1 (mod 4), and E is Q-isomorphic to one of the elliptic curves: n  e  n  a.4  A  32(b-l) n+2  _243«"+6(b-l)p2n+6  0.2  GI G2  -e  • 3 ~ py/4p b  1  -  n  £ • 2 • 3 ~ psj4p b  1  n  3  £  p  _ **+2(b-l) 2  - 3*  where e G {±1} is the residue of3  b  3  p  2  8 2£+6(6-l) n+6 3  p  p modulo 4.  l  8. there exist integers £ > 2 - b and n > 0 such that 4 + 3 p is Q-isomorphic to one of the elliptic curves: e  n  is a square and E  Chapter 3.  Elliptic Curves with 2-torsion and conductor  HI  -e • 3 ~ p^/4 + 3 p  H2  e-2-3 -^p*J'4  l  e  i  3  where e € {±1} is the residue of3  s  A 24 £+6(f.-l)pn+6  £+2(b-l)pn+2  2832«?+6(b-l)p2n+6  3  n  l3  0,4 n  + 3p  b  a  2(J,-l) 2  «2 b  104  23 p  p  3  p modulo 4.  b  1  9. there exist integers m, e {4,5}, £ > 2 — b and n > 0 such that 2 square and E is Q-isomorphic to one of the elliptic curves: a  n  e • 3 - pv  12  -e-2-3 - pV'  ft  1  b  /2m  1  + V 3  2 m  2  + 3V  m-2 2(b-l)p2  22m3<»+6(b-l)pn+6  «+2(b-l)pn+2  2m+632«+6(b- 1 )p2n+6  3  where e G {±1} is the residue of 3  b  3  p modulo 4.  1  10. there exist integers m e {4, 5}, £ > 2 — b and n > 0 such that 2 square and E is Q-isomorphic to one of the elliptic curves:  JI J2  e • 3 - p /2  -  m  v  3p e  n  2  -e-2-3 - p\/2 -3V b  1  m  A _22m3<f+6(b-l)pTi+6  2 3  _ £+2(b-l)pn+2  1  3  where e £ {±1} is the residue of3  b  n  04  2  i  — 3^p fs a  m  m- 2(b-l)p2  a h  e  A  (Z4  2  + 3 p" is a  m  2  m+632 ?+6(b-l)p2n+6 l  p modulo 4.  1  11. there exist integers m e {4, 5}, £ > 2 - b and n > 0 such that 3 p square and E is Q-isomorphic to one of the elliptic curves: e  KI K2  e • 3 - p^3 p 1  e  - 2  n  m  -e-2-3 - pv V-2 f t  1  / 3  m  where e e {±1} is the residue of3  b  m  04  A 227713^+6(6-l)p?i+6  3£+2(b-l)pn+2  _ 7n+632i"+6(b-l)p2n+6 2  p modulo 4.  1  12. there exist integers m e {4,5}, £ > 2 - 6 and n>0 such that 3 - 2 p square and E is Q-isomorphic to one of the elliptic curves: e  LI  e • 3 ~ p^3  L2  -e • 2 • 3 ~ ps/3  1  b  -  e  l  i  2p m  n  - 2 p" m  m  n  is a  04  A  _2«i--232(b-l)p7i+2  22m3»?+6(b-l)p27i+6  £+2(6-l)p2  _2 + 3 ^+ ( - )p™+6  «2 b  is a  _2"i-232(b-l)p2  a-2 b  - 2  n  3  TO  6  2  6  t>  ;1  Chapter 3.  Elliptic Curves with 2-torsion and conductor  where e e { ± 1 } is the residue of3  105  23 p a  !3  6  p modulo 4.  b  1  23. there exist integers £ > 2 — b and n > 1 swcn that p — 4 - 3^ is a square, 3 = — 1 (mod 4), and 2? is Q-isomorphic to one of the elliptic curves: n  e  a  «2  Ml M2  - e • 3 - ps/p t-2- 3 - p /p b  1  b  1  e  n  3  1  y  A  4  „ ^+2(fe-l) 2  -4-3 -4-3  n  3  where e £ { ± 1 } is the residue of3  p  2  2(6-l) n+2  432^+6(b-l)pn+6  _ 83^+6(6-l) 2n+6 2  p  p  p modulo 4.  b  1  14. there exist integers rn e {4,5}, I > 2 — b and n > 0 such that p square and E is Q-isomorphic to one of the elliptic curves:  n  NI N2  t • 3 ~ /p t-2- 3 - p^p l  Py  b  l  A  -  n  n  -  _ m-23^+2(6-l)p2  23 23 m  e  m  3  where e G { ± 1 } is the residue of3  e  is a  22m£2e+6(b-l)pn+6  2  e  m  A  a  «2 b  —2 3  2(6-l)p?i+2  _2m+6^e+6{b-l)p2n+6  p modulo 4.  b  1  15. there exist integers £ > 2 — b and t e {0,1} such that ' ^ 4  3  + 1  is a square,  3 p = 1 (mod 4) and E is Q-isomorphic to one of the elliptic curves: e  a-2 Ol  a  f+2(b-l)p2t+l  3  -3"-y V ' p +  £  02  4  3  2  4 32f+6(b-l)p3+6t  + 1  -2.3"-y+ /^±i  3  1  £  A  4  2(b-l)p2t+l  2  83*!+6(b-l)p3+64  v  where e £ { ± 1 } is fne residue of3  b  p  1  t + 1  modulo 4.  26. fn<?r<? exist* integers rn £ {4,5}, £ > 2 — b and t € {0,1} swcn f/W " 3 2,  <+1  square and E is Q-isomorphic to one of the elliptic curves:  a-2 PI P2  -3 -y y 6  £  +  a 2 m 3  ;  + i  2  A  4  m-23<?+2(6-l)  - .2.3"-y+y^fti  3  2t+l  2(6-l) 2t+l p  e  where e G {±1} is the residue of3  b  p  1  t+1  modulo 4.  2  2  2m32f+6(b-l) 3+6t p  ?n+6 t!+6(b-l) 3+6i 3  p  is a  Chapter 3.  106  Elliptic Curves with 2-torsion and conductor 2 3@p a  s  17. there exist integers £ > 2 - b and t € {0,1} such that  4  3  1 p  is a square,  3 p = 1 (mod 4) and E is Q-isomorphic to one of the elliptic curves: e  0,4  A  Ql  3^+2(6-l)p2t+l  _2432<»+6(b-l)p3+6t  Q2  _32(6-l)  where e £ {±1} is the residue o / 3  6  p  l  t + 1  2i+l  283^+6(6-1)3+6<  modulo 4.  18. there exist integers m <E {4, 5}, £ > 2 - b and t e {0,1} swcn that " ' J ~ 2  3  1  is A  square and E is Q-isomorphic to one of the elliptic curves:  a  a  2  Rl  c.^-y+y "' ;-  R2  e.2-3 -y  2  b  3  1  2  )  _22m32t!+6(b-l)p3+6t  _ 2(6-l)p2t+l  2m+63f?+6(b-l)p3+6t  p  / '" '-  + 1  A  4  2m-23«+2(b-l) 2t+l  3  1  3  where e € {±1} is the residue of3  b  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:  a  a  2  A  4  SI  - -3 -y y^  32(1—1) 21+1  S2  e.2.3"-y+y^  3«+2(b-l)p2t+l  +  b  243*!+6(b-l)p3+6t  e  where e 6 { ± 1 } is the residue o / 3  b  1  p  t + 1  2  832«+6(b-l)p3+6t  modulo 4.  20. there exist integers m 6 {4,5}, £ > 2 — b and t€ {0,1} SUCH f/wf ^~2  square and E is Q-isomorphic to one of the elliptic curves:  a  a  2  TI T2  -3 -y+y ";+ 2  b  £  2  A  4  2m-232(b-l) 2t+l p  -2-3 -y+y 7 ' b  e  3C  3«+2(b-l) 2t+l  3  where e € { ± 1 } is t«e residue of3  p  b  1  p  t + 1  modulo 4.  22Tn3<?+6(b- 1 )p3+6t 2771+632^+6(6-1)^3+6*  is a  Chapter 3. Elliptic Curves with 2-torsion and conductor 2 3@p a  107  6  21. there exist integers m G {4,5}, £ > 2 - b and t G {0,1} such that square and E is Q-isomorphic to one of the elliptic curves:  Ul U2  2 m  3 p  04  A  2rn-232(b-l)p2t+l  _22m3»f+6(6-l)p3+6t  _3«+2(b-l) 2t+l  2m+6 2£+6(6-l)p3+6t  -e • 2 • 3 « - y + V  i D  where t G { ± 1 } fs the residue o/3 - y fe  +  is a  3  modulo 4.  1  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: A  04  - y  y ^  VI  - - 3  V2  e • 2 • 3"-y+y 2 ^  b  c  +  _32(b-l) 2(+l  2432«+6(b-l) 3+6t  3^+2(6-1) 2t+l  _ 8 (!+0{b-l)p3+6t  p  where e G {±1} fs the residue of3  p  b  1  2  3  modulo 4.  t+1  23. inert? exfsf integers m G {4,5}, £ > 2 — b and t G {0,1} swcra that ~ "* fs fl square and E is Q-isomorphic to one of the elliptic curves: 3t  a  A  04  2  WI  ^ - y - V ^  W2  e2.3"-y+  1  / 3 V  -2  m _ 2  3  2 ( 6 _ 1 )  p  2  22m 31^+6(6- 1 )p3+6t  2 t + 1  3<?+2(6-l)p2t + l  '- '"  2  _2'»+6 2<?+6(b-l)p3+6t 3  p  where e G { ± 1 } fs fne residue o/3 y b  +  modulo 4.  1  Jn f«e case f/wf 6 = 2, f.e. = 2 3 p , we furthermore could have one of the following conditions satisfied: 3  2  2  24. there exist integers m G {4,5}, n > 0, and s G {0,1} swc/z that square and E is Q-isomorphic to one of the elliptic curves: 02 XI X2  e  . *+  e-2-3 sJ -^ s+l  2  V  A  04 2  3  2  m-2 2, l 2 3  +  p  32s+lpn+2  2?7r+6^34-6Sp2n+6  P 3  fs A  108  Chapter 3. Elliptic Curves with 2-torsion and conductor 2 3 p a  where e 6 { ± 1 } is the residue o / 3  s + 1  l3  5  p modulo 4.  25. there exist integers rn G {4, 5}, n > 0, and s € {0,1} such that '"~ " square and E is Q-isomorphic to one of the elliptic curves: 2  a  a  2  YI  2  Y2  A  4  m-2 2. 3  l 2 p  5 +  _32s+l n+2  _ .2• 3 ^ p y / ^ -  is a  p  2m+6 ^3+6s^2n+6  p  e  where e 6 { ± 1 } is the residue of3 p  modulo 4.  s+1  26. there exist integers n > 1 and s € {0,1} such that ' ~ is a square, p = — 1 (mod 4) and E is Q-isomorphic to one of the elliptic curves: 4p,  1  n  3  a ZI  e  8 + 1 P  A  4  32s+lpTi+2  .3 * + ^ ^  e• 2• 3  Z2  a  2  y^l=i  - 3  where e G { ± 1 } is the residue of3 p  2  _ 2  3  3  +  6  p  s  2  n  +  6  2833+6s ?x+6 p  + V  s  4  modulo 4.  s+1  27. there exist integers n > 1 and s G {0,1} swch that ^Ap - is a square and E is Q-isomorphic to one of the elliptic curves: 4  a  _  ZI Z2  e• 2•  A  a.4  2  3 p^J " s+1  p  3  2 s + i y  2433+65^71+6  325+lpTl+2  4  —2 3 8  3  where e G { ± 1 } is the residue o / 3  s + 1  3 + 6 s  j>  2 n  +  6  p modulo 4.  28. there exist integers m G {4, 5}, n > 1, and s G {0,1} such that square and E is Q-isomorphic to one of the elliptic curves: a AA1 AA2  C•  2  _  30+lpy/*^.  _ .2.3'+V^  2  m - 2  a  4  3  2  s+1  ~  2  A S  + l  p  32.5+1^71+2  C  where e G { ± 1 } is the residue of3 p  p  modulo 4.  2  22m^3+6spTi+6 2 +^33+6.Sp2n+6 m  is a  Chapter 3. Elliptic Curves with 2-torsion and conductor 2 3 p a  l3  109  s  29. there exist integers m e {5}, s e {0,1} and t G {0,1} such that square and E is Q-isomorphic to one of the elliptic curves: A  04  a-2  22m 3+6Sp3+6t  2m-2g2s+lp2t+l  BBl  3  2^71+6 Q3+6S p3+6t  2s+lp2t+l  BB2  3  where e G {±1} is the residue of3 p s+1  is a  modulo A.  t+1  30. there exist integers m e {4}, s e {0,1} and t € {0,1} SMC/I frtflf - ^ - is a square and E is Q-isomorphic to one of the elliptic curves: 2  a  a  2  A  4  CC1  2m-232s+lp2t+l  _ 22m23+6Sp3+6t  CC2  _32s+lp2t+l  2"i+6^3+6.Sp3+6£  wrtere e G {±1} is fne residue of3 p s+1  modulo 4.  l+1  Theorem 3.25 T«e elliptic curves E defined over Q, of conductor 2 3 p , and having at least one rational point of order 2, are the ones such that one of the following conditions is satisfied: 4  b  2  1. there exist integers m > A, £ > 2 — b, and n > 0 such that 2 3 p square and E is Q-isomorphic to one of the elliptic curves: m  2  Al A2  -e • 3 - p v 2 3 V + 1 e-2-3 - py/2 3 -p +1 b  1  /  1  m  m  t  22m32f+6(b-l)p 7i+6  2m-2 e+2(b-l)pn+2 3  n  3  where t G {±1} is the residue of3  + 1 is a  n  A  a.4  a b  e  2  2m+63t'+6(b-l)pn+6  2(fc-iy  p modulo-A.  b  l  2. there exist integers £ > 2 - b and n > 0 such that A • 3 + p is a square and E is Q-isomorphic to one of the elliptic curves: e  o  a  2  BI B2  e-3 - p^/A-3 +p -e-2-3 - p^A-3* b  l  b  e  l  n  +p  n  n  A  4  3  «+2(b-l) 2  3  2(6-l) n+2  p  p  2  4 2f?+6(b-l)pn+6 3  283<?+6(6-l)p2n+6  Chapter 3.  Elliptic Curves with 2-torsion and conductor  where e G { ± 1 } is the residue of3  110  23 p a  l3  5  p modulo 4.  e+b  l  3. there exist integers m > 4, £ > 2 — b, and n > 0 such that 2 3 square and E is Q-isomorphic to one of the elliptic curves: m  o Cl C2  1  4  /  m  2m-2 e+2(b-l) 2  n  t-2-3 - p^/2 3 b  1  m  3  +p  e  n 3  where e € { ± 1 } is the residue of3  22m 2£+6(b-l)pn+6  p  3  2(6-l) n+2  2tn+6 t?+6(t>- l)p2n+6  p  3  p modulo 4.  b  l  4. there exist integers £ > 2 — b and n > 0 such that 4 • 3 — p E is Q-isomorphic to one of the elliptic curves: e  0  DI D2  1  n  1  3  -3  ~p  e  n  where e € { ± 1 } is the residue of 3  _2432t°+6(b-l) Ti+6 p  p2  2 ( b _ 1 )  is a square and  A  e+2(b-i)  -p  i  -e-2-3 - p^4-3 b  n  a.4  2  e-3 ~ py/4-3 b  p"  + 2  2  83«+6(b-l)p2n+6  p modulo 4.  e+b  l  5. there exist integers m > 4, £ > 2 — b, and n > 0 such that 2 3 square and E is Q-isomorphic to one of the elliptic curves: rn  o El E2  4  -p  - p  2m-2 e+2(b-\)p2  -e •-3 ~ p /2 3 1  m  e  n  y  b  1  m  e  2  p  b  is a  n  m+6 e+6{b-l)p2n+6  _32(b-l) n+2  where c e { ± 1 } is the residue of 3  —p  _22m32£+6(b-l)pn+6  3  n  e  A  a  2  e • 2 • 3 - p^2 3  h  is a  n  A  a  2  -e • 3 - p v 2 3 ^ +p b  +p  e  3  p modulo 4.  x  6. there exist integers m > 4, £ > 2 — b, and n > 0 such that 2 p m  n  + 3 is a e  square and E is Q-isomorphic to one of the elliptic curves:  o Fl F2  1  e• 2•3  m  b _ 1  04  A'  2m-232(6-l)pn+2  22m3<"+6(b-l)p2n+6  £+2(b-l) 2  2m+632i'+6(6-l)pn+6  2  -t • 3 - p^/2 p b  + 3  n  e  p\/2 P + 3 m  n  e  where e € {±1} is the residue of3  3  b  p  p modulo 4.  1  7. there exist integers £ > 2 — b and n > 0 such that 4p  n  is Q-isomorphic to one of the elliptic curves:  — 3 is a square and E e  Chapter 3. Elliptic Curves with 2-torsion and conductor 2 3 V Q  0,4  A _243<"+6(b-l) 2n+6  2  GI G2  e • 3 ~ p^4p b  -e  1  -  n  •2 •3 - p ^ 4 p b  1  & -  n  UI  5  2(6-l) n+2  a  3  /  p  p  _3«M-2(t.-l) 2  3  e  where e G { ± 1 } is the residue of3  b  p  1  n + 1  modulo 4.  S. fTzere exz'sf integers I > 2 — £> and n > 0 such that 4 + 3 p is Q-isomorphic to one of the elliptic curves: e  a-2  HI H2  4  -e-2-3''- p\/ + 1  2 ( f > - i y  3<!+2(b-l) n+2  V  3  4  3  p  where e G { ± 1 } is the residue of3  is a square and E  A  0,4  e-3 -V\/ + 3 V 6  n  2832t'+6(b-l) 2n+6 p  p modulo 4.  b  x  9. there exist integers m > 4, I > 2 — b, and n > 0 such that 2 square and E is Q-isomorphic to one of the elliptic curves:  4  2  11 12  -£-3  ~ p /2 1  +  m  y  e-2-3 - pv h  1  3p e  n  3<:+2(b-i)pn+2  2ira+6g2>:+6(b-l)p2n+6  b  p  p modulo 4.  1  - 3p  m  a  Jl J2  -t  4  • 3 - py/2  -  3p e  e-2-3 - p^2 h  1  - 3 ^  m  _22m3<?+6(6-l) n + 6  _3(?+2(b-l) n+2  2m+632*?+6(6-l)p2n+6  3  p  where e G { ± 1 } is the residue of3  b  p modulo 4. e  a  KI K2  -e  e  -  n  e •2 •3 - p /3 p b  1  l  s  n  -  2  m  2  m  n  —2  rn  A  04  2  • 3 - py/3 p l  is a  1  11. there exist integers m > 4, I > 2 — b, and n > 0 such that 3 p square and E is Q-isomorphic to one of the elliptic curves:  b  n  m-2 2(b-iy  2  n  e  A  a  2  m  is a  p  10. there exist integers m > 4, I > 2 — b, and n > 0 such that 2 square and E is Q-isomorphic to one of the elliptic curves:  l  n  22?n 3 ^ + 6 ( 6 - l ) n + 6  3  where e G { ± 1 } is the residue of3  b  e  m-2 2(6-l) 2  2  +3V  / 2 m  + 3p  A  a  a h  m  _2«J-232(b-l) 2 p  3«+2(b-l) n+2 p  22m^e+6(b-l)  pn+6  _2m+6 2»?+6(b-l)p2n+6 3  is a  Chapter 3. Elliptic Curves with 2-torsion and conductor  23p ct  f3  112  5  where e € {±1} is the residue o / 3 p modulo 4. b  l  12. there exist integers m > 4, £ > 2 — b, and n > 0 such that 3 — 2 p square and E is Q-isomorphic to one of the elliptic curves: e  a  4  —2" - 3 ( - )p + l  LI L2  e-2-3 -W3 -2 p" f c  £  m  2  b  1  n  22?Ti ('+6(l)-l)p2n+6  2  3  £+2(6-l) 2  3  where e e {it 1} z's the residue o / 3  2  is a  n  A  a  2  m  _2m+6^2e+6{b-l)pn+6  p  p modulo 4.  b  1  13. there exist integers £ > 2 — b and n > 0 such that p - 4 • 3 is a square and E is Q-isomorphic to one of the elliptic curves: n  a.4  A  _ f!+2(fc-l) 2  4 2f!+6(6-l)pn+6  a-2  Ml M2  t • 3 - pyjp h  -t-2-  l  - 4 •3  n  3 ~ pyjp h  x  £  3  -4-3  n  e  t  3  where e e {±1} z's the residue of3 p  p  2(b-l)pn+2  2  3  _ 8 <?+6(6-l)p2n+6 2  3  modulo 4.  e+b  14. there exist integers m > 4, £ > 2 — b, and n > 0 such that p — 2 3 square and E is Q-isomorphic to one of the elliptic curves: n  4  2  _ m-2 <?+2(b-l) 2  -e • 3 p \ / p - 2 3* t-2- 3 - p\/p" - 2 3* b_1  ft  n  ]  m  2  3  m  3  where e e { ± 1 } is the residue of 3  b  e  p  2(ft-l)pn+2  2  2m 2<?+6(b-l)pn+6 3  _ m + 6 « + 6 ( b - 1)^271+6 2  3  p modulo 4.  1  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 Ol 02  £  e  .  3  .2• 3  6  A  04  2  6 - y + y 4.3^+1  2  m <?+2(b-l) 2t+l  2  4 2*!+6(b-iy+6t  2(b-l) 2t+l  2  8 2«?+6(b-iy+6t  3  - V + y ^ ± l  where e e { ± 1 } is the residue of3  e+b  is a  A  a  a  NI N2  rn  3  p  p  p modulo 4.  1  t  3  3  Chapter 3. Elliptic Curves with 2-torsion and conductor 2"3 V /  113  5  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: 0-4  A  2m e+2(b-l)p2t+l  227ncj2£+6(b-l) 3+6t  2(b-l) 2(+l  2m+6 <?+6(b- l)p3+6t  a-2  PI  3  P2  3  where e G { ± 1 } is the residue of3  b  p  1  p  3  J9  t + 1  modulo 4.  17. there exist integers £ > 2 — b and t £ {0,1} such that ' " 4 3  is a square and  1  p  E is Q-isomorphic to one of the elliptic curves:  Ql  e-so-y+y ;4 3  2m i+2(b-l)  1  3  e • 2. 3 -y+y - ;&  Q2  A  a.4  a2  4  3  21+1  _ 4 2^+6(b-l) 3+6t 2  3  where e G { ± 1 } is the residue of3  p  28 <"+6(b-l) 3+6t  _ 2(b-l) 2t+l  1  3  3  p  p  p modulo 4.  e+b  1  t  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: a.4  A  2m »+2(b-])p2/.+l  _22moj2<f+6(b-l)p3+6t  _ 2(b-l) 2t+l  2?n+6 <f+6(b-l) 3+6t  a  2  Rl  ^-y+y "' ;-  R2  £.2-3"-y+y '" ;-  2  3  2  1  3  3<  1  3  where e G {±1} is the residue of3  b  1  p  p  t + 1  3  p  modulo 4.  19. there exist integers £ > 2 - b and t G {0,1} such that  j  s a  square and E  is Q-isomorphic to one of the elliptic curves: a  SI S2  e  .  3  A  04  2  b - y + y 4 ^ i  3  -e • 2 • 3 ^ - y + y ^ f i  w«ere e G { ± 1 } zs ifte residue of3  b  3  p  «?+2(b-l)  p  l  2(b-l) 2£+l  t  2t+l  modulo 4.  2  2  4 f+6(6-l)p3+6t 3  8 2f+6(b-l)p3+6t 3  Chapter 3.  114  Elliptic Curves with 2-torsion and conductor 2 "3 p c  l3  s  20. there exist integers m > 4, £ > 2 - b and t G {0,1} such that + 2m  is a  3  square and E is Q-isomorphic to one of the elliptic curves: <2  A  2m-2g2(6-l)p2t+l  22m3<?+6(6-l)p3+6t  3<+2(6-l)p2i+l  2m+632(?+6(b-l)p3+6«  «2  4  TI T2  £  - 2 - 3  b  - y + y 2 y  where e G { ± 1 } is the residue of3  b  1  p  modulo 4.  i + 1  22. f/zere exist integers m > 4, £ > 2 - b and t G {0,1} such that " 2  3<! p  fs  square and E is Q-isomorphic to one of the elliptic curves: a  0-2 _  Ul U2  e  £  .  3  6 - y  - 2 - 3  6  +  2m-232(6-l)p2t+l  y 2 ^ 3 i  - y + y  2  m  ;  3  A  4  _3«?+2(6-l)  '  where a £ {±1} is the residue of3  y  b  +  2t+l  _ 2m3«+6(6-l) 3+6( 2  p  2m+632£+6(b-l) 3+6t p  modulo 4.  1  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  a-2  VI  e  -3"-y+  i x  / ^  3  - -2.3 -y+yy  V2  A  4  _ 2(b-l)p2f+l  3  3«+2(b-l)  b  2t+l  24 «+6(b-l)p3+6t 3  _ 8 2f+6(b-l)p3+6£ 2  3  e  where e G { ± 1 } is the residue of3 p b  i  modido 4.  23. there exist integers m > 4, £ > 2 - b and t G {0,1} such that ^~ "' 3  is a  2  square and E is Q-isomorphic to one of the elliptic curves: a  a  2  WI W2  _ . e  36-1^+1^35^  £•2 •3  6  - y + y  3  ';  2  m  A  4  _2"i-232(b-l) 3  «+2(b-l)  2t + l  2t+l  2  2m3<?+6(6-l)p3+6t  _2m+632<?+6(b-l) 3+6i p  Chapter 3. Elliptic Curves with 2-torsion and conductor  where e G { ± 1 } is the residue of3  p  b  1  2  a  l3  s  modulo 4.  t+1  In the case that b — 2, i.e. N — 2 3 p , 4  115  23 p  we furthermore could have one of the  2  following conditions satisfied: 24. there exist integers n > 0 and s e {0,1} such that '~ Q-isomorphic to one of the elliptic curves: 4p7  a  XI  e  X2  a  2  _2433+6«p2n+6  p  £-2.3 +V s  V  / 4  T  -3  1  w«ere e £ {±1} is the residue of3 p s  is a square and E is  A  4  32s+l n+2  . 3-+  1  y  2 s +  2833+6.5^71+6  modulo 4.  n+1  25. there exist integers n > 1 and s € {0,1} such that t^l+A j Q-isomorphic to one of the elliptic curves:  s a  a  a  2  YI  A  4  3 +V 2!  Y2  e •2 •  3  s  +  1 P  x  f  p  " +  3  4  square and E is  2433+6.5^,77+6  2  2s+lp77+2  2833+6.5^277+6  u;/zere c G {±1} is the residue of 3 p modulo 4. s  26. there exist integers m > 4, n > 1 and s € {0,1} such that '"+P" is a square and E is Q-isomorphic to one of the elliptic curves: 2  a  a  2  ZI Z2  A  4  22m^3+6Spn+6  2 ~3 p m  e• 2-  3 yJ '"+ s+1  2  pn  2  2s+1  2  2m+6^3+6Sp2n+6  32s+lp7i+2  P  ztf/iere e G { i l } zs the residue of3 p modulo 4. s  27. there exist integers m > 4, n > 1 and s G {0,1} swc/z f/zaf and E is Q-isomorphic to one of the elliptic curves: a  AA1 AA2  e  a  2  . 3-+  - £ . 2 . 3 ' + ^ ^  2  g  P  zs A square  A  4  V  _22m<j>3+6.$pTi+6  _ 2 +l n+2  2m+623+65p2n+6  2 - 3 m  3  2  S  2 s +  p  Chapter 3.  Elliptic Curves with 2-torsion and conductor  116  23 p a  l3  s  where e e { ± 1 } is the residue of3 p modulo 4. s  28. there exist integer n > 1 and s G {0,1} such that Q-isomorphic to one of the elliptic curves:  A  (I4  «2  BBl BB2  is a square and E is  -3  2 s +  V  2  g2.s+lpn+2  £ • 2 • Z' pyJ>£f± +1  4 33+6.5pn+6  — 2 3 8  3 + 6 , 5  j9  2 , l + 6  where e G { ± 1 } zs tTze residue of3 p modulo 4. s  29. fizere exzsf integers m > 4, n > 1 arzd s G {0,1} sz^c/z f/zat " g " zs a square and E is Q-isomorphic to one of the elliptic curves: 1 p  a  2  A  4  CC1  _ m-2 2»+l 2  2 2 m ^ 3+6 s pn+6  CC2  32.S+I n + 2  _2^+633+6s^2n-(-6  2  3  p  z^/zere e G { ± 1 } zs z7ze residue of3 p modulo 4. s  30. f/zere exz'sf integers m > 4 and s, £ G {0,1} swc/z rTzrtf - ^ ~ zs a square and E 2  is Q-isomorphic to one of the elliptic curves:  a  a-2 DDI DD2  2  -£ • 2 • 3"+V+y £g±! s  2  3  where e G { ± 1 } is the residue of3 p  A  4  77i.- 32.s+l 2t+l p  2  2.s+l 2 +l p  2 m 33+65^3+64  2'm+633+65 3+6f.  t  p  modulo 4.  i+l  31. there exist integers m > 4 and s, t G {0.1} such that " ~ 2,  is a square and E  1  i  is Q-isomorphic to one of the elliptic curves:  ^•3  EE1 EE2  4  A  m-232.5+1^24+1  _ 2m-33+6Sp3+6t  a  a-2 s +  v y^ +  2  _ . 2• 3«+y+y2==i e  where e G { ± 1 } zs the residue of3 p s  t+1  _32s+l 2t+l p  modulo 4.  2  2  m+633+6s 3+6t p  Chapter 3. Elliptic Curves with 2-torsion and conductor  117  23 p a  l3  s  T h e o r e m 3.26 The elliptic curves E defined over Q, of conductor 2 3 p , 5  b  and hav-  2  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 3 p is Q-isomorphic to one of the elliptic curves: e  + 1 is a square and E  n  a  a  A  4  2  A2  2 • 3 "Vv/SV + 1 - 4 • 3b" V \ / 3 V + 1  3<"+2(6-l) n+2  4 • 3 ( -V  2l23<?+6(6-l)p7i+6  Al'  - 2 • 3 ~ p^3 p  3^+2(6-1)^71+2  2632^+6(6-1)^277+6  . 2(b-iy  2l2g<!+6(6-l) n+6  Al  ft  b  A2'  1  i  2  +1  n  2632£+6(6-l) 27i+6  p  4 • 3 -V\/3y + 1 6  4  p  fc  3  p  2. there exist integers £ > 1 and n > 0 such that 3 + p e  is a square and E is  n  Q-isomorphic to one of the elliptic curves: (a) £ is even; a  a  2-3 - pv 3*+P -4-3 - p^/3 +p  Bl'  -2-3 ~ py/3  B2'  h  1  /  l  n  e  h  l  3  4  +p /  3  p  n  2  p  1232^+6(6-l)p7i+6 p  p  + 2  63<!+6(6-l) 2n+6  263<?+6(&-l) 2n+6  2(6-l) n+2  4•3'  +p  f  2  p  3  n  4-3 " pv 3 1  2(6-l) n+2 . £+2(6-l) 2  n  l  6  A  4  2  Bl B2  fe  21232^+6(6-1)^77+6  ( -V b  (b) £ is odd; CI C2  a-2 2-3 - py/3 +p -4-3 - p\/3 +p'  CI' C2'  -2-3 - p^3 4 • 3 - p /3  h  i  i  fe  h  1  e  h  1  3  2  +p +p  s  n  2632^+6(6-iy+6  p  4 • 3 ( - )-p +  1  h  1  n  2  2l23<»+6(6-l)p2n+6  e+2(b-i) 2  n  e  A  4  £  1  a f.+ 2(b-l) 2  n  3  2632^+6(6-1)^71+6  p  4 •32(b-l) n+2 p  123<"+6(6-l)p27i+6  2  3. there exist integers £ > 1 and n > 0 such that 3 — p  n  is a square and E is  Q-isomorphic to one of the elliptic curves: (a) £ is even; a  a DI D2 DI' D2'  2-3 ~ psj3 l  n  _ 2(6-l) 77+2  263<?+6(6-l) 2n+6  n  4 . «+2(6-l) 2  _ 212 3 2 , 5 + 6 ( 6 - 1 ) n+6  _32(ft-l) n+2  263^+6(6- l) 2n+6  4 . 3«+2(b-l) 2  _2l232«?+6(6-l)pn+6  -p  l  -4-3 - p\/3^-p f c  1  -2-3 - pv 3* b  1  /  4-3 - p 3^ 6  1  /  v  A  4  2  h  3  -p  n  ~P  n  p  3  p  p  p  p  p  p  Chapter 3.  Elliptic Curves with 2-torsion and conductor 2 3@p a  118  5  (b) £ is odd; a El E2  2-3 -V3 -P" - 4 - 3 - p^/3 ~P  El'  -2 • 3 - V \ / 3  E2'  4 • 3 -V\/3  3^+2(6-l) 2  (  b  A _26^e+6(b-i)pn+6  0,4  2  6  l  l  p  n  h  b  _  3  2  p  3  n  3  123f+6(6-l) 2n+6 p  _2 3 ^+6(b-l)pn+6 6  p  _ 4 . 2(6-l) n+2  ~P  £  4  «+2(b-l) 2  -P  n  f  . 2(h-l) n+2  2l23«+6(6-l)p2n+6  p  4. fhere exz'sf integers £ > 2 — b and n > 0 swc/t f/iaf p  2  — 3 is a square and E is  n  f  Q-isomorphic to one of the elliptic curves: (a) £ is even; a _ e+2(b-i) 2  ai FI F2 FI' F2'  1  - 3  n  l  3  -4-3 "V\/p -3' - 2 • 3 " p v P - 3^ 4 • 3 ^p^p - 3 b  n  b  1  /  b  A  4  2 • 3 ~ p^p b  2632£+6(6-l)  p  4 • 3 (fc-i) n+2 2  p  !  -2  n  n  £  2  4  3  n+6  ^p  2 n + 6  632f+6(t—l)n+6  _2l23«+6((>-l)p2n+6  . 2(f.-l) n+2 3  1 2  £ + 6 ( b _ 1  p  (b) £ is odd; 04  A  32(6-1)^+2  _263«+6(b-l)p2n+6  0-2 GI  2 • 3 ~ py/p h  1  - 3  n  f  G2  -4-3 ^ pv P" - 3  GI'  -2 • 3 "V\/p - 3* 4 • 3 ~Wp - *  G2'  b  1  /  b  f  n  b  n  3  21232^+6(6-l) n+6  - 4 . 3^+2(6-l) 2  p  p  3  263^+6(6-l)p2n+6  2(b-l) n+2 p  - 4 . 3^+2(6-1)^2  2  1232(?+6(6-l)p7 +6 l  5. fhere exz'sf integers £ > 1 and t € {0,1} swc/t f/iaf -± - is a square and E is 3  1  Q-isomorphic to one of the elliptic curves: (a) p = 1 (mod 4); a HI  2.3 -y+y^i  H2  -A-3 - p  HI'  -2-3 -y  H2'  b  b  l  +  t  ^^  +  i  /  3  v  ^i  4-3 -y+y ^i 6  (b) p = - 1 (mod 4);  A  04  2  b  3  2t+l  3«+2(b-l)  26 2*:+6(b-l)p3+6t 3  2  123«+6(6-l)p3+6(  2*+l  2  632tM-6(6-l) 3+6t  4 . 2(6-l) 2t+l  2  123«+6(6-l)p3+6t  4 • 3 ( " )p 2  b  1  3*+2(6-l) 3  p  24+1  p  Chapter 3. Elliptic Curves with 2-torsion and conductor  a  0-2  11  2 - 3  - y +  b  i x  . 36-y+y  12  _  11'  -2-3 -y+y^  12'  4  3f±l  -V+  f e  / ^i  1  3  v  l3  A  4 . 3^+2(6-1)  3  2t+l  263^+6(6-1)^3+64 2l232*:+6(6-l)p3+64  2(b-l) 2t+l p  . 3^+2(6-1)24+1  4  s  4  p  h  4-3  a  32(6-l) 2t+l  / 2 £ ± i  119  23 p  6. there exist integers £ > 1 and £ € {0,1}  2  63t"+6(6-l)p3+64  21232^+6(6-1)^3+64  f/wf - — - is a square and E is  Q-isomorphic to one of the elliptic curves: (a) p = 1 (mod 4); a  02 2 . 3  JI  - y +  b  i v  /  ^ i  3  J2  - .3  JI'  -2.3 -y V^f  f  -y y^  e  +  4  b  +  i  4-3 -y+y ^i  J2'  f e  3  A  4  _ 32(6-1)^24+1  263^+6(6-1)^3+64  4 . 3 ^ + 2 ( 6 - 1 ) 24+1  _ 1232^+6(6-l) 3+64  __32(6-l)p24+l  263<?+6(6-l)p3+64  4 . 3<!+2(6-l)p24+l  _2l232£+6(6-l)p3+64  2  p  (b) p = - 1 (mod 4); a  a  2  K I  2-3 -y+y ^  K2  ^•s^-y+y ^  KI'  -2.3 -y+y^  K2'  4.3 -y+yy  b  3  6  h  A  4  31+2(6-1)^24+1  3  _  . 32(6-1)^24+1  4  3  _2632<?+6(6-l)p3+64  £+2(6-l)p24+l  2123^+6(6-1)^3+64 _2632t"+6(6-l)p3+64  - 4 • 32(b-l)p24+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: a  a  2  LI L2  S ^ P N /  8  •3 V  + 1  -2-3 - pv 8-3V+ 1 b  1  /  A  4  2 . £+2(6-l) n+2 3  3  2632^+6(6-l)p2n+6  2(6-l) 2 p  93«+6(6-l)pn+6  2  LI' L2'  S ^ P ^ / S  • V 3  2 • 3 - p^8-3 p b  l  e  n  +1  + 1  2 . 3«+2(6-l)pn+2 3  2632^+6(6-l)p2n+6  2(6-iy  293^+6(6-  l)pn+6  8. there exist integers £ > 2 — b and n > 0 such that 8 • 3 + p is a square and E is Q-isomorphic to one of the elliptic curves: e  n  Chapter 3.  Elliptic Curves with 2-torsion and conductor  3 ~ pyj8-3  M l M 2  +p  i  n  -2 • s ^ y /  3  n  6  1  /  8  -3*+P  A 2  2  93^+6(b-l) 2n+6  . «+2(fc-l) 2 3  p  32(b-l) n+2 p  n  s  632M-6(b-l) n+6  3  2  2-3 - pv  3  2(b-l)pn+2  • *+P  8  M l ' M 2 '  .  2  l3  0,4  2  l  a  / + 2 ( 6 - i y  a b  120  23 p  p  p  2  632^+6(b-l) n+6  2  93<!+6(b-l) 2n+6  p  p  9. there exist integers £ > 2 — b and n > 0 such that 8 • 3 ^ — p E is Q-isomorphic to one of the elliptic curves:  n  0,4  A  2 •3«+2(b-Dp2  _ 632f!+6(6-l) n+6  a  2  N I N 2  3  - W 8 - 3 ^  b  -P  n  -2-3 -V\/8-3«-p 6  N2'  2  _32(6-l) n+2  n  2  p  _  2 • 3<+2(6-l)p2  N I '  2-3 " p\/8-3 b  1  _ 2 ( b - l ) n +2 3  -p  <  is a square and  n  2  p  p  93f+6(b-l) 2n+6 p  2  6  3 *+6C>-l) n+6 2  p  93«+6(b-l) 2n+6 p  10. there exist integers £ > 2 — b and n > 0 such that 8p + 3 is a square and E is Q-isomorphic to one of the elliptic curves: n  2  O l 0 2 O l '  or  3 " p\/8p 1  1  -3 "V\/8p b  n  +3  +3  n  2 •3 ~V\/8p b  2  + 3*  n  -2-3 * p\/8p b  A  a.4  a b  e  e  f  . 2(b-l) n+2 3  p  3f+2(b-l)^  2 3^+2(b-l)p^ • 3 ( - )p + 2 b 1  n  2  2  63*!+6(b-l) 2n+6  2  932£+6(b-l) 7i+G  2  63f+6(b-l) 2n+6  2  932<:+6(b-l) n+6  p  p  p  p  + 3*  n  11. there exist integers £ > 2 — b and n > 0 such that 3 p — Sis a square and E is Q-isomorphic to one of the elliptic curves: n  a PI  3 " pv 3V - 8 1  /  P2  - 2 -3 - py/3 p  PI'  -3 ~ pyj3 p  P2'  b  b  1  l  l  l  A  04  2  b  - 8  n  n  2 •S^WSV  - 8 1  - 8  -2-3 ( -V 2  b  2  3*?+2(b-l) n+2 2  _ 932( +6(b-l) 2n+6 2  V  b  p  ,  p  - 2 •3 < ~  63<?+6(6-l) 77,+6 p  2  63«?+6(b-l) n+6 p  _ 932«+6(b-l) 2»2+6  3«+2(b-l) n+2 p  2  p  12. there exist integers £ > 1 and n > 0 SMC/Z that 3 — 8p is a square and E is Q-isomorphic to one of the elliptic curves: (  n  Chapter 3.  121  Elliptic Curves with 2-torsion and conductor 2 3 p cc  a  a-2  Ql Q2 Ql' Q2'  3 ~' py/3 h  i  - 8p  l  -2 • 3 ~ py/3 - - 8p h  l  e  -t^py/V  2  -2-3 - p"+  - 8p  e  _ 9 2^+6(6-l) n+6  p  - 8p  1  p  p  2(b  n  2 • 3 - py/3 b  A 263<'+6(6-l) 2n+6  3^+2(6-l) 2  n  n  3  6  4  • 32(fc-l) n+2  -2  n  l3  1)  3  p  26 €+6(6-l) 2n+6  2  3  £+2(6-l) 2  p  -2 3 9  p  (  M+6  fc_1  V  1+6  13. there exist integers £ > 2 - b and n > 0 such that p - 8 • 3 is a square and E is Q-isomorphic to one of the elliptic curves: n  a  Rl R2 Rl' R2'  3 - px/p x  -8-3  n  1  -2-3 ~ p^/p b  l  A  0,4  2  h  e  -2 • 3^+2(ft-D 2  S^py/p"  632f+6(b-l) n+6 p  p  -8-3*  n  2  3  p  _ 93^+6(6-l) 2n+6 2  _ e+2(b-i) 2  -8-3  e  23  2 • 3 pvV -8-3 6_1  2(b-l) n+2  i  26 2£+6(6-l) n+6 3  p  3  p  2(b-l) n+2 p  _ 9 £+6(b-l) 2  14. there exist integers £ > 2 - b and t e {0,1} such that  8  3  3  +  p 2  Ti+6  is a square and  1  p  E is Q-isomorphic to one of the elliptic curves: a  2 . 3^+2(6-1)  SI S2  2-3 -V  SI'  _  S2'  A  04  2  b  3  + 1  / ' + 8  v  3 f  1  6 - y + y 8 ^ t i  2-3 -V b  + 1  / 8 3 V  p  + 1  32(6-1)  2t+l  2  632<M-6(b-l) 3+6( p  293£+6(6-l) 3+6t  2t+l  p  2 . 3^+2(b-l) 2t+l  2632^+6(6-1)^3+6*  2(6-l) 2t+l  29 ( +6(6-l) 3+6(  p  3  ,  3  p  15. there exist integers £ > 2 - b and t € {0,1} such that  p  8 , 3  1  is a square and  E is Q-isomorphic to one of the elliptic curves: a  TI T2 TI' T2'  36-y+y 2.36-I  3  8.3^-1  t+l/8-3«-l  ^ V P b - y + y 8.3^-1  2-3 -y+ b  a.4  A  2 . 3^+2(b-l)p2t+l  _26 2«+6(6-l)p3+6t  _32(6-l) 2t+l  293«+6(6-l) 3+6t  2  i  v  / 8  '  3 f  -  1  3  p  p  2 . 3*?+2(b-l) 2t+l p  — 2 3 + ( )p + ' 6  _ 2(b-l) 2t+l 3  2£  6  b_1  3  6  293^+6(6-l) 3+6t  p  p  16. there exist integers £ > 2 - b and t e {0,1} such that -^ - is a square and E 8  is Q-isomorphic to one of the elliptic curves:  3  Chapter 3. Elliptic Curves with 2-torsion and conductor  122  2 3^p a  5  A  a  2  3 -y+y^ b  UI U2  -2-3 - p y/^f b  1  t+1  L  UI' 2 . 3  U2'  6  - y  +  i  v  . 2(6-l) 2t+l  2  / 5 ± 3 i  3  p  2  63€+6(b-l)p3+6t  3<?+2(fc-l)p2t+l  2932«+6(b-l)p3+6t  2 • 32(b-Dp2t+l  263^+6(6-l)p3+6i  €+2(6-l)p2t+l  2932*;+6(6-l)p3+6t  3  27. there exist integers £ > 2 — b and t € {0,1} swch that ^-T- - is a square and E 3  is Q-isomorphic to one of the elliptic curves: a VI V2  A _ 63€+6(6-l)p3+6i  _ «+2(fc-l)  2932^+6(6-1)^3+6*  p  -2-3 -y+y^ h  2t+l  3  2. 2(6-l) 2(+l  VI' V2'  . 0,4 2 • 32(b-D 2i+l  2  3  2-3 -y+y^  p  _ £+2(6-l)p2t+l  b  3  2  _2 3^+ ( -!)p + 6  2  6  h  3  6 t  932£+6(6-l)p3+6(.  18. there exist integers £ > 2 — b and t G {0,1} such that -~  is a square and E  3  is Q-isomorphic to one of the elliptic curves:  WI W2  W2'  4  +  2  .  3  b  - v  +  p  3^+2(6-1)  b  - 3  • 32(6-l) 2t+l  -2  s"-y V^ -2-3 -y+y^  WI'  A  a  a-2  2t+l  -2-3 ( - y 2  V ^  b  i  t + i  3«+2(6-l)p2£+l  6 - y + y 3 ^ 8  2  63«+6(6-l)p3+6t  -2 3 9  2 £ + 6 ( b _ i  y  + 6 (  263^+6(6-1)^3+61 _2 3 ^+6(6-l)p3+6i 9  2  In the case that 6 = 2, i.e. N = 2 3 p , we furthermore could have one of the 5  2  2  following conditions satisfied: 19. there exist integers n > 0 and s 6 {0,1} such that Q-isomorphic to one of the elliptic curves:  f  s a  square and E is  Chapter 3.  Elliptic Curves with 2-torsion and conductor  a XI X2  2• 3  y" /e±i  4 .  3  3»+lpy/lZ±± 4 .  X2'  2633+65^71+6 21233+65^271+6  2s+l n+2  3  Py  123  s  A  2s  p^^-  4  2•  l3  4  _.  XI'  a  a 3 +y  2  s + 1  23 p  p  y  2633+65^77+6  2s+\ n+2  21233+65^277+6  2 s +  3  p  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  a-2  325+1^71+2  YI  2*^3+ 6s jyZn+6  - 4 • 3 *+V 2  Y2  32s+l 77+2  YI' Y2'  A  4  _ 2633+65^271+6  P  4  .  3  ^  2l233+6s^n+6  - 4 • 3 +V  2l233+6Spn+6  2s  ^  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  a-2  ZI Z2  0 0  A  4  32s + l p 2 t + l  2633+6Sp3 + 6£  _ 4 . 2 5 +lp2t+l  2l233+6s^3 + 6/. -  3  (b) p = - 1 (mod 4):  a  A  _ 25+lp2t+l  2633 + 65^3+6*  0.2  AA1 AA2  4  0 0  3  2l233+6SpT?.+6  4 . 32s+lp2t+l  22. E is Q-isomorphic to one of the elliptic curves:  a  a  2  BBl  0  BB2  0  A  4  _ 2(6-l)p2 3  4  . 2(6-l)p2 3  2  6 6(6-l)p6 3  _ 12 6(b-l)p6 2  3  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°3 p l3  124  5  (a) p = 1 (mod 4): 2  CC1  0  CC2  0  A  0,4  a  2636(f)-l)p3+6t  _32(b-l) 2t+l 4  _ 1236(b-l)p3+6i  . 2(6-l) 2t+l 3  2  p  (b) p= - 1 (mod 4): 04  A  2(6-l)p2t+l  _26 6(fe-l)p3+6t  _4 . 3 2 ( 6 - 1 ) ^ + 1  21236(6-1)^3+6*  0,2 DDI  0  DD2  3  0  3  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:  EE1 EE2  2•3 2-3  ' y/^  s +  +y  s + 1 P  yJ^  3  3  2• 3  /^=f-  s+1  2933+6sp2?i+6  8  .£ is a square and E is  A  3 p 2s+l  2  _32s+lpn+2  _2 3 +6spn+6 6  3  2933+6Sp2n+6  Ps  FF1'  - 3 '  FF2'  2-3 pyJ^=f-  +  2633+65^71+6  2  04  FF1  s+1  p  3  2«+lpn+2  02  -2 •  633+6s n+6  29 3+6sp2n+6  p  2s+1  25. there exist integers n > 1 and s e {0,1} such that Q-isomorphic to one of the elliptic curves:  FF2  2  32.s+lpn+2  2 • 2• 3  2 s  P  EE1' EE2'  A  04  0,2  1  P ^  2 •  3  p  2s+1  2  _32s+l n+2 p  _ 6 3+6spn+6 2  3  29 3+6.Sp2n+6 3  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  a  -2 • 3 2 •  l3  +1  2 s  —2 3 + p + 9  2 s  7  3  3  6s  2n  6  2633+65^71+6  +y  —2 3 + p +  2s+l n+2  2 • 3 +Vy ^ S  26g3+6Spn+6  +V  3  -2 • 3  125  5  A  2s+lpn+2  y py/z^  GG1' GG2'  a  0,4  2  GG1 GG2  23 p  9  p  3  6s  2n  Theorem 3.27 The elliptic curves E defined over Q, of conductor 2 3 p , e  b  2  6  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 3 p is Q-isomorphic to one of the elliptic curves: e  a  a-2  Al  2 • 3 -'p\/ V + - 4 • 3 - p^3 p + 1 3  b  A2  b  Al' A2'  1  e  1  - 2 • 3 - p /3 p b  1  i  b  26 f+6(fe-l)pn+6 2l232f+6(6-l)p2n+6  1  263^+6(6-1)^77+6  2(6-l) 2 p  .  4  3  p  3  4-3 - p\/3V + l  A  4  2(b-iy  3  + 1  n  y  + 1 is a square and E  4 . f+2(b-l)n+2  3  n  n  e+2(b-l) n+2  3  p  21232^+6(6-  l)p2n+6  2. there exist integers £ > 1 and n > 0 such that 3 + p is a square and E is e  Tl  Q-isomorphic to one of the elliptic curves: (a) £ is even; a BI  a  2  2-3 - px/3 b  l  +p  e  n 3  B2  -4-3 - pv 3 +P  BI'  -2-3 -V\/3 +P  fe  1  /  £  n  A  4  f+2(6-l) 2 p  4 . 2(6-l) n+2 3  B2'  fe  f  n  4-3 -W3*+jD b  2632^+6(6-1)^77+6  3  p  2(6-l) 2 p  2l23«+6(6-l)p2n+6 2 6 3 2 ^ + 6 ( 6 - 1)^77+6  2l23<?+6(6-1)^277+6  n  4 . 3^+2(6- l)p77+2  (b) £ is odd; a Cl  2•  l  n  -4-3 ~ p^3  Cl'  -2-3 - pv 3*+P A-3 ~ p^3 +p  C2'  32(6-1)^77+2  3 - p^/3 +p 1  C2  b  l  6  b  l  +p  l  1  n  /  t  A  0,4  2  b  n  4  . <+2(b-l) 2 3  p  263«+6(6-l)p2n+6  21232^+6(6-1)^77+6 263^+6(6-1)^277+6  32(6-l) n+2 p  21232^+6(6-1)^,77+6  n  4 . 3^+2(6-l) 2 p  Chapter 3.  Elliptic Curves with 2-torsion and conductor  23 p a  l3  126  5  3. there exist integers £ > 1 and n > 0 such that 3 - p is a square and E is n  Q-isomorphic to one of the elliptic curves: (a) £ is even; 02  2-3 - p /3 -p -4-3 - py/3  DI'  -2 • 3 ~ py 3  D2'  1  b  1  A  0,4  DI D2  b  e  e+2(b-i) 2  n  3  -2 3 6  p  y  -p  e  b  1  4•  p  n + 6  n  /  - 4 . 32(b-l)p«+2  - p  e  n  3 - py/3 -p b  2 f + 6 ( 6 - 1 )  1  e  n  2  3«+2(6-l)p2  123f+6(b-l)p2n+6  _2632/:+6(6-l)pn+6  - 4 . 32(fe-l) n+2  2  p  123^+6(6-l) 2n+6 p  (b) £ is odd;  El E2  2 • 3 ~ ps/3 - 4 •3 ' py/3  El' E2'  -2-3 ~ p\/3 -p 4-3 - py/3 -p  l  l  b l  i  A  0,4  a-2 b  - p -p n  _ 2(b-l)p"+2 3  4 . 3^+2(fc-l) 2  b  1  <  l  p  _ 1232#:+6(b-l) 7i+6  p  h  63£+6(6-l) 2n+6  2  n  l  2  _32(fc-l)n+2  n  p  n  2  . ^+2(b-l) 2  4  3  p  6 t?+6(6-l) 2n+6 p  3  _ 1232^+6(b-l) n+6  p  2  p  4. there exist integers £ > 2 — b and n > 0 such that p - 3 is a square and E is n  Q-isomorphic to one of the elliptic curves: (a) £ is even; a FI F2  2 • 3 -V\7p - 3* n  b  -4-3 - pv P - 3 6  1  /  - 2 • 3 ~ py/p  F2'  4 • 3 - p^/p  n  FT  b  b  l  f  - 3  n  32(b-l) n+2  e  _ 63<'+6(6-l) 2n+6  p  2  _ . *:+2(b-i) 2 4  - 3*  n  1  A  04  2  3  p  32(6-1)  2  p J l  +6  _ 63*:+6(6-l) 2n+6  n+2  2  - 4 . ^+2(b-l) 2 3  p  1232f+6(6-l)  p  2  p  1232£+6(6-l) n+6 p  (b) £ is odd;  GI G2  2 • 3 - ps/p l  A  04  0,2 b  - 3  n  !  _3*+2(6-l) 2 p  -4-3 - pv P -3 b  1  /  n  f  4 • 3 ( - )p™+ 2  fc  1  2  2  632£+6(b-l) n+6 p  _ 12 >:+6(6-l) 2rj+6 2  GI' G2'  -2-3 "V\/p -3* b  n  4 • 3 ~ pJ'p b  1  n  - 3  e  _3*:+2(b-i) 2 4 • 3 ( - )p + 2  fc  1  p n  2  2  3  p  6 2f+6(6-l) n+6 3  p  _ 12 £+6(6-l) 2n+6 2  3  p  5. there exist integers £ > 1 ana* £ G {0,1} such that -y^ is a square and E is 3  Q-isomorphic to one of the elliptic curves: (a) p = 1 (mod 4);  Chapter 3.  Elliptic Curves with 2-torsion and conductor  a  a  2  2-336-y+y -y+ y ^  HI  6  H2  _  HI'  ~2 •  H2'  .  4  1  3  3  127  5  A  2(6-l) 2t + l p  2  2  p  2  p  2  3  3  p  3  6 «+6(ft-l)p3+6i  12 2£+6(6-l)p3+6t  2(6-l) 2t + l  4 . <+2(b-l) 2t+l  4-3 -y v + /2f  f3  4  3  b  +1  a  4 . €+2(b-l) 2t+l  3 -y+y 2f±i b  (W  /3 3f±l  23 p  6 £+6(b-l) 3+6t 3  p  12 2£+6(b-l)p3+6t 3  l  - 1 (mod 4);  p =  a  a  2  11  l  A  4  3<+2(t-l)p2i+l  2-3 - p J^fh  t+1  x  12  4 3  £+2(b-l)p2£+l  2  6 2M-6(b-l)p3+6i  4 • 32(b-l)j,2t + l  2  12 ^+6(6-l)p3+6t  3  b  3  3  12'  6 3 2«+6(6-iy+6t 12 *5+6(6-l)p3+6t  3  _-2-3 . fc-y+y3f+_L -y+y ^i .3"-y+y ^i  11'  2 2  4 . 2(6-l) 2t+l  4  p  3  3  3  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); a  «2  JI  2-3"-y+y^  JI'  -4-3''-y ^  J2'  4-3"-y+y^i  J2  (b) p=-l  +1 /5 v  _  3<+2(b-iy(+i . 2(b-l) 2t+l 4  i 3  3  2  2  33  3  p  _ 162 2« «+ + 6 (6b(-bl )-pi 3y++66i t 2  2  33  (mod 4); a  A  4  _ 2(b-l) 2t+l  KI  K2'  _ 162 2f+6(6-iy+6i ^+6(b-l)p3+6t  2t+l  _4 . 2 ( 6 - l ) 2 t + l  2  KI'  p  *+2(b-l)  a  K2  A  4  3  2  . b-y y3^i 3  +  ^•^-y+y ^i _ . b-y+y3 4  3  p  4 . <+2(b-l)  6 ^+6(b-l) 3+6t 3  p  2t + l  3  _ 12 2«+6(b-iy+6t  ^3  4. b-y+y 3^1  2  2  _32(b-iyt + l  4 . f+2(b-l) 2t+l 3  p  3  _ 21623 f+6(b-iy+6t 2 € + 6 ( 6 - l ) 3+6t 2  3  3  7. there exist integers m > 3, £ > 2 — b and n > 0 such that 2 3 square and E is Q-isomorphic to one of the elliptic curves: m  p  n  + 1 is a  Chapter 3. Elliptic Curves with 2-torsion and conductor 2 3 p a  a  l3  128  6  A  04  2  l) 2n+6  LI L2  2 • 3 - V \ / 2 3 y + 1 2»ng*+2(6-l)pn+2 - 4 • 3 -Vv 2 V + 1 4 - 3 ( - y  LI' L2'  2m3«+2(H) n+2  22m+632^+6(t)-l)p2n+6  . 2(b-iy  2771+12 C j j ? + 6 ( b - l ) p n + 6  b  m  b  -2 •3  /  2  m 3  m  1  1  4 •3 - p / 2 3 y +1 b  m  1  4  v  b  p  3  i  - p\/2 3V +  f c  22m+6 2e+6(b-  2m+12 e+6(b-l)pn+6  i  3  3  8. there exist integers m > 2, £ > 2 — b and n > 0 such that 2 3 + p is a square and E is Q-isomorphic to one of the elliptic curves: rn  Ml  2 • 3 - /2 3 1  m  n  Py  M2  -4-3 pv 2 3 +p  Ml'  -2 • 3 - /2 3 4 • 3 - p^/2 3  M2'  b_1  b  /  l  £  m  m  1  m  2 fc 1  3  2  277i ('+2(b-l)p2  n  e  n  3  1  n  2771+123^+6(6-1)  2  4 • 3 (''- )p + 2  n  1)^71+6  22m+6 2i+6{b-  4 • 3 ( - )p +  n  +p +p  e  Py  b  2m3^+2(b-l)p2  +p  i  n  A  04  0.2 b  e  2  2n+6  2m+632<£+6(b-l)pn+6  277i+123<?+6(6-l)p27(+6  9. there exist integers m > 2, £ > 2 — b and n > 0 such that 2 3 - p" is a square and E is Q-isomorphic to one of the elliptic curves: m  a NI  /  1  m  -p  e  277i3«+2(6-l)p2  n  N2  - 4 • 3 - py/2 3  -p  NI'  ~2-3 - p^2 3  -p  N2'  b  b  1  m  m  1  e  i  /  1  m  _  n  - p  e  4  .  3  _ 2m+6 2e+6(b-l) 2  2(6-l) n+2  _  n  4  pn+6  3  2171+123^+6(6-1)^271+6  p  277i3<+2(6-l)p2  n  4 • 3 - py 2 3 b  A  04  2  2 • 3 ~ py 2 3 b  _22m+632«+6(6-l)p7i+6  . 32(6-l) 7i+2  2  p  77i+123«+6(6-l)p2n+6  10. there exist integers m > 3, £ > 2 — b and n > 0 such that 2 p" + 3 is a square and E is Q-isomorphic to one of the elliptic curves: m  o  a-2  01 02  or 02'  A  4  2 •3 - pv 2 » + 3 -4-3 - py/2 p +3  2771.32(6-1)^71+2  22m+63<M-6(b-l)p27i+6  4 . 3«+ (<>-i)p  277i+1232(?+6(6-l)pn+6  - 2 - 3 " p \ / 2 P + 3^ 4 - 3 - p V ' 2 P + 3^  277132(6-1)^71+2  b  1  /  m  m  n  n  e  J  b  l  b  b  1  1  e  T n  m  n  n  2  4 . 3^+ (b-l) 2  2  2  2  2m+63£+6(b-l)p27i+6  2771+1232^+6(6-1)^71+6  p  11. there exist integers £ > 2 - b and n > 0 such that Ap — 3 is a square and E is Q-isomorphic to one of the elliptic curves: n  e  Chapter 3. Elliptic Curves with 2-torsion and conductor  PI P2 PI' P2'  - 3*  2 • 3 ~ p\j4p  n  -4-3 - pv P b  1  / 4  ~3  n  A  4 • 32((>-iy»+2  -2 • 3 - p v ¥ - 3 l  1  _  €  /  x  3  p  4 • 3 ( - )p +  4 • 3 - p\/4p™ - 3 b  . «+2(6-l) 2  4  2  f  e  129  5  _2l0 f+6(h-l)p2n+6  2  l  l3  a.4  a b  2°3 p  fa  1  n  2  - 4 . 3^+2(i>-l) 2 p  3  2  14 2M-6(b-iy+6 3  —2 3 + ( 10  £  6  b_1  a  2  Ql Q2  2 • 3 -V\/2  m  -4-3 - pv 2  m  Ql' Q2'  -2-3 " p 2 + 3 V 4 • 3 - p\/2 + 3 V  b  1  /  b  1  b  1  /  + 3p e  n  +3V  2  n  A  4  m 2(6-l)p2 3  . «+2(b-l)pn+2  4  6  4- 3 p is a  rn  a  2n  214 3 2 ^ + 6 ( 6 - l ) p n + 6  12. there exist integers m > 2, £ > 2 — b and n > 0 such that 2 square and E is Q-isomorphic to one of the elliptic curves:  f t  )p +  3  22?n+63*-+G(b- 1 )pn+6 2m+1232«+6(b-l)p2?j+6  m 2(6-iy  22m,+63<?+6(b-iyi+G  4 . ^+2(b-l) n+2  2m+1232/:+6(b-l)p2n+6  m  2  3  v  m  3  p  13. there exist integers m > 2, £ > 2 — b and n > 0 such that 2 square and E is Q-isomorphic to one of the elliptic curves:  rn  a  2 • 3 - p\/2 - 3 p - 4 • 3 - py/2 - 3p  Rl' R2'  - 2 • 3 ~ p- /2  1  b  m  1  h  m  1  m  4 •3 - p 2 1  /  n  e  s  b  f  m  v  2  n  _  3  n  e  2  _4.  - 3p  n  p  . £+2(b-l) n+2  4  - 3p e  m 2(b-l) 2 3  m 2(b-iy 3  3  n  A  a.4  2  Rl R2  b  — 3 p is a  «+2(b-iy+2  _ 2 2 m . + 6 3 ^ + 6 ( b - 1 )pn+6 2  m+12 2f+6(b-l)p2n+6 3  _22?n+63^+6(b-l)pn+6 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 3 p" — 2 square and E is Q-isomorphic to one of the elliptic curves: t  a  SI S2 SI' S2'  l  b  4•3  b _ 1  A 22m+63*?+6(b-l)pn+6  4  i  n  - 2™  -4 • 3 - p \ / 3 V - 2 -2-3 - pv 3V-2 b  a  _2m 2(6-l)p2  2  2 • 3 ~' py/3 p h  1  1  pv  /  / 3  3  m  m  3>_+2(b-iy+2  4.  _ m 2(b-iy  m  V -2  m  2  4.  3  3<+2(4-iy+2  is a  _2m+1232*?+6(b-l)p2?i+6 2  2m+63^+6(b-l)pTi+6  _2"i+1232f+6(b-l)p2n+6  15. f/zere exzsf integers m > 2, £ > 2 — b and n > 0 S M C A Z f/wf 3 - 2 p " is a square and E is Q-isomorphic to one of the elliptic curves: £  m  Chapter  3.  Elliptic Curves with 2-torsion and conductor  0-2 TI  a  2 • 3 ~ py/3 h  1  -  e  2 p m  -4 • 3 - p / 3  -  2 p  TI'  -2-3^ pV 3^ -  2p  T2'  h  1  e  y  1  4 • 3 - p%/3 h  m  /  l  -  t  n  m  2 p m  .  4  n  22m+63<?+6(6-l)p2n+6 _2m+1232i?+6(6-l)pn+6  p  3  2 27Ti+63i?+6(6-l)p2n+6 2  p  . <?+2(b-l) 2  4  130  s  _ m+1232(?+6(6-l)pn+6  i+2(b-\) 2  3  _2")32(b-l)pn+2  n  0  A  3  T2  a  4  _2m 2(b-l)p-n+2  n  2 3 p  16. there exist integers m > 2, £ > 2 — b and n > 0 such that p — 2 3 square and E is Q-isomorphic to one of the elliptic curves: n  a  «2  Ul  2 •3 - p^p b  l  -  n  2 3*  U2  -4 • 3 ~ V v V -  Ul'  -2 • 3 ~ p /p  U2'  b  b  1  s  4 • 3 - py/p b  1  2 3 m  2™3  -  n  -  n  e  f  2 3 m  A  4  _ m3^+2(6-l)p2  22m.+6 2t+6{b-l) pii+6  4 • 3 ( - )p"+  _2m+123<?+6(6-l)p2n+6  2  m  e  is a  m  2  b  1  2  3  _2m.3*!+2(b-l)p2  22771+632^+6(6-l)pn+6  4 • 3 ( - )p +  _2 +l 3^+ ( - )p +  2  h  1  n  2  m  2  6  b  1  2n  17. there exist integers m > 2, £ > 2 - b and t G {0,1} such that ' " 2  6  fs a  3 + 1  square and E is Q-isomorphic to one of the elliptic curves: a-2 VI  a  2.3 -y+y "' ; 2  ft  3  +i  2m.3^+2(6-l)p2t+l  V2  4 • 3 -vt+V2"'3f+i  VI'  2-3 -y y  V2'  4-3 -y+y  6  b  +  2 m  3j  + 1  2 m 3 f + 1  b  A  4  4 • 3 C>-i)p *+i 2  2  22777.+632 ?+6(b-l)p3+6( l  2777+123<?+6(b-l)p3+6t  2777.3f+2(b-l)p2t+l  22777+632£+6(6-l)p3+6t  4 •32(b-l) 2*+l  2777,+ 123<?+6(b-l)p3+6f  p  18. there exist integers rn > 2, £ > 2 - b and t £ {0,1} such that  2m3  J  1  is a  square and E is Q-isomorphic to one of the elliptic curves: a.2 WI W2 WI'  a  2-3 -y+y "' ;2  b  3  1  4-3 -y+y '" '2  b  3  2.3 -y "T 4-3 -y+y "' 'b  + 1  / 2  1  1  A  4  2m3«+2(b-l)p2f+l  _22m+632£+6(6-l)p3+6t  _4 . 3 ( b - l ) p 2 t + l  27T7.+123J?+6(b-l)p3+6t  27773('+ (b-l)p2t+l  _22m+632<?+6(b-l)p3+6t  2  2  V  W2'  b  2  3  1  -4 • 3 (<>-iy+i 2  2 + 3^+ ( - )p + m  12  6  b  1  19. there exist integers m > 2, £ > 2 - b and t 6 {0,1} such that +  3  2m  square and E is Q-isomorphic to one of the elliptic curves:  3  6t  is a  Chapter 3.  Elliptic Curves with 2-torsion and conductor  .  2  3  6 - y + y  2™+3<  X2  4-3 -y+ y '" '  XI'  2-3 -y V 7 '  i  b  b  2  2  +  4-3 -y+ / '" 6  X2'  i 2 x  131  s  2m32(b-l)p2t+l  22m+63£+6(b-l)p3+6(  2t+l  2m+1232(?+6(b-l)p3+6t  2m 2(6-l)p2t+l  22m+63l+6((b-l)p3+6t  4 . 3<+2(fc-l)p2t+l  2m+1232f+6(6-l) 3+6«  3  3  +3f  l3  A  4 . 3^+2(6-1)  + 3  a  04  0.2  XI  23 p  p  20. there exist integers m > 2, £ > 2 - b and t e {0,1} such that "* 2  is a  3  square and E is Q-isomorphic to one of the elliptic curves: a  YI Y2  2.3  -y+y^  6  2 m  2  - ' 3  p  .3 6 - y + y  2  A  4  4-3 -y+y  YI' Y2'  f t  a  2  2--3'  4 . 3 6 - y + y ^  _  m32(fa-l)  . <!+2(6-l) 2i+l  4  2  2t+l  3  p  m32(6-l)p2t+l  _4 .  3*+2(Hy[+l  _ 2m+63<M-6(6-  iy+6t  2  2771+1232^+6(6-1)^3+61  _2 + 3^+ (' - )p + * 2m  6  6  ,  1  3  6  277i+1232i?+6(6-l)p3+6t  21. there exist integers m > 2, £ > 2 — b and t € {0,1} such that  ™ is a  3  2  square and E is Q-isomorphic to one of the elliptic curves:  3 -y 4.3"-y V /"  a  A  __2"i-32(6-iyt+i  2277i+63<"+6(6-l) 3+6(  «2  ZI  2  Z2'  b  +1  y  +  Z2 ZI'  •  3  ' - / "  3f  _ .3 -y+'yte 2  4  b  . 3 6 - y + y  ^  4  4 .  3  f+2(6-l) 2t+l p  _277732(6-1)  2i+l  4 . <?+2(6-iy*+l 3  p  _27n+l 232^+6(6-l) 3+6t p  22m+63<?+6(6-iy+6t _2m+1232<?+6(6-l) 3+6t p  In the case that 6 = 2, i.e. N = 2 3 p , we furthermorexould 6  2  2  have one of the  following conditions satisfied: 22. there exist integers n > 1 and s e {0,1} such that Q-isomofphic to one of the elliptic curves:  is a square and E is  Chapter 3. Elliptic Curves with 2-torsion and conductor  AA1  2•  3 pyf " s+1  p  l3  s  A 26^3-f-6sp2n-f 6  2s+l n+2  +1  3  p  3  AA2  y  21233+65^71+6  2,+ l n+2  2633+65^271+6  . 2  21233+65^11+6  4•3 2  2•  AA1'  a  04  a-2  132  23 p  3 pyf^l±l s+1  3  AA2'  4  s +  p  3  S +  l 2 p  23. there exist integers n > 1 and s 6 {0,1} such that ~ -^ is a square and E is Q-isomorphic to one of the elliptic curves: p  3  a BB1 BB2 BB1' BB2'  2• _  _ 2 l 2  26 3+6s n+6  4 . 2s+l ?i+2  — 2 3 + - p "+  3 pyj£ff± s+1  3  . 3'+  4  2• 4.  S +  3  s+l  p  3  3  S + I p  2  3  3  6  s  2  6  26 3+6s n+6 3  4 . 2.s+l n+2  s'+^y^  p  1 2  p  _ 2  3 pyj^±l  24. there exist integers s,t  A  04  2  —2  p  p  1 2  3 + 3  6 s  p  e {0,1} such that E is Q-isomorphic  2 n  +  6  to one of the  elliptic curves: (a) p = 1 (mod 4): 04  A  _32s+l 2t+l  6 3+6s 3+6t  a  2  CC1 CC2  0 0  2  p  4•3  2 s +  y  3  p  _2l 3 +6.s 3+6t  t + i  2  3  p  (b) p = - 1 (mod 4): a DDI DD2  04  A  2s+l 2/.+ l  _26 +6. 3+C(  _4 . 2s+l 2i+l  2l2 3+6s 3+6t  2  0 0  3  3  3  p  3  5p  3  p  p  25. E is Q-isomorphic to one of the elliptic curves: a  a  A  0 0  2(6-l) 2  _ 6 6(b-l) 6  4  2  EE1 EE2  3  p  -4 • 3 < -V 2  6  2  2  3  p  12 6(b-l) 6 3  p  26. f/iere exz'sfs an integer t e {0,1} swc/z fTiaf E zs Q-isomorphic to one of the 1  elliptic curves:  Chapter 3. Elliptic Curves with 2-torsion and conductor  133  23 p a  )3  s  (a) p = 1 (mod 4): 04  A  32(6-l)p2t+l  _2636(6-l)p3+6t  .  2l236(b-l)p3+6t  a  2  FF1  0  FF2  0  _  4  3  2(b-l) 2t+l p  (b) p = - 1 (mod 4): 0-2  0,4  A  GG1  0  _32(6-l)p2t+l  26 6(6-l)p3+6t  GG2  0  4 • 32(b-l)p2t+l  _ 1236(6-l)p3+6t  3  2  27. f/zere exzsf integers n > 0 and s e {0,1} SMCTZ (AM.* g Q-isomorphic to one of the elliptic curves: 4 p  a  a  2  HH1 HH2 HH1' HH2'  2•  y+i [&^± py  4-3' + p 5  1  /4p V  "" 3  - 2 - 3 ^ ^ 4-3  s + 1  pv  / 4 P  3^  1  A  4  4 • 32s+l)pn+2  _2  1 0  3 + 3  4 • 32«+l)p"+2  —2 3 + p +  2 s + 1  V  3 J^p-  112 III' 112'  2-3 +V s  4•  3  /2lL V  ^  F ^  i  6s  2n  2 3 + p"+ 14  3  6s  " is a square  1 > n + 2  A  m 2. +y  4 . 325+1^77+2  2771+1233+65^271+6  2  3  2 3 M  6  6  22m+633+6s^n-f 6  s+1  Px  3  10  a.4  02  2•  p2n+6  2l433+6Spn+6  2 s  28. there exist integers m > 2, n > 0 and s e {0,1} swc/z rTzflt and E is Q-isomorphic to one of the elliptic curves:  III  6 s  -4 • 3 +V -4 • 3  1  is a square and E is  1  s  2 s + 1  p  2  4 . 325+1^77+2  22771+633+65^71+6 2m+1233+6s^2n+6  29. there exist integers m > 2, n > 0 and s e {0,1} szzc/z that " and E is Q-isomorphic to one of the elliptic curves: 2  P 3  ' zs fl square  Chapter 3. Elliptic Curves with 2-torsion and conductor  a 2 3 V  a-2  JJ1 JJ2  2  .  y  i  +  p  m  l3  5  A  2 3 m  4 . 3 * + ^ ^  22m-\-6^3-\-6sp7i-\-G  2 s +  325+1^+2  _4.  - 4 - 3 ^ 2 = = ^  jjr JJ2'  cc  4  / ^ £ L  y  134  23p  2 s +  2m+12  ^3+6Sp2n-ir6  22771+633+6.5^71+6  V  —4 • 3 «+lpn+2 2  2m+l2^3+6Sp2n-\-6  30. there exist integers m > 2, n > 0 and s € {0,1} such that " ' ' fs a square and E is Q-isomorphic to one of the elliptic curves: p  a  02  KK1  _  2 . 3 ^ ^  KK2  - 4 . 3 ' + ^ ^  KK1'  "2 • S-'+Vv ^ 4 • 3* V y ^  KK2'  72  7  +  2  m  2  3  3  2  A  4  + l  S  p  22m+6^3+6.Sp7i+6  2  4 . 325+1^+2  _2m+1223+6.Sp27i+6  _ rn. 2 +l 2  22771+633+6.9^71+6  4 . 32.5+1^72+2  _ 2m+1233+6.s^2n+6  3  2  p  S  31. there exist integers m > 2 and s,t G {0,1} such that ™ + fs a square and E 2  1  is Q-isomorphic to one of the elliptic curves:  a  «2  LL1  2  .  3  .  +  v  +  y 2 = ± i  LL2  _..  LL1'  - .3'+y+y™  LL2'  4.3,+y+ym  4  3  s + y +  y r^i  2  2  A  4  m 2s+lp2t+l 3  4 .  3  2s+lp2t+l  2 3 m  4 .  3  2 s +  V  t + 1  2s+lp2t+l  2  2m+6 3+6Sp3+6t  2  m+12 3+6.Sp3+6i  2  277i.+6 3+6.Sp3+6t  2  m+12 3+6.Sp3+6t  3  3  3  3  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:  a . ,+y+y2^i  a  2  MM1  2  3  _ 2m+6 3+6Sp3+6t 2  3  3  MM2  -4-3'^V+V2^  MM1'  - .3^y+y^i  MM2'  2  A  4  m 2s+lp2t+l  _4 . 2 s + l p 2 ( + l 3  2  m 2s+lp2t+l 3  2  4-3^V+V^  i  _4 . 2 s + l p 2 i + l 3  2  m + 1 2 3+6.ip3+6t 3  _ 2777+6 3+6.Sp3+6t 2  2  3  m+1233+6Sp3+64  Chapter 3.  Elliptic Curves with 2-torsion and conductor 2 3 p a  ,3  135  s  Theorem 3.28 The elliptic curves E defined over Q, of conductor 2 3 p , 7  b  and hav-  2  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 • 3 + p is a square and E is Q-isomorphic to one of the elliptic curves: e  a  0,2  Al  l  A  4  2-3 - pyj2-3 h  +p  e  n  2 • 3^+2(b-i) 2  28 2£+6(b-l) n+6  4 . 32(6-1)^+2  2l3 <?+6(b-l)p2n+6  ~4-3 "Vv 2-3*+p  Al'  -2 • 3 ~ py/2 • 3 + p  A2'  4 - 3 " p v 2 - 3 +p  4 . 2(b-l) n+2  2 - 3 - V \ / 2 - 3 ^ +p  2(b-l) n+2  Bl  6  /  b  l  e  1  /  2•3  n  6  -A-3 - p^j2-3  Bl'  -2-3 - p^/2-3  b  l  b  l  n  n  4-3 - p\/ 1  2  3 €  ( " )p b  1  n  2  832t!+6(b-l)pn+6  2l33>:+6(6-l)p2n+6 2  p  73^+6(b-l) 2n+6 p  8 2•( b3< < -V -l) n+2  2l432^+6(b-l) n+6  8 • 3'  2l4 2J!+6(b-l) n+6  3  +P  p  3  2  p  +2  +p  e  b  3  +p  l  f+2  3  n  B2 B2'  n  n  f  3  p  A2  b  n  6  p  2  p  + 2  ( - y b  i  73<?+6(b-l) 2n+6 p  3  p  2. there exist integers £ > 2 — b and n > 0 such that 2 • 3 — p is a square and E is Q-isomorphic to one of the elliptic curves: n  CI C2  -4-3 - pV 2-3^-p™  cr  ~2-3 - py/2-3  2 • 3 " p\/2 • 3 1  b  b  -p  e  1  n  -p  e  n  C2'  4 - 3 - p v 2 - 3^-p™  DI  2-3 " pV 2-3^-p  b  1  2  /  }  A  a.4  «2 b  /  _ 2  .  3^+2(6-l)p2  D2  1  /  - 4 • 3 - py 2 b  1  /  3  3  D2'  4-3 " pV 2-3^-p  b  b  1  /  n  8  p  8 • 3*+ ( "V 2 b  _2(b-l)pn+2 8 • ^+ ( ~ )p  M + 6 ( b _ 1 )  + 6  133f!+6(b-l)p2n+6  2  p  p  n  -2-3 -V\/2-3*-p"  2l33*?+6(b-l)p2n+6 -2 3 p™  p  . «+2(6-l) 2  _32(b-l) n+2  •3 - p  DI'  2  - 4 . 2(b-l) n+2  r l  e  _283 £+6(6-l)pn+6  . 2(b-l) n+2  4  3  b  (  7 f+6(b-l)p2n+6  2  3  _ 14 2«+6(b-l)pn+6 2 2  3  7 «+6(6-l) 3  2n+6  3  2  3  b  1  2  _2l42t:+6(b-l)pn+6 3  3. there exist integers £ > 2 — b and n > 0 such that 2p + 3 is a square and E is Q-isomorphic to one of the elliptic curves: n  Chapter 3.  Elliptic Curves with 2-torsion and conductor 2 "3 p c  a El  1  e  E2  -4-3 - pv 2p  El'  -2-3 - p^/2p h  1  /  1  n  n  4 . 3*+2(b-l) 2  +3  2  4 • 3 - p /2p" + 3  FI  2-3 - pv 2p fe  1  /  +3  FI'  - 2 • 3 - py/2p  n  -4-3 - p /2p 1  3  1  1  /  2  14 <i+6(b-l)p27i+6  p  2  n  2  2  p  8 • 3 ( " )p"+ 2  n  b  1  2  732<5+6(6-l) n+6 p  3  3«+2(b-l) 2  e  4- 3 - p v 2 p + 3^ b  1  83t5+6(6-l)p2n+6  1332«+6(b~l) n+6  p  b  p  2  £+2(6-l) 2  8 • 3 ( - )p +  +3  n  2  p  2  e  y  b  f  +3  n  p  3  p  v  b  3  83<'+6(b-l) 2n+6  13 2f+6(6-l) n+6  2  . 2(6-l) n+2  4 . 3«+2(b-l) 2  e  F2 F2'  p  e  1  2  p  + 3^  E2'  b  A  2 • 32(b~l) n+2  +3  n  b  136  5  CI4  2  2-3 - ps/2p b  l3  2  732<?+6(b-l) n+6 p  143<!+6(b-l) 2n+6 p  4. there exist integers £ > 2 — b and n > 0 such that 2p — 3 is a square and E is Q-isomorphic to one of the elliptic curves: n  0,4  A  . 2(6-l) n+2  _ 83«+6(b-l) 2n+6  a  2  GI G2  2•3 ~W2p -4 • 3 ~ p^2p  GI'  -2-3 " p\/2p -3  b  b  1  b  n  n  4- 3 - ps/2p  HI  2-3 ~ p /2p b  l  _ /5+2(b-l) 2  - 3*  . 2(b-l) n+2  HI'  -2 • 3 - pv 2p" - 3* 4-3 - psV2p -3  H2'  b  b  1  _4 . f+2(b-l)p'  -3  - 4 • 3>'- psj2p  n  8  3  n  p  . 2(b-l) n+2 3  p  p  p  3  13 2<f+6(b-l) n+6 3  p  7 2^+6(b-l) n+6  2  3  p  _ 14 (?+6(6-l) 2n+6 2  p  3  7 2f+6(6-l) n+6  2  p  8  1332(5+6(b-l) n+6 2  2  _3«+2(h-l) 2  e  p  _ 8 f5+6(b-l) 2n+6  2  p  3  /  1  2  3  H2  1  p  2 • 32(b-Dp«+2  e  s  2  p  e  n  3  _4 . 3 ^ + 2 ( b - l ) 2  f  - 3  n  1  2  e  1  G2'  b  - 3« - 3  n  e  3  p  _ 14 f!+6(b~l) 2n+6 2  p  3  5. there exist integers £ > 2 — b and n > 0 such that 2 + 3 p is Q-isomorphic to one of the elliptic curves: e  11  a4  2-3 ~ pv 2 + 3 V - 4 • 3 ~ p^2 + 3 V  4. 3 < + 2 ( d - i y + 2  12  b  1  l  11'  -2-3 - pv 2 + 3 V  12'  4 • 3 - pv 2 + 3 V  Jl  2-3 - pv 2 + 3 V  b  1  b  ft  /  1  2 • 3 ( -V  4  -4-3  Jl' J2'  -2-3 - pv 2 + 3 V /  2  1332<?+6(b~l) 2n+6  2  13 2^+6(b-l) 2n+6  p  b  3  8 f?+6(b-l) n+6  2  ^+2(b-l) n+2 8• 3 ^ p p  b  1  /  p  2  3  2  3  p  14 ^+6(b-l) n+6  p  8• 3 ^ p  2  3  2  M-2(6-l) n+2 2  p  7 2(?+6(b-l) 2n+6  2  2  4-3 - pv 2 + 3 V  3  p  3  3  p  2  . <H-2(b-l) n+2  pA/2 + 3 V 1  83*M-6(b-l) n+6  p  2  /  J2  b  3  /  1  b _ 1  2. 2(t-l) 2  /  b  is a square and E  A  2  a  n  p  7 2f+6(b-l) 2n+6 14 £+6(6-l) n+6 3  2  p  3  p  Chapter 3. Elliptic Curves with 2-torsion and conductor 2 3 p a  l3  137  5  6. there exist integers £ > 2 - b and n > 0 such that 3 p -2 is a square and E is Q-isomorphic to one of the elliptic curves: e  KI K2 KI' K2' LI L2 LI' L2'  2 • 3 - p^/3 p -4-3 ~ py/3 p b  1  i  1  -2 -2  n  e  n  „ . 2(6-iy 2  4 . 3«+2(6-l)pn+2 _ . 2(b-l) 2 4 . £+2(6~l) n+2 3<?+2(b- l)pn+2  -2 • 3 - V \ / 3 V - 2 4-3 "V\/3V - 2  2  b  b  e  n  - 2 • 3 ~ p /3 p 4 • 3 - p^3 p l  i  s  b  1  i  n  p  83^+6(b-l) n+6  2  2  -2 3 ^ 7  2  p  + 6 ( b _ 1 )  2 T l + 6  143£+6(b-l) n+6  p  3  p  _ 1332<?+6(b-l)p2Ti+6  2  3<M-2(b-l)n+2 _ . 2(b-l) 2 8  p  2  2  -2 -2  n  p  3  _ 1332<?+6(b-l) 2n+6  p  -8-3 ^V  s  b  3  3  2-3 -V\/3V-2 - 4 • 3 " p /3 p -2 l  8 f+6(b-l) n+6  3  2  b  b  A  <Z4  0-2 b  n  p  p  _ 732f+6(6-l) 2n+6 2  2  p  143«+6(6-l)pn+6  7. there exist integers £ > 2 - b and n > 0 such that 3 - 2p is a square and E is Q-isomorphic to one of the elliptic curves: e  a.4  A  -2 • 3 2 ( 6 - 1 ) ^ + 2  283^+6(6-l) 2n+6  4 . 3«+2(b-l) 2  _ 1332 »+6(b-l) 7i+6  0.2  Ml M2  2-3 ~ p^3 b  1  -2p  e  n  Ml' M2'  -4-3 "V\/3^-2p" - 2 • 3 ~ py 3 - 2p 4 • 3 -^p /3 - 2p  - 2 • 32(b-l) n+2  NI N2  2 • 3 " W 3 * - 2p -4-3 - p\/3' -2p  3«+2(b-l) 2 -8 • 32(6-l) "+2  NI' N2'  -2-3 -V\/3' -2p  b  b  1  /  b  e  i  n  y  b  b  p  n  !  p  <  2  283^+6(b-l) 2n+6 p  _2l332«+6(b-l) n+6  p  _2 3 ^+6(b-l 7  2  ?  n  4 • 3 "V\/3 - 2p  3  b  £  n  «+2(b-l) 2 p  _ . 2(b-l) n+2 8 3  p  ) n+6 p  2l4 «+6(b-l) 2n+6 3  b  p  p  p  n  p  p  4 . 3<!+2(6-l) 2  n  1  n  p  _ 2732*5+6(6-l) n+6 p  2  14 (!+6(b-l) 2n+6 3  p  8. there exist integers £ > 2 - b and n > 0 such that p — 2 • 3 is a square and E is Q-isomorphic to one of the elliptic curves: n  Chapter 3. Elliptic Curves with 2-torsion and conductor  Ol 02  or 02' PI  n  - 4 • 3 -V\/p - 2 • 3 n  b  4 •3 ~ b  1  -2-3'  /  W V  -2-3'  2 • S^VvV - 2 - 3 *  P2  - 4 • 3 ~ py/p  pr  -2 • 3 - p^p  P2'  e  • 3 - pV P"  -2  b  h  1  l  s  A  4 •3 " V v V b  3  -2  • S^ ^" ^ 2  1  2  2  832^+6(b-l)pn+6  4 • 32(ft-i)p«+2  _2l33^+6(b-l) 2n+6  2(b-l) n+2  _273«+6(b-l)p2n+6  3  p  - 8 • 3<+2(6-l)p2 32(6-1)^71+2  p  2  1432<;+6(b-l)pn+6  _273^+6(6-l) 2n+6 p  2l4 32£+6(b-l)pn+6  ~2-3  1  p  _2l33«+6(b-l)p2n+6  p  e  - 2•3  n  2832t?+6(b-l) n+6  p  4 . 2(c.-l) n+2  e  -2-3  n  • 3^+2(b-D 2  -2  2-3 - p\/p -2-3* 1  6  l3  CI4  «2 b  138  2°3 p  i  _ . ^+2(b-iy 8 3  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  A  2 . 3^+2(6-1) 2t+l  2832^+6(6-1)^3+64  a-2 Ql  Q2  2.3 -V+ h  1  4  / ^±i 2  v  -4.3 -y+y2^i b  Ql'  Q2' Rl  R2 Rl' R2'  2133^+6(6-1)  4 • 32(6-l)p2t+l  . b-y+1^2.3^+1 4-3 -y+y ' ' 2  3  2 3 +1  b  2-3 V '-;; 4-3-y -y+y b  2•  b  +  2 2 3 3 +1+i  3 -y+y b  2_3i±i  4-3 -y+y ^ti b  2  2 . 3<?+2(b-l) 2t+l  2832/5+6(6-l) 3+6t  p  p  4 . 2(b-l) 2t+l 3  3  p  2  32(6-1)  p  2t+l  3 . €+2(6-l) 2t+l 3  p  p  p  p  3  133«+6(6-l) 3+6t 273(5+6(6-l) 3+6t  2(6-l) 2t+l  g . *+2(b-l) 2t+l  2  1432i?+6(6-l)  3+6t  273(5+6(6- l ) 3 + 6 t p  2l432^+6(6-l) 3+6t p  10. there exist integers £ > 2 - b and t e {0,1} such that '  2 3  E is Q-isomorphic to one of the elliptic curves:  3+6t  1 p  is a square and  Chapter 3.  Elliptic Curves with 2-torsion and conductor  a  a  2  SI  11.  h  1  t+1  2  3  1  S2  4-3 -y+  SI'  2-3 -y+y - ;-  1  / ' 2  v  2  b  S2'  4-3 -y+  i  TI  2-3 -y+  i  b  b  3 C  -  1  3  v  / - '-  x  / ' ;-  2  3  2  3  4-3 ~y+ b  i v  /  3  - '-  2 3  3  '-  2 . 3^+2(6-1)  2l3 «+6(6-l) 3+6t 3  2t+l  3  1  2  273*5+6(6-1)^3+64  1  p  _2l432t5+6(6-l) 3+6t p  _ 2(6-l)p2i+l 3  1  g . 3*5+2(6- l ) 2 4 + l p  p  133«!+6(6-l)p3+6t  2  p  8 . 3^+2(6-1)^24+1  2  73«+6(6-l) 3+6( p  _ 1432(5+6(6-l)p3+6t 2  there exist integers £ > 2 — b and t G {0,1} such that y-  is a square and E  2s  is Q-isomorphic  to one of the elliptic curves:  a Ul  2-3 -y+y^ - y + y 2 ± ^  - 4 - 3  Ul'  -2.3 -y+y^  U2'  4 . 3 6 - y + y ^  b  2  .  3  6 - y + y 2 + ^  _  VI'  - 2 . 3  4  . 3  b  b  - y + y 2 ± ^ - y + y 2 + ^  4-3 -y+y^ b  A  4  . 2(6-l) 2t+l  2  3  283»?+6(6-l) 3+6t  p  . 3<5+2(b-l) 2t+l  4  b  V2  V2'  a  2  b  U2  VI  p  _ 832£+6(6-l) 3+6t  . 2(6-l) 2t+l  4  3  T2'  / 2 v  _2832f5+6(b-l)p3+6t  _ 2(b-l) 2l+l  2-3 -y+  i  2t+l p  1  TI'  b  5  A  _ 4 . 32(fr-l) 2t+l  _  4-3 -y+y - ;2  1  1  T2  b  /3  p  1  6  a  4  2 . 3<!+2(b-l)  2-3 ~ p J - '\J  139  23 p  p  p  21332*5+6(6-l) 3+6t p  . 2(6-l) 2t+l  2  3  . 3«+2(6-l) 2t+l  4  2  p  p  p  2  p  732*5+6(b-l) 3+6t p  2l4 «+6(b-l) 3+6t  <5+2(6-l) 2t+l  27 2*5+6(6-l) 3+6t  . 2(b-l) 2t+l  2l4 *5+6(b-l) 3+6t  )p  p  3  p  22. there exist integers £ > 2 — b and t 6 {0,1} such that is Q-isomorphic  3  2t+l  g. 32(6-1  8  p  2l3 2i5+6(b-l) 3+6t  3*5+2(6-1 ) 2 t + l  3  83f5+6(b-l) 3+6t  to one of the elliptic curves:  3  3  p  p  3  p  is a square and E  Chapter 3.  Elliptic Curves with 2-torsion and conductor  a  a.2 WI  2-3 -y+y^  f,  i  b  - y + y ^  4 . 3<'+2(f,-l)p2(+l  wr  - 2 . 3  b  - v + y ^  - 2 - 32(6-iyt+i  _2l332£+6(t>-l)p3+6t  2  XI  2-3 -y+ / ^  3*!+2(b-3)p2t + l  X2  -4.3 -y+y2f^  _ . 2(6-iy +l  XI'  - 2 . 3  i  3  v  b  b  8  3  -2 3 7  t  2  3*M-2(b-iyt+l  - y + y ^ 3  8  7  2  2  y  + 6 i  143«+6(6-iy+6t 3  3  In the case that 6 = 2, i.e. N — 2 3 p ,  2 £ + 6 ( b _ i  _27 2<+6(b-iy+6t  _ . 2(6-iyt+i  4-3 -y+y ^ b  3  83«!+6(6-iy+6t  _2l3 2)?+6(b-l)p3+6t  4 . 3^+2(6-1) 2t+l b  s  283*?+6(6-iy+6t  t+i  - 4 . 3  W2'  l3  A  2  W2  X2'  a  4  -2-3 < - y  6  140  23 p  2  143*!+6(b-iy+6t  we furthermore could have one of the  following conditions satisfied: 13. there exist integers n > 0 and s 6 {0,1} such that Q-isomorphic to one of the elliptic curves: a YI Y2 YI'  a  2  2-3- +V ir 4-3 +v / ; s  /2j  2-3  i  v  s  2p  +l  v  2-3 + p s  1  Y2'  4-3 +V  V  ZI  2• 3  V  s  s + 1  p  V  /2P  /2j  3  lf  / 2 P  3  +1  + 1  ZI'  2.3 + pV ^  Z2'  4• 3 * + ^ ^  s  1  / 2 P  3  /2j  y  + 2  p  2  21333+65^71+6  n+2  2833+6^277+6  p  2 s  4.3- +V  V  2s+1  2s+1  + 1  ±i  3  2 a +  8•3 3  p  y  2  5 +  833+6Sp2n+6  2  + 2  733+6s n+6 p  2l433+6.Sp2n+6 2733+6s 7i+6 p  2  8 •3 - y  2  21333+6.^71+6  2  y  2 s +  2 s + 1  is a square and E is  A  4-3 +>  ±i  Z2  5  2-3  + 1 3  4  2 s +  4-3  2 p  + 2  14. there exist integers n > 0 and s £ {0,1} such that Q-isomorphic to one of the elliptic curves:  2l433+6s 2n+6 p  is a square and E is  Chapter 3.  141  Elliptic Curves with 2-torsion and conductor 2 3 p a  a 2 •  A  2 • 3 +y  2833+6s n+6  AA2  -4 •  AA1'  -2-3'^py/^  pT,  2  3^py/^  2•  BB1 BB2  s+1  4 • 3'  AA2'  2s  3 pyJ +  + 1 P y  4 .  BB2'  2833+65^71+6  2  1333+6.Sp2n+6  2  p  32.s+lpn+2  27 3+6Sp2n+6  8 • 3 +y  2l433+6s 7i+6  3 pyj^  32s+l n+2  2733+6Sp2n+6  3 ^ ^  g. 2,+ l 2  21433+65^71+6  s+l  3  2s  s+l  4 •  3  4 . 32.s+l n+2  3 pyl^±l  2•  2l3 3+6Sp2n+6  2s+lpn+2  3  2s+1  - 4 - 3 ^ p ^  BB1'  p  2• 3 p J^  s  04  2  AA1  f3  p  p  3  p  15. there exist integers n > 0 and s 6 {0,1} such that ^-g-- is a square and E is Q-isomorphic to one of the elliptic curves: 2  a CC1 CC2  2  .  3 ' + ^ ^  -4.3'+V^  - 2 - 3  2  CC2'  4  DDI  2  .  .  3  ^  ^  3 ' + ^ ^  DD2  -4 • 3 * + i / £ £ 2  DDI'  -2 " 3 ' + ^ ^  p >  4.  3-^py/^  s  +  2833+65^77+6  V  4 . 32.S+I n + 2  -2 • 3  CO'  DD2'  A  0,4  2  2 s +  — 2  -8  — 2  p  2  + 1  p  2 s +  3  +  C  s  p  2  n  +  6  7  3  3  +  6  s  p  2  n  +  6  21433+6.5^71+6  2  32s+lp7,.+2  -8 • 3  3  _2l333+6.Sp2n+6  2,+ l n+2  • 3 "  3  2833+6.5^71+6  V  4 . g2s+lpn+2  3  1  _2733+6.s 2n+6 p  2l433+6.Sp7i+6  V  Theorem 3.29 The elliptic curves E defined over Q, of conductor 2 3 p , 8  b  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 I > 2 — b and n > 0 such that '~ Q-isomorphic to one of the elliptic curves: 3tp  1  2  is a square and E is  Chapter 3.  Elliptic Curves with 2-torsion and conductor  ai  a  2  A2  A  . 3^+2(6-l) n+2  _ 93 «+6(6-l) 2n+6 2  p  8  -4 • 3»~ V25^=1  3  2  p  2 . 3^+2(6-l) n+2  4-3"-V / V'3  v  2  8 • s^py/ '";-  B2  3  fc  p  - 4 . 3 ' - ^ ^  _ . 2(6-l) 2  B2'  8 . 3 6 - ^ 3 ^  g . 3«+2(fe-l) 77+2  3  cr  D2 DI' D2'  8 • 3 "V^/ + " b  3f  p  "4 • 3 " b  1532f5+6(6-l) 2n+6 p  zs a square and E is  p  2  215 3^+6(6-1) 2n+6  b  2 • 3<+2(6-Dp2  29 2«+6(6-l) n+6  8 • 32(6-l) n+2  2l53£+6(6-l) 2n+6  2  1  3  p  A  - 4 - 3 ^ ^  8 • 3 - pyJ ' /' 3 +  2  3  8 • 3 - p / '+ "  6  2  8 • 3 ( -!)p™+ 2  +  93€+C(b-l)pn+6  2932<+6(6-l) n+G  2 • C>-i)p"+  4-3-Vv^  2 n  2  p  1  +  b  2  3  h  3  6  A  2 • ^+ C'-i)p  b  _  2  2  04  2  C2' DI  5  p  CI C2  _2i 3 ^+ ( -i)p  p  2. there exist integers £ > 2 — b and n > 0 such that Q-isomorphic to one of the elliptic curves: a  p  293(?+6(b-l) n+6  Bl'  2  p  p  g . 3^+2(6-1)^71+2  1  3  2l53£+6(6-l) n+6  6  -2 • 3 ( -V  1  15 £+6(6-l) n+6  3  -8 • 3 < " V 2  p  _29 2»!+6(6-l) 2n+6  p  A2' Bl  s  2  _ . 2(6-l) 2  Al'  /3  04  Al  142  23 p  2  ?  8 • 3*+V  p  p  293^+6(6-1)  2n+6  2l532(5+6(b-l) 7i+6  8 • 3«+V 2 • 3 C'-i) ,n+  3  p  2  293<5+6(6-l) 2n+6 p  2l532f+6(6-l)  Of _  n+6  T]  3. f/tere exzsf integers £ > 2 — b and n > 0 swc/z tTzzzf — j - is a square and E is 2  Q-isomorphic to one of the elliptic curves:  Chapter 3. Elliptic Curves with 2-torsion and conductor  _2932<?+6(6-l)p7i+6  b  2  2-3' ( - y  E2'  8 - 3 " - ^ ^  -8 • 32(h-i) «+2  Fl  4•3"  F2  -8-3-Vv ^  F2'  t  p  _23^+( -)p+  - 4 . 3 ^ ^  + 2  2l53<;+6(b-l) 2n+6 9  El'  Fl'  143  5  2. <+2(fc-iy  -8 • 32( -!)p«+ i  2  6 fe 1  7  -4.3-Vy ^ 8-3 "Vx/^ 7  b  - 2 • 32(b-l) «+2 p  8•  2  2  -2-3 ( - V+ 2  8• 3  fc  e  +  1  ^p  2  93^+6(b-l)p2n+6  _2l532<?+6(b-l)pn+6  3 V>-Vp e+2  6  p  3  Vv ^  n  2l5 £+6(b-l) 2n+6  p  7  b  /3  A  3  E2  a  (24  a-2  El  23 p  2  293<J+6(b-l)p2n+6  _2l532*?+6(b-l)pn+6  2  •n _o£  4. there exist integers I > 2 — 6 and n > 0 such that — — is a square and E is Q-isomorphic to one of the elliptic curves: p  2  a-2  GI  4 . 3 ^ / ^  G2  _ .3 -v7^  GI'  _ .3 -V  . 2(b-l) n+2  2 3  b  8  b  4  / V  ^  G2'  8-3 - pV ' T  HI  4-3 -V ";  H2  -s^-Vy ^  HI'  -4 • 3>"  H2'  fc  1  / E:  b  /p  2£  3f  7  Vv ^ 7 2  _293«+6(b-l)p2n+6  p  2^2,2t+fi(b-\) n+&  _ .3£ 2(b-l) 2 +  8  p  p  . 2(b-l) n+2  _29 e+&(b-l) p2n+6  _ . £+2(b-iy  2l532/i+6(b-l)p7a+6  - 2 • 3< < -V-  2932f+6(b-l)pn+6  8 • 3 C>-iy+2  _ 153< +6(b-l)p2n+6  2 3  8  V  S-^PV  A  04  3  p  3  +2  6  2  ,  2  7  -2 • 3* ( -V  2932/5+6(b-l) n+6  ^  8 • 3 ( -V+  _2l53*?+6(b-l) 2n+6  +2  2 fe  6  2  p  5. i7zere exist integers £ > 2 — b and t e {0,1} such that is Q-isomorphic to one of the elliptic curves:  p  z s a  square and E  Chapter 3.  Elliptic Curves with 2-torsion and conductor 2 3 p' Q  a  /3  A  0 4  2  144  5  11  2 . 3<5+2(b~l)p2t+l  2932f+6(b-l) 3+6t  12  8 • ( - yt+i  2l5 f+6(b-l) 3+6(  2 . f?+2(b-l) 2i+l p  2932«+6(b-l) 3+6t  2 t + 1  2  11'  2  f e  i  3  -4-3'-V+V3gi  12'  .  8  3  b  -y  +  l  v  3  /3itl  8 •3 ^p 2  4-3 -y+y2i±i b  Jl _  Jl'  -^-y+y^  J2'  s-^-y+y ^  8  . 3  - y  J2  b  2  y 3 i t i  +  p  +  3  2  g  3  . f+2(6-l) 2t+l  g  3  . 2(b-l) 2t l p  . 2(b-l) 2t+l 3  p  . C+2(b-l) 2t+l 3  p  p  3  p  p  153f:+6(6-l)p3+6t  2  93«5+6(b-l) 3+6t p  2l532(?+6(b-l) 3+6t p  293<+6(b-l) 3+6t p  2l532(5+6(b-l) 3+6t p  6. there exist integers £ > 2 — b and t e {0,1} such that is Q-isomorphic to one of the elliptic curves: a  KI  4-3 -y V ^  K2  -s-^-y+y^i ^.^-y+y ^  KI' K2' LI  +  3  1  3  8  .  3  b -  y  A  04  2  b  . 3*5+2(b-l) 2t + l  2  p  2 . 3*5+2(6-l) 2t+l  _2932(5+6(b-l) 3+6(  t ,  1  2  p  -8 • 3 ( -V -2-3 ( - y 2  2  f c  b  i  t + 1  + i  L2 LI'  -4-3 -y V^  -2 • 3 (f-iyt+i  L2'  8.3 -y+V^i  3 . £+2(b-l) 2( +l  +  g  . 3^+2(6-l) 2(+l p  8  b  +  b  p  2l53*5+6(b-l) 3+6t  2  + y 3 ^ i  b  _2932f5+6(b-l) 3+64  - 8 • 3 ( - )p «+i  4-3 -y V^ _ . 3 - y + y 2^1 b  is a square and E  2  3  p  p  p  2l53^+6C>-l)p3+6t 293^+6(6-l) 3+6t p  _2i 3 5  2  2 ( !  + ( -i)p + 6  ( >  3  6 t  93«+6(b-l) 3+6( p  _2l532«+6(b-l) 3+6t p  In the case that b — 2, i.e. N = 2 3 p , we furthermore could have one of the 8  2  2  following conditions satisfied: 7. there exists an integer n > 0 such that to one of the elliptic curves:  p  g " is a square and E is Q-isomorphic 1  1  Chapter 3. Elliptic Curves with 2-torsion and conductor 2 3@p a  Ml M2  Ml'  2 •3 +y+  4 . 3«+  2 s  8-3 -4 • S'+Vy^ 2 • 3 8• 3  A  a.4  «2  2s+1  p\/^  s + 1  2s+1  M2'  8 •  3 pJ +  NI  4 •  3  N2  s+1  pn  1  8  .  3  2 , + l  p p  29 3+6Sp2n+6  2  3  2  2l533+6s n+6  n+2  2933+6Sp2n+6  p  2  p  2  15 3+6SpTi+6 3  x  3 pyJ +*  2•3 p 8•3 p  3 p J '+  2-3 p 8-3 p  2s+1  J "+  s+1  8 •  p  1  Py  s+1  2s+1  pn  NI' 8 •  s+l  p,  2s+1  i  x  2 2  n+2  9g3+6.5p77+6  2l5|T{3+6.5p2n+6  2  2933+6Spn+6  n+2  2l5g3+6.Sp2n+6  2s+1  N2'  145  s  8. there exists an integer n > 0 swcfr ffrfl/ ^-g— is a square and E is Q-isomorphic to one of the elliptic curves: 1  2• 3 p -8 • 3 +V  —2 3 p  2 • 3 +V" -8-3 p  —2 3  2s+1  Ol  9  n+2  2s  02 Ol'  - 4 - 3 ' + ^ ^  02'  8 - 3 ^ ^  PI  4-3' V +  / £ V  ^  _  PI'  - 4 - 3 ^ ^  8  .  8  3  - + I  P  N  2s  2s+1  P2  P2'  A  (I4  a-2  -2-3 8• 3  / ^  2 s + 1  2 s  p  +y  2s+1  •y+'Py/*^  2n+6  215 33+65^77+6 9  +2  3+6:s  p  2n+6  2  2l533-i-6s n+6  2  2933+6.s 7i+6  p  p  —2 3 15  + 2  -2 • 3 p 8• 3 p 2s+1  3+6s  3+6s  p  2n+6  2933+6.5p77+6  2  _2 3 + - p + 15  n+2  3  6 s  2n  6  9. there exist integers s,t € {0,1} such that E is Q-isomorphic to one of the • elliptic curves: «2  Qi  0  Q2  0  Rl  0  R2  0  A  0.4  2•3 p -8 • 3 p -2 • 3 p 8•3 p 2s+1  2;+1  2;+1  2s+1  2t+1  —2 3 9  3+6s  p  3+6t  2t+1  2l533+6sp3+6£  2t+1  2933+65^3+6*  24+1  —2 3 15  3+6s  p  3+6t  Chapter 3. Elliptic Curves with 2-torsion and conductor 2 3 p cc  3.4  f3  146  s  Proofs of 2 3 p and 2"3V a  / J  We sketch a constructive proof that the curves listed i n 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 i n 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 2 3 p . 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 w i 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 i n Chapter 2. Rather than get bogged down i n all the details of computing the conductors of the curves we w i 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. a  l3  6  In what follows we refer to the elliptic curve y = x + ax + 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.II 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. 2  v {a) 2  v (b) v (A) 2  2  A.1.I  A.l.II  n  = r i + l (if m > 1) > n + 1 (if m = 0)  m + 2r - 2 x  2m + 6ri  2n 2rj + 1  3  A.l.Ill n +2 2n + 1 6n + 9  2  Chapter 3. Elliptic Curves with 2-torsion and conductor 2 3 p a  l3  5  It follows directly from Theorem 2.1 that for curves i n A.I.I  v (N) = { 2  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  ifr  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,  x  = 0 , m = 2, a = - 1 (mod 4), b = 1 (mod 4)  147  Chapter 3. Elliptic Curves with 2-torsion and conductor 2 3 p a  /3  148  s  and for the curves in A.l.II we have '6  v (N) 2  =  /  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 3  if ri = 0, m > 4, a/2 = 1 (mod 4)  0  if rj = 0, m = 6, a/2 = - 1 (mod 4)  1  if r i  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 ri = 0, m = 4,5, a/2 = - 1 (mod 4)  if  7i -  = 0,777 > 7,  = 2, m > 1 .  a/2  =  -1  (mod  4)  6/4 = - 1 (mod 4).  A s 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 Diophantine equation which p satisfies, then, if 3 appears as a coefficient of d in DE then v (N) = 2, otherwise 2  3  {  0  if r = 0 and I = 0  1  if ra = 0 and I ^ 0  2  ifr  2  2  =l.  Similarly, we find that if p appears as a coefficient of d? in D E then v (N) = 2, p  Chapter 3. Elliptic Curves with 2-torsion and conductor  23 p a  /3  5  149  otherwise  VP(N)  if r  3  = < 1 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, i n 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 i n A . l across the appropriate theorems in Sections 3.2 and 3.3. Notice that, by taking £ = 0, we get curves of conductor 2 p . This is how the curves in the theorems of Section 3.1 were originally found, though the proof we gave there d i d not reflect this. a  2  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 i n subsequent chapters. 4.1  U s e f u l Results  In this section we collect together, for the convenience of the reader, all the results that we w i l l need to solve the Diophantine equations in the next section. The first result we will need is Catalan's Conjecture, which is now a theorem of Mihailescu [52]. In this work we w i l l refer to it as "Catalan's Theorem", or simply as "Catalan". Theorem 4.1 (Mihailescu) The only solutionlo x  n  in positive integers x,y,m,n  - y  m  with n,m  the diophantine  equation  = 1  > lis given by 3 — 2 = 1. 2  3  Some results of Cohn and Ljunggren that we w i l l make use of are the following. Theorem 4.2 (Cohn [19]) Let k be odd. The only solutions to x  2  positive integers x, y and n > 3 are k  x  y  6a + 1 4a+ 5 10a+ 5  5• 2 ° 7-2 ll-2 3  2 q  5 a + 3  150  n  3- 2 3-2 3-2  2a  a  2 a + 1  3~ 4 5  + 2  k  = y  n  in  151  Chapter 4. Diophantine Lemmata with a > 0.  Theorem 4.3 (Ljunggren [48]) The equation x — 3y = 1 has no solution in pos4  2  itive integers.  The main results that we w i l l use i n 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 equation  x + y = Cz n  n  2  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 x +3y n  a  n  = C^z  3  has no solutions in coprime integers (x, y, z) with \xy\ > 1, unless  (|x|,|y|,a,n,|Cz |) = (2,l,l 7,125) 3  I  or, possibly, (a, C) = (1, 2).  Theorem 4.6 (Bennett [3]) Suppose that a < b are positive integers with ab = 23 a  /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 ax ' — by = ± 1 are given by 7  71  The proofs of the above theorem rely heavily on results on ternary diophantine equations coming from the theory of Galois representations and modular forms. In a few cases of the proofs of the results in the next section we w i 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.  Chapter 4.  Diophantine  152  Lemmata  Consider the equation 3 x + 2 y = d with £, m, q fixed integers. We want to show this equation has no solutions for x,y, d, with \xy\ > 1, i n 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 EE 1 (mod 4). Then, as in [5], we associate to this solution the elliptic curve e  E  •Y  2  a A c  q  m  q  2  + XY = X  + —^-X  A =  2 -' 3 (ab )"  3  (  +  2  2 ~ b X, m  2  q  whose discriminant is 2m  2  e  2  and conductor is N(E )  =  aAc  2 3l[p, a  p\ab  E  where a € {—1,0}. We then associate to E a Galois representation p ' ' which is irreducible (see [5] corollary 3.1 , this is where \xy\ > 1 and q > 7 is required). The representation p " arises from a cuspidal newform of weight 2 and level N = 2 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 w i l l be considering in the next section? Well, the above result applies to the equations a b c  a A c  q  ,b,c  q  Q + 1  d = 3 V + 2 and d = 3 + 2 p m  2  2  e  m  n  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 P ;„(n) denote its smallest prime factor and P (n) denote its largest prime factor. This notation w i l l be used throughout the rest of this section. m  max  Chapter 4. Diophantine Lemmata Lemma 4.7  153  1. The solutions to d - 2 3p 2  rn  e  = 1  n  withTO,£, 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 = 3 ±2 with£>  1 and (d, m) = (p + 1,0),  e  (c) p = - =± with £ odd and (d, rn) = (4p + 1, 3), 3  2  (d) p = ^rp- with £ even and (d, m) = (4p — 1,3), (e) p=  with £ odd and (d,m) = {8p - 1,4),  (f) p = 2 - 3 m  (g)  e  = 2m  p  2  (h) p = ' " ~ ' 2  ± 1 withTO> 3 and d = 2p + 1,  ± 1 with m>hand  2  + 1  (d, £) = (2p +1,0),  wff/iTO> 5, > 1 and d = 2 ~ +1, wrtere 3* | m - 2. m  l  (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 d - 2 3p 2  zfif/i m,£,n>  rn  e  n  = -1  0 and d > 1 ynwsi /zaue ^ = 0 and satisfy one of the following 1  (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 orTO> 3, since 3 and 5 are quadratic non-residues modulo 8. If rn = 0 then the equation can be written (d+l)(d-l) = 3V. It follows d + 1 and d — 1 are coprime and so d +l =3 d-l=p  n  £  f d - 1 = 3* or < ld+l=p".  Chapter 4. Diophantine  154  Lemmata  We w i l l consider such cases many times throughout the proofs in this section so to save space we w i l l collapse them into one single pair of equations: (d±l  = 3  e  \d+l=p . n  Subtracting the two equations gives ±2 = 3 ^ - p . n  If n = 0 then £ = 1 and d = 2; if n = 1 then p = 3 T 2. Finally, if n = 2 then p = 3 =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 d — 1 = 2 3 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 e  2  e  2  m  e  (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 3 V m  where gcd (d + 1, d — 1) = 2, so one of the following three cases must hold: (d±l <  = 2-3V  [dTl  = 2m  \d±l  = 2-p  „  {d+l=3  1  \d±l  n  A  e  o  r  = 2-3  e  \  (4- ) 1  \dTl=p  n  Case 1 of (4.1): Subtracting the equations and dividing through by 2 gives 3p e  n  - 2~ m  = ±1.  2  (4.2)  Suppose 3 p — 2 ~ = — 1, then consideration modulo 3 implies m is even. Writing m — 2 = 2k the equation can be written as 3 p = (2 + l)(2 - 1), thus e  n  m  2  e  2 ± 1= 3 k  2 Tl=p  e  k  n  n  k  fc  Chapter 4. Diophantine Lemmata  155  The second equation has no solutions by Catalan's theorem. Thus (4.2) is 3 p — 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 2(m-2) ( e^ 3^-1 | _ ( f instance, [BeOO] Lemma e  2  n  m  =  2  m o d  1  a  3  n  dS  Q  2  m  s e 6 /  o r  3.1). O n the other hand, if n = 2 then £ is even (look modulo 3). The equation can then be written (3 / p) - 2 ~ = 1 £  2  2  m  2  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 p - 2 ~3 n  m  2  e  = ±1.  If n = 1 then p = 2 ~ 3 * ± 1 which is case (ii)(f) of the lemma. If n = 2 then p = 2 3 * ± 1. A simple inspection modulo 4 reveals that the negative sign cannot occur, therefore m  2  2  m _ 2  2  p  ™- 3 2  =  2  e  +l.  M o v i n g 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. N o w suppose m > 5, so the equation is 3*-2 "- p = ± l . T  (4.3)  2  The right-hand side must be positive and £ must be even by considering the equation modulo 8. Letting £ = 2k we may write (3 + l ) ( 3 - 1) = 2 " p . fc  f c  m  2  Since gcd (3 + 1,3 — 1) = 2 we are in one of the two cases: fc  fc  3  f c  ±l = 2  f 3 ± l = 2p or < f c  3 =F 1 = 2 - p fc  m  3  [3 =F 1 = 2 ~ . k  m  3  Chapter 4. Diophantine  156  Lemmata  The first case has no solutions, so we must be i n the second case. Subtracting the two equations and dividing through by 2 gives p — 2 = ± 1 , and so m _ 4  p=2 rn  ± 1 with m > 6.  4  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 2 ~ p = 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 m  2  2  m  =  - y  3 < ± i .  If m is even then the equation is (2 - ~ ^ p) — 3 = ± 1 which has no solutions by Catalan's theorem. Therefore m is odd and the equation can be written as (  m  2  2  2  e  2x - 3 = ± 1 , £  2  where x = 2^ ~ ^ p. 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: n  3  2  2x - 3 2  = -1.  e  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 p = 3* ± 1 has only the solution (p, rn, £) = (11,3,5). This completes the proof (1). m _ 2  2  2) Considering the equation d - 2 3-'p = - 1 modulo 3 implies £ = 0. Therefore, the lemma follows from Lemma 3 i n [37]. • 2  Lemma 4.8  m  n  1. The solutions to d - 2p 2  rn  n  = 3  e  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) = {3 > + 2,3), e  (b)  2  = 2f^±l wifft £/2 odd and (d, m) = {3 l + 2,4), i 2  P  (c) p = 2 ~ m  2  ± 3 l with m>3andd  = 2T  l 2  3' , i 2  P  (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 d + 3 = 2p 2  e  rn  n  with rn > 2, £ > 1 and n > 0 satisfy one of the following: (i) n — 0, m = 2 and £ = 1, (ii) n = 2,p = -^p- £ odd, m = 2, and d = 2p - 1, 3  /  (m) ?i odd, n = 1 or P (n)  > 7, p = 1 (mod 3), m — 2 and £ odd.  mm  3. The solutions to d + 2p 2  m  n  = 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 = ± ( 4 p - 3 ^ ) , 2  (c) p = 3 / - 2 - , m>3,£ £  2  m  2  even, and d = ±(2p -  3 l ), l 2  (iii) n = 2 and (a) p = 5 and (m, £, d) = (3,6,23), (b) p = 7 and (m,  d) = (7,8,17).,  (c) p = 21^=1, £ / 2 odd, m = 3,andd = ±(4p 2  2  -  3 ' ), e 2  P r o o f . 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 P (n)  > 7 then there are no solutions.  max  Write the equation as 3*(±l) + 2 V = d g  where q = P (n) max  r  2  > 7 and y = p l . This is a particular form of the equation n q  3V  + 2 y" = d . m  2  Let E j, be the elliptic curve associated to a solution (x, y, d) = (a, b, c) of this equation as described i n case (v) of [5]. It follows from Lemma 3.3 in [5] that E ^ has conductor a  a  fi  c  N(E )  I 3»  if m = 6  [6p  if m > 7.  = \  aAc  F  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±3  =2  k  [dT 3 = fc  fd±3  = 2p  \dT  3 = 2™- .  k  2 "V m  n  fc  1  Suppose we are i n the first case. Subtracting the two equations and dividing through by 2 gives 2 - V ± 3 = 1. m  fc  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 + 3 (l) = 2 - ( - l ) , t  where t = P . {n) ma x  fc  i  and x = ±p / . n  m  2  3  By Theorem 1.5 in [6] it follows that k =  t  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 P (n)  = 5 then there are no solutions.  max  Recall £ = 2 by assertion 3 so the equation can be written as d — 2™ x = 9 where x = p / . Letting 0 < a < 4 be the residue of m modulo 5 we may write 3  n  5  b  (2 d) = ( 2 ^ x ) 2m  2  If a > 3 we may scale through by 2 ~  1 0 b  5  + 2 "9. 4  where 4a = 106 + r, and 0 < r < 9 is  even, to get 2 ™d\ 2  f2 ^x\  2  m  l  5b  2  5  2 9. r  2  I  2b  The point is that solutions to our original equation correspond to rational points of a particular form on a genus 2 hyperelliptic curve: Y = X + 2 9, 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. 2  Assertion 6: If m > 2 and P , (n) ma x  5  2s  = 3 then the only solution is (p, n, ra, £) =  (5,3,9,2). In this case the equation can be written as d = 2 x + 3^ where x = p / . 2  m  3  n  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 _ 2("'+ «)/ x ~ *-6)/3 2  v X  3(  4  A a  n  d  n D  _ ~  2 d Q  Chapter 4.  Diophantine  160  Lemmata  the equation becomes D  2  = X  3  where 2a, b € {0,2,4}.  + 2 S 2a  b  We are only interested in solutions where X and D are of the form above, i n particular (X, D) is to be a {3, oo}-integral solution. The table i n 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 = 2 where a > 1 then p = 5 and (m, I, d) = (6,4,41). a  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 + 2 p ){d-2 p ) s  t  s  = 3. e  t  It follows that d + 2 p* = 3 s  d-2 p s  e  = 1,  t  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: 2 S +V = (3 + l)(3 - 1). fc  fc  It follows that 3 + l = 2V fe  3  f c  -l = 2  . f 3 ± l = 2p or < [3 Tl = 2 . f c  f c  i  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 second 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 i n the lemma. Since £ is even (assertion 2) the equation can be written as (d + 3 ) ( d - 3 ) = 2 p, fc  fc  m  where £ = 2k. It follows that fd±3  = 2p  k  \ciT3  f c  (d-3  = 2 m  =2  k  \ci+ 3 = 2 - V -  1  fe  m  Eliminating d in the first case gives ± 3 = p - 2 ~ . Thus p = 2 "~ ± 3 . In the second case eliminating d gives — 3 = 1 — 2 ~ p, that is fc  rn  k  2  m  7  2  fc  2  2 '- p = 3 + l . m  2  fc  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 d = 2 2  m  + 3 which only has the e  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 ) ( d - 2p ) = s  2  -3 . e  As we've done many times before in these arguments we eliminate d: 4p = 3 + l. s  e  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 d = 4x 2  where x = p l . n 3  -  3  3, e  Changing variables we may write this as —^  3  - 2 • 3" 4  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\  _ [4x\  2  5  _  8  a  where x = p l and a G {1,3, 5, 7, 9}. Again, there are no solutions of the desired form to these hyperelliptic curves as shown in Chapter 5. This completes the proof of (2). 3) If n = 0 then the equation is n b  d = 3 2  e  2. m  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)}.  A s 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 p > 1 + 10 = 11. Furthermore, since m > 2 then considering the equation modulo 4 implies £ even, and further considerations modulo 8 imply m > 3. 2  Assertion: If m > 2 and P (n) solutions. max  rn  n  > 3 then the equation d + 2 p 2  m  n  = 3 has no e  Chapter 4. Diophantine  Lemmata  163  Since £ is even we may factor the equation (d-3 )(d  + 3) =  k  -2"y\  k  where £ = 2k. This leads to the two cases fd±3  = ±2  k  fd±3  \d -f 3 = +2 ~ p k  =  k  m  x  \d =F 3  n  ±2p  n  = -f 2  fc  m _ 1  .  Eliminating d i n the first case gives fc  _ m-2 n  3  2  p  =  1 (  and by Theorem 1.2 in [3] there are no solutions, since P (n) ing d in the second case gives max  3 k  =  n p  > 3. Eliminat-  2 ~. m  2  If Pma.x(n) > 5 then by Theorem 1.5 in [6] it follows that k = 1, i.e. ^ = 2, which contradicts £ > 3. N o w assume P (n) = 3. We return to the original equation and write it as mBX  d  = 2x  2  m  + 3,  3  e  where x = p ' / . The solutions to this equation correspond to the rational points on the elliptic curves r  3  Y  =X  2  + 2 3  3  2 a  6  of the form ya+spn/'A X  =  2d a  — - 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 = 2 for a > 0. Factoring the equation, as we d i d i n the proof of the assertion above, we are i n one of two a  cases: fc _ ™ - y -  3  2  =  i  o  r  fc _ n  3  p  2 "- , T  =  2  (4.4)  Chapter 4. Diophantine  164  Lemmata  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 + l ) ( 3 - l ) = 2 - V , s  s  m  where k — 2s, i.e s = £/4. We are in one of two cases: f3" + l = 2  m _ 3  p  f3 ±l = 2 "  n  s  [3 -1 = 2  m  3  \ 3 = F l = 2p . s  s  n  The first case clearly has no solutions (under our assumptions on p and n). Eliminating 3 in the second case gives s  ±1  2~ m  =  -p ,  4  n  from which it follows from Catalan's Theorem that n = 1, and so p = 2 ~ ± 1 . It follows from assumptions on p that rn > 6. Therefore, the equation in (4.4) becomes m  3 - 2 ~ p= m  fc  4  1,  2  where m > 6 and 4 | k (look modulo 16). Considering the equation modulo 5 implies 2 ~ p = 0 (mod 5), thus p = 5 and we are in case (ii)(c) of the lemma. N o w 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 — p = 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 m  fc  2  n  (3'+p )(3 -p ) = 2 " , J  s  s  m  2  where k = 2t and n = 2s. It follows that J3  4  +p* = 2 ~ m  \3* - p  5  = 2,  and so by eliminating p we get 3 — 2 s  f  3  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  Lemma 4.9  165  Lemmata  1. The solutions to d - 23 2  rn  e  = p  n  with m > 2, £ > 0 and n > 1 satisfy one of the following: (i) n — 2 and  (a) p = 2 " 3 - 1, m > 3, and d = p + 2, m  2  f  (b) p = 3 - 2 - , m>3,andd e  m  (c) p = 2 ~ m  -3 ,m>  2  = 2~  l  + p,  5, and d = 2 ~  - p,  2  e  m  rn  1  (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) (iv) n = 6, and (p,m,£, (v) P„n„(n)  <E {(5,3,3, 29), (7, 7,4,13)}, d) — (5, 9,1,131),  >7,m<5and£>l,  (vi) n = 1. 2. TTze solutions to d +23 2  m  e  = p"  with m > 1, £ > 0 and n > 1 satisfy one of the following: (i) n = 2 and (a) p = 2 ~ 3 m  2  + l,m>3,andd  e  (b) p = 3 + 2 - , e  m  (ii) n = 3and(p,m,£,d)  2  =  p-2,  m > 3, and d = p - 2 • 3 , e  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) P {n) mm  > 7 andm < 5,  (y) n = 1. 3. The solutions to d +p 2  n  = 23 rn  e  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) P \ (ri) m n  > 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 P (n)  > 7 then there are no solutions.  max  Write the equation as ™tf(l)1  + y<I  2  where q = P (n) max  > 7 and y = p l .  d  2  =  This is a particular form of the equation  n q  2 3 V + y" = d . m  2  Let E, 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 E ,b,c h conductor lAc  a s  a  N(E )  f 3»  if m = 6  [6p  if  = \  aAc  "  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. A p p l y i n g the change of variables n/3  p  X =  22s 32* n  and D  j 2 3 ' 3 s  3 i  Chapter 4. Diophantine  167  Lemmata  to the equation d = p + 2 3* where m = 6s + a and £ = 6t + b with 0 < a, b < 5, gives the equation 2  n  m  D =X 2  +23 .  3  a  b  The rational points of the desired form on these elliptic curves can be found by inspection of the tables i n 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)}. These pull back to the following solutions of the original equation: (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 d = p + 2 3* with 5 | n correspond to rational points on D = X + 2"3 , 0 < a, b < 9, with x coordinate of the form 2  2  b  n  m  fe  X = 2  2.  5 3  2t •  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 i n the lemma. Considering the equation d = 2 3 + p modulo 8 implies m > 3. Since n is even we may factor the equation as 2  m  (d + p )(d k  e  n  - ) k  P  = 2 3' m  where n — 2k. It follows that we are in one of two cases  Chapter 4. Diophantine  Lemmata  168  Eliminating d i n the first case gives 2 ~ 3 — p = 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 3 — 1. In the second case, eliminating d gives m  m _ 2  3 zf e  2  k  e  , ?  = 2 -.  k  m  P  (4.5)  2  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 P (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 ( n ) > 5. So we must have that k = 2 , a e N . If k = 1, i.e. n = 2, then we are are in case (i) of the lemma. So suppose k = 2 > 1. If the equation in (4.5) is 3^ + p = 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 — p = 2 ~ . If m = 3 then the equation is 3 — p = 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: max  a  ma31  a  m  k  2  e  e  m  k  2  k  ( '/ +p / )(3'/ -p / ) = 2 2  f c  2  2  f c  2  m  3  2 )  from which it follows that _  3^/2 _|_ pk/2 3  */2_  p f e  /2  2'"-3  =  2  ;  and so, eliminating p / we get 3 ^ — 2 ~ = 1. By Catalan's Theorem (m,£) £ {(5,2). (7 4)}, only one of which pulls back to a solution of the original equation; fe  2  2  m  4  ;  (p,n,m £-d) f  = (7,4,7,4,113).  This proves assertion 4. Finally, if n is odd it clearly follows from the equation d? — 2 3 = p 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 (n) > 7. The case when £ = 0 is rn  m a x  e  n  Chapter 4. Diophantine  Lemmata  169  treated i n [37] so we may assume £ > 1. The solutions with 3 | n correspond to points on the elliptic curves y = x - 2 2> 2  3  a  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 i n part (ii) of the lemma. The solutions with 5 | n correspond to points on the hyperelliptic curves y =x 2  5  23 a  b  of the form  z^' 2  x =  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 i n the lemma. Since n is even we may factor the equation as {d + p )(d-p ) k  =  k  -2 3 , rn  e  where n = 2k, it follows that we are i n one of the two cases d-  = -2  k p  \d±p  d +p = 2 ~3, k  m  = 2-3  k  1  e  or <  \d + p = 2 ~K  e  k  m  Eliminating d in the first case gives p — 2 ~ 3 = 1 which has no solutions for k > 3 by [3], thus k = 1 or 2. If k = 1 then p = 2 ~ 3 + 1 and we are in case (ii)(a) of the lemma. If k = 2 then we may factor the equation as k  m  2  i  m  (p+l)(p-l)  =  2 -3, m  2  e  2  e  Chapter 4. Diophantine  170  Lemmata  where, as usual, eliminating p gives ± 1 = 3 - 2™ , so e  (m,£)  4  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. Suppose P (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 > 1 then considering the equation p = 3 + 2 ~ modulo 3 implies rn > 4 and even, and modulo 4 implies £ is even. A s usual, factoring and applying Catalan gives (p, n, m, £) = (5,4, 6, 2). This proves assertion 5. max  a  k  e  m  2  Finally, if n is odd and (n, 15) = 1 then either n = 1 or P (n) > 7 (and m < 5 as we've argued using [5] many times before). This proves (2). 3) A s we've seen in (1) and (2) there are no solutions with both m > 6 and P ( n ) > 7. The solutions with 3 | n correspond to rational points on the elliptic curves y = x + 2 3 of the form max  m a x  2  3  a  b  X  =  23' 2s  2t  and these can be determined by consulting the tables i n Appendix B. Similarly, the solutions with 5 | n correspond to rational points on the hyperelliptic curves y = x + 2 3 of the form 2  5  a  b  X  =  2 3 ' 2 s  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  Lemma 4.10  171  Lemmata  1. The solutions to d - 2 2  V  = 3  m  with m > 1,£ > 0 and n > 1 satisfy one of the following: (i) n = 1 and (a) p  2 m / 2 =  ^ + \ £>i,d  (W P = 3* ± 2 / m  2 + 1  = 2 - / + 1, 2  , * > 1, d = p ± 2 / , m  2  (c) £ = 0,m odd. (ii) n = 2 and W (p,m,£,d) p = 2 m  € {(5,6, 2,17), (7,8,4,65)}, 2  - 1,TO> h,£ = 0 and d = p + 2,  (iii) n = 3and (p,m,£,d) = (17,7,0,71). 2. The solutions to 2 _ m  d  2  =  _en s  p  with m, £ > 0 and n > 1 satisfy one of the following: (i) n = 1 and («) (p,m,£,d) (W p = 2 / m  e {(5,6,1,7), (7,10, 2, 31)}, 2 + 1  - 3*, m > 4 even, £ > 1, and d = ±(2 /  m 2  (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 d +2 2  m  = 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 = ^"V  1  i m l m > 3 odd, and d = 3 / p - 2, £  2  - p),  Chapter 4. Diophantine  172  Lemmata  (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 i n [37] so we only need to consider £ > 1. In this case it follows that m is even. A s i n the proofs of the other lemmata it follows from [5] there are no solutions with both P (n) > 7 and m > 6. Also, there are no solutions i n 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). max  Assertion 1: The solutions with P (n)  > 7 must have £ = 1.  max  Since m is even we can factor the equation as (d+2 )(d-  2) =  k  3p,  k  e  n  where m = 2k. One of the following cases must hold: jd + 2 = 3 p k  \d-2  f e  e  jd±2  n  k  = l,  = 3  e  \d^2 =p . k  n  Eliminating d i n the first case gives 3 p — 2 = 1 which has no solutions with P x ( " ) > 7 by Theorem 1.2 in [3]. Eliminating d in the second case gives e  n  k+l  m a  3 -p e  = ±2  n  k  +  \  (4.6)  from which £ = 1 follows by Theorem 1.5 of [6]. This proves assertion 1. Assertion 2: There are no solutions with P  m  (??-.) ^ 7. It follows from remarks above that such solutions must have m 6 {2,4} and £ = 1. Furthermore, it follows from (4.6) that a x  n  p  =  2  m/2+l  +  3 >  and so p — 7 or 11, which contradicts P (n) n  max  > 7. This proves assertion 2.  Assertion 3: The solutions with n even are the ones with n = 2 as stated i n the lemma .  Chapter 4. Diophantine  173  Lemmata  Considering the equation modulo 4 implies £ is even, so we may factor the equation as  (d + 3 ^ V / ) ( d - 3 ^ V ) = 2 , l  2  / 2  m  from which it follows that (d±3 /y/ f  |dqF3'/y  =  2  / 2  2m  1  = 2.  Eliminating d give "& l p l — 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 z 2  n 2  m  2  from which it follows that 2  (m-2)/2  2  ( m - 2 ) / 2 -j-  ±  j x  3^/2  =  =  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. A s in the proofs of the other lemmas it follows from [5] there are no solutions with both P (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. max  Chapter 4. Diophantine  174  Lemmata  Assertion 4: The solutions with P (n) > 5 must have £ = 1. Furthermore, there are no solutions with P (n) > 7. max  matX  This follows by a similar argument as i n assertions 1 and 2 of part (1) above. If 5 | n then solutions to d - 2 = —3p would correspond to {2,oo}integral points on hyperelliptic curves y — x + 2 3 with x of the form x = (—3p™/ )/(2 ) 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 2  m  n  2  5  5  2h  4  2fc  (d+2 / )(d-2 / ) = m  2  m  -3 p,  2  e  and so one of the following cases must hold: d + 2 /2  tfpn  m  d - 2™/2  =  =  _  (  d  ±  or < [d+  1 (  2  m/2  ± e  =  2 /2  3  n_  m  =  Tp  Eliminating d i n the first case gives 3 p — 2 / — 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 = 2 / - 3. This completes the proof of (2). 3) For the case when £ — 0 see [37]. A s usual, we can apply [5] to show there are no solutions with both P (n) > 7 and rn > 6. Also, local considerations 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. e  m  2 + 1  rn  2+l  e  max  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 3d — 2  one of the following:  2  m  = p  n  with m > 0 and n > 1 satisfy  Chapter 4. Diophantine  175  Lemmata  (i) n = 3 and (p, m, d) = (11,8,23), (ii) P ax(n) m  > 7 or n even, and m = 1,  f/n) n = 1. 2. The solutions to 3d  2  - 2  = —p with m > 0 and n > 1 satisfy one of the  m  n  following: (i) n = 3 and (p, m, d) = (5, 7,1), (ii) ra = 1. 3. The solutions to 3d  2  + 2  = p  m  with m > 0 and n > 1 satisfy one of the  n  following:  (i) n = 3and (jp, rn, d) = (11,3, 21), (ii) P  max  (n) > 7 and m = 1,  (iii) n = 2 and m G {0,1}, (iv) n = 1.  4. TTie solutions to 3d - 2 p and n G {1,2}. 2  m  = -1  n  5. The solutions to 3d — 2 p 2  m  n  m > 0 and n > 1 sah's/i/ m e {0, 2}  = 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 P (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 y = x + 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 y = x + 2 3 , 0 < a < 5) and solutions correspond to {2, oo}-integral points on the elliptic curve of a certain form. Using the tables i n 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 max  2  2  3  a  3  3  Chapter 4. Diophantine  Lemmata  176  (whose minimal model is of the form y = x + 2 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 = 2  5  a  3  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 a x ( ^ ) < 3. If 3 |rathen the equation can be written as (9d) = ( 3 p / ) - 27 which, by the tables i n Appendix B, has no solutions of the desired form. If 4 | n then the equation can be written as "id = x — 1 which has no solutions by [22]. Therefore, ra = 1,2. A s for the case m = 2, Theorem 1.2 of [5] implies P (n) e {2,3,5}. If 5 | n then the equation can be written as (2 3 d) = ( 2 3 p ) - 2 3 which has no solutions by Theorem 5.1. If 3 | ra then the equation can be written as (2 3 d) = (2 • 3 p ) - 2 3 which has no solutions by the tables in Appendix B. If 4 |rathen by Theorem 1.2 of [7] there are no solutions. Therefore, n = 1, 2. 5) Considering the equations modulo 4 impliesra?,€ {0,1}. Suppose m = 0 then Theorem 1.1 of [5] implies -P x(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) = (3p"/ ) -|-27, and since y = x + 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. A s for the case m = 1, Theorem 1.2 of [5] implies P (n) 6 {2,3,5}. If 5 | ra then the equation can be written as (2 3 d) = ( 2 3 p ) + 2 3 which has no solutions by Theorem 5.1. If 3 | ra then the equation can be written as (2 3 d) = (2 • 3 j t W ) - 2 3 which has no solutions by the tables in Appendix B. If 4 | ra then there are no solutions since the equation 3d = 2x + 1 has only the trivial solution d = x = 1 (see [47]). m  2  2  n  3  3  4  max  4  2  2  2  n / 3  3  4  3  2  2  n/5  5  8  5  3  ma  2  3  3  2  3  max  4  3  2  2  n/5  5  8  5  2  2  2  2  3  3  4  3  4  This completes the proof of the lemma.  •  Chapter 5 Rational points on y = x ± 2 3 2  5  a  f3  In this chapter we are concerned with finding all the rational points on the genus 2 hyperelliptic curves y = x ± 2 3 where a and j3 are integers. The results obtained here were used i n the proofs of the Diophantine lemmata of Chapter 4. 2  5.1  b  a  13  Introduction a n d Statement of Results  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 : y = x ± 2 3^ 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 y = x + A and y = x + B are Q-isomorphic if A/B is a tenth power. Unfortunately, there is one curve we cannot say anything about, namely y = x — 2 3 . 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 w i l l prove the following theorem. 2  5  a  2  2  5  5  2  5  3  9  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 : y  2  = x  b  + 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 y = x ± 2 3' 2  5  Q  a  c P  e  0 0  0 1  1 1  0 0 0  2 4 5  1 1 1  (l.±2) (0,±3) (0, ± 9 ) , ( - 2 , ± 7 ) , (3, ±18)) (-3,0)  0 0 1 1 1 2 2 2 2 2 2  6 8  1 1  (0,±27) (0, ±81), (18, ±1377)  0 5 8 0 2 4  1 1 1 1 1 1 1 1 1 1 1 1  ("1,±1) (3, ±27)  3 3 4 4 4 4 4 4 4 4 5 5 5 5 6 6 6  5 6 8 0 1 0 1 2 3 4 5 6 8 0 1 2 5 0 2 4  1 1 1 1 1 1 1 1 1 1 1 1 1 1  C(Q)\  {oo}  (-1,0),(0,±1)  (7, ±173) (0,±2),(2,±6) (0,±6),(-2,±2) (0, ±18), ( - 3 , ± 9 ) , (6, ±90) ( - 3 , ±27) (0,±54) (0,±162) (1,±3) (1,±5) (0,±4) (l,±7),(-2,±4) (0,±12) (-2, ±20) (0,±36) (6, ±108), (-2, ± 4 ) (0,±108) (1, ±324), (9, ±405) (-2,0), (2, ± 8 ) (-2, ± 8 ) (1, ±17), (-2, ±16) (-6,0), (-2, ±88) (0,±8)  178  3  a  c P  e  C(Q)\{oo} (0,±216) (0, ±648)  6 6  6 8  1 1  8 8 8  0 2 4  1 1 1  (0,±16) (0,±48) (0,±144)  8-, 6 8 8  1 1  (0,±432) (0, ±1296) (1,0) (3,0) (3, ±15) (3, ± 9 ) (9, ±81) (2, ± 4 )  0 0 1 1  0 5 2 4  3 4 4  8 0 2  5 5 5 5 5 7 8  0 4 5 6 8 4 1  -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1  8  5  -1  (10, ±316) (2,0), (6, ±88) (6, ±72) (6,0) (9, ±189) (18, ±1296) (33, ±6255) (4, ±16) (12, ±432)  (0, ±24), (4, ±40) (0, ±72), (12, ±504)  Table 5.1: Theorem 5.1: A l l points on C : y = x ± 2 ° 3 2  5  /5  Chapter 5. Rational points on y = x ± 2 3 2  5.2  5  Q  179  /3  Basic T h e o r y of Jacobians of 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 i n 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 discussion 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 y = 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 w i 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 determining 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 : y = x + x + 1 has infinitely many Q-rational points. Some examples of rational points on E are: 2  2  3  (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 i n determining the group structure of C(k), called the Mordell-Weil group. In our example above, E(Q) ~ Z , with generator P = (0,1), and the points listed above are P, 2P, SP, 4 P 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 will briefly sketch how the Jacobian is constructed from C and state some of the basic facts that we w i l l use.  Chapters.  5.2.1  Rational points on y = x ± 2°3^ 2  180  5  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 y = 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 2  of Aut(A;/fc) on C(k) as follows: if a G Aut(fc//c) and P G C(k) then P  n  =  (x , y"). The set of fc-rational points on C can be defined as a  C(fc) = { P 6 C(fe) :P  a  5.2.2  = P for all a G Aut(fc/fc)}.  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) — 2~2p P- 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 m  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 D = D for all a G Aut(k/k). Note that if D = 2~2, mp(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). a  P  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]/(y -f(x)), 2  181  Chapter 5. Rational points on y = x ± 2 3 2  h  a  13  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 y — 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; 2  k[C] = {F e k{C) : F = g/h for some g,he  k[C] with h{P) + 0}.  P  This is a discrete valuation ring with (normalized) valuation denoted by ordp, maximal ideal denoted Mp generated by i p , a uniformizer (see, [69] chapter 2). We extend ordp to k(C) by ordp(g/h) — ord (g) - ordp(h). The point of all this is that for g s k(C) we can write P  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  y  if 7j  9^  0  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)=  P(9)(P)-  old  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).  182  Chapter 5. Rational points on y = x ± 2 "3 2  5  €  13  This is clearly an abelian group. This is going to be the main object we are concerned with. We w i l l see i n a little bit that there is a more natural way to view this object when C has genus 2. If D\ and D are degree 0 divisors then 2  we write D\ ~ D , and say D\ and D 2  2  are linearly equivalent, if D\ — D  2  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) i n 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(oo) 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, i n 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 y = x ± 2 3' 2  5  a  183  3  N o w 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) representative 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 {P , Q } be two points in J(k). There i s a u n i q u e M ( x ) G k[x] of degree 3 such that y = M(x) passes through the four points P i , Q\, P , Q This curve intersects C at another 2 points P 3 and Q 3 and so 2  2  2  {Pi,Qi}  2  + {P ,Q2} + {P3,Q } = 0. 2  3  In other words {Pi,Qi} + {P ,Q } = {P^,g }. 2  5.2.5  2  3  (5.3)  2-torsion i n the Jacobian  From the identity 5.2 it follows that the elements of the form {(#], 0), (6 ,0)}, where 6\ and 6 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 2  2  precisely all the 2-torsion elements. Thus, there are 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°}.  = 1 5 elements of  Chapter 5. Rational points on y = x ± 2 3^ 2  5  184  Q  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 , Q ) = (P, Q) or (Q, P ) for all a € Aut(fc/fc)}. 17  CT  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 conjugate over k(Vd), A s an example consider the curve C : y = x + 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) 2  5  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 x + 1. Using (5.3) we can compute 5  {oo, (0,1)} + {oo, (-1,0)} =  {00,00}  + {(0,1), (-1,0)} = {(0,1), (-1,0)}.  Over the quadratic field Q(i) (where i = v —l) we have 7  (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 i n 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 y = x ± 2 3 2  5  a  185  13  where J(k) rs is the torsion subgroup of J(k) (which is finite) and r is the rank of J(k). Computing the torsion subgroup J(k) rs is a computationally straightforward task. J(k) embeds into J ( F ) for each prime p for which C has good reduction (p does not divide the discriminant of / ) . The finite groups J(¥ ) are easy to compute and so piecing together the information at different primes we can usually, in practice, determine the structure of J(k)t . 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) . For all the curves we w i l l be considering, we w i l l use M A G M A to compute the torsion subgroup. ta  t0  tors  P  p  ors  tors  There is no known effective procedure for computing the rank, however there are a number of heuristics for computing bounds on the rank. In practice 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 determining the rank. This w i l l be the case in all the curves we consider (except for y = x - 2 3 , 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 y — x + 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 y = x + 1 are (0, ± 1 ) and (—1,0). Of course this is certainly well known; it is a special case of Catalan's theorem. 2  5  3  9  2  5  2  5  5.2.8  Computer Representations of Jacobians  A n y element {(u\,v\), (u ,v )} 6 J(k) can be represented uniquely by a pair of polynomials (a(x),b(x)) € k[x} , where a(x) = (x — u\)(x — 112) andy = b(x) is the unique line through [u\, v\) and (u , v ) (take y — b(x) to be the tangent line to C if (u\,vi) = (u2,v )). This is equivalent to requiring f(x) — b(x) 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 2  2  2  2  2  2  2  <f> : J(k)  —> k [x] x 2  ki[x],  186  Chapter 5. Rational points on y = x ± 2°3^ 2  5  where the image is the set of all (a, b) such that a is monic and a | / — b . A n algorithm for adding two elements i n J(k) by adding their corresponding images (ai,6i), (02,62) in k [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 2  2  4> • J(k) <—> k [x] x k\[x], 2  5.2.9  Some Examples (Using M A G M A )  Let's come back to our example y = x + 1 (where k = Q). The elements in J(Q) and their corresponding representations are as follows. 2  5  O = {00,00  (1.0)  {00,(0,1)  (x,l)  {00,(0,-1)  (x,-l)  {00, (-1,0)  (x + 1,0) (x +  {(0,1), (-1,0)  2  X, X +  1)  (x + x, —x — 1)  {(0,-1),(-1,0)  2  (x ,l)  {(0,1), (0,1)  2  (x ,-l)  {(0,-1), (0,-1)  2  ( x - 2 x + 2,2x-3)  {(l + i , - l + 2 i ) , . ( . l - i , - l - 2 i )  2  ( x - 2 x + 2 , - 2 x + 3).  {(l + i , l - 2 i ) , ( l - - i , l + 2i)  2  As another example consider the curve y = x + 2 3 over Q. Some points 2  5  2  4  Chapter 5.  187  Rational points on y = x ± 2 3^ 2  5  a  on J(Q) and their corresponding representations are as follows. { o o , ( 0 , 1 8 ) } . — ( x , 18) {oo, (-3,9)} i — • (x + 3,9) {oo, (6,90)} i — • ( x - 6 , 9 0 ) {(0,18),(0,18)}^(x ,18) 2  {(-3,9), (-3,9)} i — • ( x + 6x + 9,45/2x + 153/2) 2  {(0,18), (6, -90)} i—> ( x - 6x, - 1 8 x + 18) 2  { ( - 1 + Vlli, 2 + 4 V T T i ) , ( - l - Vlli, 2 -  4Vlii)}  i—> ( x + 2x + 12,4x + 6). 2  We now show how M A G M A can be used to find the structure of J ( Q ) . > _<x>:=PolynomialRing(Rationals() ) ; > C:=HyperellipticCurve(x~5+2"2*3~4); J:=Jacobian(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 Relations: 5*P[1] = 0 { ( x , 1 8 , l ) , (x-2,-18,2), (x"2,18,2),  This tells us that J{Q)t r 0  S  (x,-18,l),  (1,0,0)  ]  — 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:=RationalPoints(J:Bound:=1000); > B : = R e d u c e d B a s i s ( R ) ; B;  Chapter 5.  Rational points on y = x ± 2 3' 2  b  Q  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 element, thus J(Q) has rank 1. Therefore J(Q) ~ Z / 5 x Z. Note, we can't conclude that A — (x — 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) = n h(Q). If A is not a generator then n > 2 and so 2  2  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:=HeightConstant(J:Effort:=2);HC; A : = J ! [ x " 2 - 6*x, - 1 8 * x + 1 8 ] ; hA:=Height(A);hA; newbound:=Exp(hA/4+HC);newbound; R : = R a t i o n a l P o i n t s ( J : B o u n d : = n e w b o u n d ) ; B : = R e d u c e d B a s i s ( R ) ; B;  0. 3 3 3 8 7 7 8 1 3 9 4 9 8 8 1 7 1 2 4 8 0 1 9 0 3 8 9 2 9 1 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 ((x  2  - 6x, - 1 8 s + 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.  > [  [  n*P  189  Rational points on y = x ± 2 3^ 2  + A  :  (x"2-3*x-18,  (x"2-6*x,  n  in  { 1 . . 4}  9*x+36,  -12*x-18,  5  ] ;  2),  2),  a  (x~2+2*x+12,  (x-6,  -90,  1)  -4*x-6,  2),  ]  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 conclusion 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 i n J ( Q ) via the embedding P i—^ {P,oo}. Suppose we have already found the torsion and free-generator of J(Q): J(Q)  X  = J(Q)tors  (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 ¥ . Let V be the reduction of V mod p, and let m be the order of V in J(¥ ). Then the divisor T = mV is in the kernel of reduction. Anything i n J(Q) can be written uniquely in the form p  p  U + n • T,  n  G  Z,  where U is an element i n 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 (the power series w i l l depend on U). A theorem 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 p  Chapter 5. Rational points on y = x ± 2 3' 2  5  Q  190  3  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 elements 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 procedure using differential forms) . Let us consider the task of finding all the rational points on the curve C 3 : y = a; + 3. First we input the curve into M A G M A and search for rational points. 2  5  > _<x>:=PolynomialRing(Rationals()); > C:=HyperellipticCurve(x~5+3); > RationalPoints(C:Bound:=1000); {@  (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 ( Q ) = { 0 0 , (1, 0), (1, ± 2 ) } . 3  > J:=Jacobian (C); > r:=TwoSelmerGroupData(J);r; > T,mapTtoJ:=TorsionSubgroup(J); > T;  > R:=RationalPoints(J:Bound:=1000);B:=ReducedBasis(R);B;  1  A b e l i a n Group o f 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 generator. 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 y = x ± 2"3 2  5  191  /3  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 returns 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 under the x-coordinate map are congruent to (x : z) modulo p , and such that the only rational points on C outside these congruences classes are Weierstrass points. We can just get a bound by using the prefix # on the command. 1  v  > 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 w i 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]. A s a second example let us consider the curve C 3 2 4 : = y = x ± 3 2 4 . (Note 324 = 2 3 .) This is the example we worked through in the previous section. We showed J (Q) ~ Z/5Z x Z 2  2  5  4  324  with torsion generator (x, 18) and free generator ( x - 6 x , -18x+18). A simple search reveals the following points in C"324(Q): 2  {oo,(0, ±18), (-3, ± 9 ) , (6, ±90)}.  Chapter 5. Rational points on y = x ± 2 3 2  5  Q  192  /3  If we try to apply Chabauty's method we find that the smallest bound returned 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 w i 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 together at different primes to conclude no other rational point can exist. We refer to this as "multiple-prime" Chabauty and consider some examples i n Section 5.5. 5.3  Data for the curves y  2  = x  5  ±  23 a  /B  Let A = ± 2 3 ^ , 0 < a. fi < 9, and C denote the curve y - x + 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 J (Q) thus concluding the rank is exactly VA- We have already successfully done this for A = 2 3 in Section 5.2.9 and moreover we found a generator for the free part of J (Q)a  2  5  A  A  2  4  A  For most of the two hundred curves we consider this works out quite well in determining J (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 , - 2 3 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< 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 i n CA(Q)- This verifies we have found CA(Q) A  3  5  7  nown  193  Chapter 5. Rational points on y = x ± 2° 3 2  5  t  13  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 ( Q ) . However, a multiple-prime Chabauty argument at the two primes works in these cases. This w i l l be done in Section 5.5. Also, there we w i l l discuss the curves of rank 2. A s shown in Tables 5.2 through 5.9 we have successfully determined the rank except in four cases: A  AG  {2 3 ,-3 ,-2 3 ,-2 3 }. 5  9  9  3  9  4  6  In the case - 2 3 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. 3  9  Let A £ { 2 3 , - 3 , - 2 3 } , and C denote the twist y = d(x + A) of CA - Over K — Q(Vd) these two curves are isomorphic from which it follows 5  9  9  4  6  ( D)  2  5  A  rkJ (K)  = r k J  A  ( Q )  A  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 f c J ^ ( Q ) were computed above. A  U  2 3 -3 -2 3 5  5  2  6  5  5  < 2 < 2 < 2  4  4  5  y = x 2  rkJ  A  2 3 y = x + 3  2  9  4  rkJ (K)  A  y =x -  9  + 23 4  A  d  )  (Q)  2  2 • 2  Thus r k J A { Q ) = 0 i n 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 ) as listed in the tables except possibly i n the cases k  n  o  w  n  1  A € {2 3 , 2 , 2 3 , 2 3 } . 2  J  4  5  G  2  6  3  A n d when A = —2 3 since we can't determine the exact rank in this case, as mentioned above. 3  9  194  Chapter 5. Rational points on y = x ± 2 3^ 2  5  C:  y  2  Q  = x + 2 3^ 5  a  a  /?  r  #LI  rank  J(Q)  C W k n o w n \ OO  0 0 0 0 0 0 0 0 0  0  0 1  2  1  1  0 1 1  Z/10 z Z/5 x Z  (-1,0),(0,±1)  1  0 1  3 4  0 2  0 2  0 2  5 6 7  1  1  1  0 0  8  1  0 0 1  0 1  9 0 1 2 3 4 5 6 7  1  0 0 1 1 2  0 Z/5 x Z Z/2 x Z Z/5 0 Z/5 x Z  1 1  1 1 1 1  1 1 1 2 2  2 2 0 0 1 1 0  1 2 2 0 0  2  1  2 0 0 1 1  Z Z Z 0 0 z z  0 1 2  0 z z  1  (1,±2) (0,±3)  11  11  (0, ± 9 ) , ( - 2 , ± 7 ) , (3, ±18) (-3,0) (0,±27)  29  (0, ±81), (18, ±1377)  17 19  2  (-1,±1)  2  (3, ±27)  19 19  8  2  0 1 2  9 0  1 1  1 1  1  z  1  Z/5 x Z  (0,±2),(2,±6) (0,±6),(-2,±2)  61 19  (0, ±18), ( - 3 , ± 9 ) , (6, ±90) ( - 3 , ±27)  29,59 29  1  11 (7, ±173)  2  1  0  0  0  2 2  2  1  1  1  0 Z/5 x Z  3  1  1  1  Z  2 2  4  1  1  Z/5 x Z  5  1  1 1  1  Z  Table 5.2: Data for y = x + 2 3 2  b  Q  /3  31 19  Chapter 5. Rational points on y = x ± 2° 3 2  5  f  195  /3  C : y = x + 2 3 2  5  Qf  /5  a  /?  r  #LI  rank  J(Q)  C ( Q ) n o w n \ OO  2  6 7  1  1  1  (0 ±54)  0  0  0  Z/5 x Z 0  8 9 0 1 2 3 4  0 1 1  0 1 1 1 0 1 1  Z/5 Z Z Z  (0,±162)  1 0 1 1  0 1 1 1 0 1 1  5 6 7  0 0 0 1  0 0 0 1 1 0 2 0  0 0 0 1 1 0 2 0 1  0 0 0 z z Z/5 z Z/5 Z  0 2  Z/5 Z  1  0 2 1  1  (0,±108)  61  1 1  1  8  1 1  Z/5 x Z Z  1  Z/5 x Z  (0,±324),(9,-±405)  19 47  9  1  1  1  Z  2 2 2  3 3 3 3 3 3 3 3 3 3 4 4 4  8 9 0 1  2  4 4 4  3 4  4  6 7  4 4 4  5  1  0 2 0 1 0 2  1  k  ;  (1,±3) (1,±5)  0 z z  29 13 31 11 11  19 11 (0,±4) (l,±7),(-2,±4) (0,±12) (-2, ±20) (0,±36)  2  19  (6, ±108)  2  17  Table 5.3: Data for y = x + 2 3 2  11  5  a  /3  (con't)  Chapter 5. Rational points on y = x ± 2 3 2  5  a  196  13  C : y = x + 2 '3 2  a  5 5 5 5 5 5 5 5 5 5 6 6 6 6 6 6 6 6 6 6 7 7 7 7 7  0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4  5  Q  /3  r  #LI  rank  J(Q)  C(Q)known \ OO  1 2 3 1 0 1 0 1 1 2 0 1 1 1 1 0 0 1 0 0 1 1 1 0 2  1 2  1 2  3 1 0 1 0 1 1 0 0 1 1 1 1 0 0 1 0 0 1 1 1 0 2  3 1 0 1 0 1 1  Z/2 x Z Z z  (-2,0), (2, ± 8 ) (-2, ± 8 ) (1,±17),(-2,±16)  0 1 1 1 1 0 0 1 0 0 1 1 1 0 2  2  3  19  z  0 Z/2 x Z 0 z z Z/5 Z Z/5 x Z Z Z/5 x Z 0 Z/5 Z Z/5 0 z z z 0 z  (-2, ±88), (-6,0)  (0,±8) (0, ±24), (4, ±40) (0, ±72), (12, ±504)  19 29,59 7,29 7  (0,±216) 17 (0,±648) 11 11 31  2  Table 5.4: Data for y = x + 2 3 ^ (con't) 2  11  5  a  Chapter 5. Rational points on y = x ± 2 3 2  5  y = x  C: 0i  /?  7  5 6 7  7  7 7 7 8 8 8 8 8 8 8 8 8  8 9 0 1  2 3 4  Q  5  2  r  #LI  rank  J(Q)  0 1  0 1  0 1  1 0  1 0 1  1 0  0 z z  0 1 1 0 0  0 1  1  0 1 1  a  0  C(Q)known \  0 0  19 19 41  Z/5 z Z/5 x Z 0 Z/5 Z  (0,±16) (0,±48)  29 29  0 1  0 1 0 0 0 1  0 0 1 0 1 0 0 0 1  0 1  0 1  0 1  0 z  59  5  0 1  0 1  0 1  0 z  29  9 9 9  6 7  1 0  1  z  11  8  0 0  0 0  9  9  00  0 0 0 0  0  0  8 9 .9  5 6 7 8 9 0 1  '*9  2  9  3 4  9 9  0 0 1 0 1 0 0  +2 3  0 z  1  1  197  /3  1  Z/5 Z  (0,±144) 11 (0,±432) 29  Z/5 0 0 z  - (0,±1296)  11  Table 5.5: Data for y = x + 2° 3^ (con't) 2  5  198  Chapter 5. Rational points on y = x ± 2"3^ 2  5  C : y = x - 2 3 2  5  Q  /3  /?  r  #LI  rank  J(Q)  0  0  1  1  1  Z/2 x Z  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  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  1  3  2(0)  0  0  0  1  4  1  1  1  z  1  5  1  1  1  z  1  6  2  2  2  z  1  7  1  1  1  z  1  8  0  0  0  0  1  9  1  1  1  z  2  0  0  0  0  0  2  1  1  1  1  Z'  2  2  0  0  0  0  2  3  0  0  0  0 '  2  4  0  0  0  0  a  z  C(Q)  k n 0  w n \ OO  (1,0)  11  (3,0)  (3, ± 1 5 )  2  (3, ± 9 )  13 11  2  11 7 17  Table 5.6: Data for y = x - 2 3 ^ 2  5  a  199  Chapter 5. Ra tional poin ts on y = x ± 2° 3^ 2  5  C : y =x - 23 2  5  a  /?  a  /?  7*  #LI  rank  J(Q)  2  5 6 7  0 0 1  0 0 1  0 0 1  0 0 z  17  8 9 0 1 2  1  1  1  z  41  0 1 1  0 1 1  0 1 1  0 z z  11 11  0 1 1  0 1 1  0 1 1  0 z z  0 0 0 1 0 1 1 1  0 0 0 1  0 0 0 z  1 1 1  z z z  2 2 2 2 3 3 3 3 3  3 4  3 3 3 3 3 4 -4 4  5 6 7 8 9 0 1 2  0 0 0 1 1 1 1 1  4  3  0  0  0  0  4 4  4 5  1 1  1 1  1 1  z z  4 4  6 7  2  0  4 4  8  0 0  0 0  0 0  0 0  9  0  0  0  0  C(Q)known \ OO  11 19  (9, ±81)  17  (2, ± 4 )  29 11  (10, ±316)  11 19 17  Table 5.7: Data for y = x - 2"3^ (con't) 2  P  5  Chapter 5. Rational points on y = x ± 2°3' 2  5  C : y  2  a 5 5 5 5 5 5 5 5 5 5 6 6 6 6 6 6 6 6 6 6 7 7 7 7 7  0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4  r  #LI  rank  1 0 1 1 2 1 2 3(1) 1 0 1 0 0 0 0 1 1 0 1 1 1 1 1 0 2  1 0 1 1 2 1 2 1 1 0 1 0 0 0 0 1 1 0 1 1 1  1 0 1 1 2 1 2 1 1 0 1 0 0 0 0 1 1  1 1 0 2  1 1 0 2  0 1 1 1  = x  5  200  3  - 2 '3^ a  J(Q) Z/2 x Z 0 z z z Z/2 x Z Z Z Z 0 z 0 0 0 0 z z 0 z z z z z  C(Q)known \ OO (2,0), (6, ±88)  19 11 (6, ±72) (6,0) (9, ±189)  2  2  (18, ±1296)  11 11  17 41 41 17 11 11 11 (33, ±6255)  2  Table 5.8: Data for y = x - 2 3 5  11  19  0 z 2  P 11  a  /J  (con't)  Chapter 5. Rational points on y = x ± 2"3 2  5  201  /3  C : y =x - 23 2  a  5  r  #LI  rank  J(Q)  a  /3  C(Q)known \ OO  P  7 7 7 7 7  5 6 7  0 1 1  0 1 1  0 1 1  0 z z  8 9  0 1  0 1  0 1  0 z  8 8  0 1  0 1  0 1  0 1  0 z  (4, ±16)  8 8 8 8 8  2  1 0 0 1 0 1 0 0 0 1  1 0 0 1  z 0 0 z  19 61  (12, ±432)  29  0 1  1 0 0 1 0 1  0 z  29  0 0 0 1  0 0 0 1  0 0 0 z --  11  0 1  0 1  0 1  0 z  11  0 1 1  0 1  0 1  0 z  11  1  1  z  29  0  0  0  8  0 0  0  0  0  9  0  0  0  0  3 4 5  9 9  6 7 8 9 0 1 2 3  9 9  4 5  9  6 7  8 8 8 9 9  9 9 9  29  11 11  Table 5.9: Data for y = x - 2 3 2  5  a  f)  (con't)  Chapter 5. Rational points on y = x ± 2 3' 2  5  a  202  3  In fact, we w i 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 is dealt with using results on ternary diophantine equations, which we also do in Section 5.5. 5  5.4  T h e f a m i l y of curves y  = x  2  5  + 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 C  :y  2  A  = x  5  + A,  where A ^ 0 is a rational number. Since CA and CR are isomorphic over Q \i A/B isa tenth power we w i 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  d  A  <  1  if A is neither a square nor a fifth power,  2  if A is a fifth power, A ^ \ ,  3  if A is a square,  A^l,  4 We have already seen that C\ has rank zero and that #C\ (<Q>) = 4. Thus for A ^ 1 we have #CA{Q) < 2n + 3. In [75] Stoll proves the following, where r A denotes the Mordell-Weil rank of the Jacobian J (Q) of CAA  A  Chapter 5. Rational points on y = x ± 2™3^ 2  203  5  Theorem 5.3 (Stoll) Let A ^ 0 be an integer such that r& = 1. Then consequently #C (Q)  < 2 and  < 7.  A  More specifically he proves the following theorem, using a refinement of the method of Chabauty and Coleman. Here p is an odd prime and v denotes p  the p-adic valuation. Theorem 5.4 (Stoll) Let A ^ 0 be an integer such that TA = 1. 1. ifv {A)  G {1,3,7,9}/or some p ^ 11,13 then n  2. ifvp(A)  = 5 for some p ^ 3,5 then HA < 1, ifvz(A)  p  < 1.  A  = 5 f«en TM < 2.  3. z/v (A) G {2,4,6,8}/or some p ^ 2,3,7, or Z/TJ (A) G {6,8}, or ifv {A) P  3  {2,6,8} then n  A  < 1, otherwise ifv (A) 3  4. if A = 1 (mod 3) fnen  7  G {2,4} fnen n  A  < 1, and if A = - \ (mod 3) f/ten  5. r / A = 1,3,9 (mod 11) fnen n  A  G  < 2, < 2,  < 1.  In the case that A = ± 2 3 the upper bounds on #C (Q) obtained by Stoll matches the number of known points on CA (Q) in the following cases: A  6  A  A G {3, 3 , 2 3 , 2 , 3,2 3 , 2 3 , 2 3 , - 2 • 3 , - 2 3 , - 2 3 , -2 3}. S  2  4  3  4  3  4  8  6  4  4  3  8  5  8  8  Thus, our results in the previous section are superceded by Stoll's results, except in one case. Notice that in the case A = 2 3 = 18 we were unable 2  4  2  to find a single prime at which the Chabauty bound is sufficient to determine C 2 (Q), thus Stoll's results now give us C 2 (Q). 1 8  1 8  The curve C 2 has 7 rational points, so the bound in Theorem 5.3 is sharp. 1 8  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 torsion 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 y = x ± 2"3^ 2  204  5  2. If A = a for some integer a ^ 1 then 2  JA(Q)tor  S  = {{(0, a), oo}, {(0, -a), oo}, {(0, a), (0, a)},  {(0,-a),(0,-a)},O}~Z/5Z. 3. If A = 6 for some integer 6 / 1 then 5  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 3 ,2 , 2 3 , 2 3 }. In the last section we verified, using a result of Stoll, the case A = 2 3 . In this section we show that for A £ {2 ,2 3 ,2 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) {(x ,y ),oo}. 2  4  5  6  2  2  0  5.5.1  G  3  4  5  6  2  6  3  0  A = 23 6  2  Let us first consider the case A — 2 3 , where 6  C (Q)kno A  Wn  2  = {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 ) =i Z/30Z x Z/30Z. 29  The reductions of T and P, denoted by T g and P 9 , have orders 5 and 30 respectively. We can input this into M A G M A as follows. 2  2  205  Chapter 5. Rational points on y = x ± 2 3 2  > > > > > >  5  a  13  _<x>:=PolynomialRing(Rationals() ) ; C:=HyperellipticCurve(x~5+2~6*3~2); J:=Jacobian(C) ; T:=J! [ x , - 2 4 ] ] ; P:=J! [ x - 4 , - 4 x - 2 4 ] ; J29:=BaseChange(J,GF(29)) ; T29:=J29!T; P29:=J29!P; Order(T29); Order(P29);  5 30  It follows that the image of the Mordell-Weil group is </> 9(J(Q)) — (P29) (P ) ^ Z / 5 Z x Z/30Z (it is a simple computation to check T £ (P )). A rational point (xo,yo) on CA{Q) has image of the form { ( x 7 J , y 7 j ) , 0 0 } in J ( F ) so we determine conditions on the integers a and b such that the element aT + bP has this image. x  2  29  2 9  29  2 9  > f o r a i n [0 . . 4] d o ; f o r > f o r b i n [0..29] d o ; f o r I f o r > i f (a*T29+b*P29) [3] l e 1 t h 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, <0, <1, <1, <2, <3, <4, <4,  0, ( 1 , 0, 0)> 1, (x + 25, 18, 1)> 29, (x + 25, 11, 1)> 0, ( x , 5, 1)> 10, (x + 7 , 3, 1) > 8, (x + 1, 16, 1) > 2 2 , (x + 1, 13, 1)> 0, (x , 24, 1)> 12, (x + 7 , 26, 1)>  This tells us that the image of aT + bP is {(xTJ, yo), 0 0 } for a and b satisfying  the following congruences.  Chapter 5. Rational points on y = x ± 2 3' 2  5  Q  206  3  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 ) , o o } , {(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<^ 9 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)): 2  (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 3 6  2  C^(Q)=_{oo,(0,±24),(4,±40)}.  I would like to thank Michael Stoll for his help with this argument. 5.5.2  A = 2 3 6  3  In this case there are no known finite points on CA{Q)- Using M A G M A we find JA(®)  =  (P)-%,  Chapters.  207  Rational points on y = x ± 2°3^ 2  5  where P = {x -24x+88,116x-584}. M a k i n g 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. Similarly, 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 C (Q)2  A  5.5.3 A = 2  5  In this case we can apply results from the theory of ternary diophantine equations to get our result. A n y rational solution to the equation y = x + 2 is of the form (x, y) — (a/e , b/e ) for some a, b, e e Z with (a, e) = (b, e) = 1. Thus a, b, e is a solution to 2  2  5  5  b  b = a + (2e ) . 2  5  2  (5.4)  5  Let g = (a.b), then g divides (2e ) , but (g,e) = 1, so g | 2 . Therefore, g = 1,2,4. Since 5.4 has no solutions with a, b, 2e pairwise coprime (see Darmon 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 we have 2  2  5  2  5  2  5  2(6/8) = (a/2) + 2  {e f,  5  2  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 , b/e ) = (2, ± 8 ) , (-2,0). 2  5.5.4  5  R a n k > 2 cases  We've shown in Section 5.3 that the curves of the form y = x + 2 3 , whose Mordell-Weil group has rank > 2, correspond to the following values of a and (3: 2  5  a  /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 y = x ± 2 3 2  5  Q  208  /J  Similarly, the curves of the form y = x — 2 3 with rank > 2 are the following: 2  5  Q  /3  (a,/3) = (1,2), (1,6), (5,4), (5,6), (7,4). For these fourteen remaining curves the classical method of Chabauty cannot be applied to bound the number of rational points since the rank is not smaller that the genus. In this case, we w o u l d 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 w i 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 problem 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 implemented 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 sophistication 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 Y = X + 2 has the form X = x/s , Y = y / s for integers x,y, s such that (x, s) = (y, s) = 1. The equation can then be written as 2  2  5  5  y = x + 2(s ) 2  5  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, s ) = (—1, ± 1 , 1 ) . These pull back to the solutions (X, Y) = (—1, ± 1 ) . 2  Chapter 6 Classification of Elliptic C u r v e s over O with 2-torsion and conductor 2 p a  2  A s we mentioned before we have broken up our attempt to classify curves of conductor 2 p into two stages. In the first stage, we showed if there is an elliptic curve of conductor 2°p 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 2 p . The novelty of these theorems is that, given a prime p, it is straightforward to check whether there are any elliptic curves of conductor 2 p (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. a  2  2  a  2  a  6.1  2  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-isomorphism, all the elliptic curves over Q, of conductor 2 p , having a rational N  209  2  Chapter 6. Classification of Elliptic Curves of conductor  2p a  210  2  point of order 2 over Q. Each theorem corresponds to a value of N. The results obtained are presented i n the form of tables analogous 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 following properties of E: 1. A minimal model of E of the form y + a\xy 2  = x  3  + a^x  2  + a x + ag, 4  where the a; are i n 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 i n 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 w i 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—Tb. 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: y/n = 1 mod  -v/ra > 0  4  if n is odd  (6.1)  ifrais even .  Theorem 6.1 The elliptic curves E defined over Q, of conductor p , and having 2  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  211  2p a  2  1. p — 7 and E is Q-isomorphic to one of the elliptic curves: A  minimal model  T  Al A2  [1,-1,0,-2,-1] [1,-1,0,-107,552]  2 2  3 -15 -15  BI B2  [1,-1,0,-37,-78] [1,-1,0,-1822,30393]  2 2  255 255  2  7  3  3  7  7 7  3  3  3  9 3  9  2. p = 1 7 and E is Q-isomorphic to one of the elliptic curves: minimal model C2  3  4  273 17 18863 17 33 17 -33 4  2  [1,-1,1,-1644,-24922]  Cl  [1,-1,1,-199,-68272]  1 7  8  2  J  J  4  [1,-1,1,-199,510]  C4  J  2  [1,-1,1,-26209,-1626560]  C3  A  T  17  7  17  7  3  2  1 7  io  1 7  3. p — 64 fsflsquare and E is Q-isomorphic to one of the elliptic curves: ai Al  1  A2  1  a  2  Py/p— 64— 1 4  a  a4 -P  p\Jp— 64 — 1 4p  1^21  6  0  2  2  p  4  VP — 04  A  Kodaira  J (P-16) P (256-p) P J  2  P  2  -P  7  8  io;iT J  I ;I5 0  2  Theorem 6.2 The elliptic curves E/Q of conductor 2p with a rational point of 2  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. p = 2 + 1, w/zere k > 5, and .E zs Q-isomorphic to one of the elliptic curves: 1  k  1  a p(2p-l)-l  p^p-l)  1  p(2p-l)-l  -p (p-l)  BI  1  -p(p-2)-l  -p' (p-l)  B2  1  -p(p-2)-l  ai Al A2  4 4 4  4  16  3  4  J  16  p (p-l) 2  4  J  2fc-8 8  (2"p+l)  2 -V  (2 +' p+l) 2 -'ip  2fc-8 8  (p -2*) 2t-8 2  6  0 -p (p-l)(2p-l)  2  3  16  p  2  p  1  :1  fe  2  p  2  p  _ fc-4 10 2  hk-8', I2  2fc-8 2 t  f c  16  0 -p (p-l)(p-2)  :J  2  4  Kodaira  A  • a  (24  2  p  3  l 2 f c - 8 ; I2  k-4 4  2  p  These are Fermat primes; it is necessary that k be a power of 2 for 2 + 1 to be prime. fe  Chapter 6. Classification of Elliptic Curves of conductor 2°p  212  2  \T \ = 4 for Al and Bl, and | T | = 2 for the other two. 2  3.  2  p = 2 - 1, where q > 5 is a prime, and E is Q-isomorphic elliptic curves:  2  to one of the  q  a p(2p+l)-l  CI C2 DI D2  1  a  a.4  2  p (p+l) J  A  6  p(2p+l)-l  -p (p+l)  16  0 -p"(p+l)(2p+l)  4  4  16  1  -p(p+2)-l  p"(p+l)  1  -p(p+2)-l  4  1  J  4  2,,-S 2 (2''+"p+l)  p  2  p  J  16  l<,-4;i;  2<>~ "p  2  2,-8 8  2  „-4 10  J  4  2,-8 8  2""V  0 p (p+l)(p+2)  16  -p'(p+l)  4  2  Kodaira  j  (p +2«y z  p  12,-8; I2  2,,-b -2 ( '+2«+- y 'l-4 4 2  p  p i  P  2  i -4;i Q  p  \T \ = 4for CI and DI, and | T | = 2 for the other two. 2  2  4. there exists m > 7 such that p — 2  = d ,for some d = 1 (mod 4), and .E zs  m  2  Q-isomorphic to one of the elliptic curves:  El E2  ai  a  1  pd-1  1  pd-1  a  04  2  -2 ~y  4  2  A  6  2  0  m  m-4 2 p  4  6  3  Kodaira  J  " V 1  _ m-6 8  2 -pd m  2 m  2  (p-2'"-*) 2 (  2  2 -12 ™ + 2_ m  p  2  a  l 2 m - 1 2 ; Ii  p  3  p )  ™-6 2  T • T* 771— 6 j - 2 L  p  | T | is 2/or tofn curves. 2  5. there exists m > 7 such that p + 2 = d ,for some d = 1 (mod 4), and i? is Q-isomorphie-to one of the elliptic curves: m  ai  FI F2  1 1  a  (24  «2 pd-1 4 pd-1 4  2  A  j  2m-12 7  (p+2'"- ) 2(m-6)  m-6 8  (p+r ) 2"-<V  6  m-6 2  0  p  _ m-4 2 2  2  2  2  _ m-6 3  p  2  p  Kodaira 2  p  2  r f  p  3  l2m-12;Ii  p  + 2  J  I7H — 6 i ^2  | T | zs 2/or both curves. 2  6. there exists m > 7 swcn zTzat 2  m  — p = d ,for some d = .1 (mod 4), and £ z's 2  Q-isomorphic to one of the elliptic curves:  GI G2 2  a\  a  1  -pd-1 4 -pd-1 4  1  a 2  m-6 2  0  p  _ m-4 2 2  A  <76  4  2  p  2  m-6 3 p  2  Kodaira  i  2m-12 7 p  2m-12 m + 2_  2  r f  2  m-6p8  ( 2  2  ,n-6 2 p  l2rn-12; IJ  p  p  )  J  JTM — 61 ^2  These are Mersenne primes; it is necessary that q be a prime for 2 — 1 to be prime. q  4  Chapter 6.  Classification of Elliptic Curves of conductor  213  2°p  2  | T | is 2 for both curves. 2  7. there exists m>7  such that " ~ z's a square integer, say pd 2  1  = 2  2  p  - 1 with  m  d = 1 (mod 4), and E z's Q-isomorphic to one of the elliptic curves: 1  «i  HI H2  1 1  11  1  12  1  a  «2  -pd-1  m  d - l 4 p d - l 4 2  2  2  m-6  p  2  2  0  _ ,n-4 3 2  p  2  3  6  (  m-6p3  2  l2m-12;IH 1  2 m -  2  2  3  Ir„_ ;III 6  3  l2m-12; HI*  12  2 _ j ^  ( "' +  m-6p9  )  6  2  p  2  5  2 _  2"-  2  Kodaira  a  2 „ . - 1 2  „ , +  2  (1-2"- )  _ 2m-12 9  -2 "- p (i 7  p  (l_2'"-' )  _ 2m-12 3  - 2 — V  4 p  6  j-invariant !  0  2 ~ p  4 -pd-1  A  6  Im-6;IH*  m-6  |T 1 z's 2 /or a/7 four curves. 2  8. there exists m > 7 swc« f/zaf " ^ 2  + 1  z's a square integer, say pd  = 2  2  + 1 loz'f/t  m  d = 1 (mod 4), and E is Q-isomorphic to one of the elliptic curves:  Ol  a  1 1  KI  1  K2  1  p d - l 4 p d - l 4  Jl J2  a  CI4  2  pd-1 4 pd-1  2 "- p  0  -2™-<p  _ m-6p2(i  T  4  A  6  2  2  G  - " p  (2"'- + l ) 2  J  l 2 m - 1 2 ; III  2 m - 1 2  ( 2 " ' + l)m-6 + 2  m-6p3  J  U-6-.III  2  0  m-6p3  m  Kodaira  j-invariant 12p3  2  2  2  2  - 2  2  2m-  _ ,n-6p5 2  3  2  (  2m-12 9  2  ™-2  p  2  d  2  2 r n -  ( "' + 2  m-6p9  +  2  2  +  1 )  3 l2rn-12; H I *  12 1  )^  i,„- ;Hi*  m - B  6  | T | z's 2/or all four curves. 2  Theorem 6.3 The elliptic curves E/Q  of conductor 4p  with a rational point of  2  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 Appendix C. 2. p — 4 is a square and E is Q-isomorphic to one of the elliptic curves: 0-2  Al A2  pJp-4 -Ipyjp - 4  04  -P  2  P  3  IT2I  2 2  A  j-invariant  Kodaira  2V  256(p-l)'  IV; 1^  - 2 V  p p (16-p) 4  P  2  : i  3  IV*; 15  214  Chapter 6. Classification of Elliptic Curves of conductor 2 p a  2  Theorem 6.4 The elliptic curves E/Q of conductor 8p with a rational point of 2  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 — 2 is a square for m = 4 or 5 and E is Q-isomorphic to one of the elliptic curves: m  a  Al  p^p -  A2  -2psjp  A  (24  2  _  2  m  -  2  2  m  m-2  P  j-invariant  2 p  2  p  2 m  2  7  2  Kodaira  - '"(p-2'"- ) 2  2  a  I J, HI*; II  P  2 ~ (2 " -p) b  _ m+6 8  3  1 2  p  m  ,  +/1  P  J  2  | T | fs 2 for both curves. 2  3. p + 32 is a square and E is Q-isomorphic to one of the elliptic curves: a  04  2  BI  PVP + 32  B2  .  — 2p^/p  :7  2p 3  2  --  + 32  \T2\  A  2  2 p  2  -2"p  j-invariant  10  4(p+8)  7  Kodaira  J  P  -2(p+128)-  i  8  P  n*; 15  4  Theorem 6.5 The elliptic curves E/Q of conductor 16p with a rational point of 2  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 = 2 4-1, where k > 4, and E is Q-isomorphic to one of the elliptic curves: k  a  «4  2  Al  -p(2p - 1 )  A2  2p{2p-  A3  -2p(p+l)  3  A  4  2 p  j-invariant  2k+4  (2"p+l)  p  2  P (P"1) 2  2  2  2  2  2 "V  fc+8  T* • T* 2fc-4' 2  2fc-8 2 p  1  (2 «p+\f 2"-"p (p+2 -'f  p  7  4fc  4  2  _  2  f c  +  8  p  i o  T*  2(2fc-8)j,  (2 "-p )' 2 -«p'< k  2  1  T* • T*  2h  2  k+  P  Kodaira  J  8  k+  2  1)  -2p(p - 2)  A4  (P-1)P  m  J  • T*  II* ;IJ  3. p = 2 — 1, where q > 3 is a prime, and E is Q-isomorphic to one of the elliptic curves: q  Chapter 6.  Classification of Elliptic Curves of conductor  BI  0,4  -p(2p+l)  (P + 1)P  2  B2  2p(2p+l)  P  B3  -2p(p-l)  P (P+1)  B4  -2p(p+2)  4  3  4 , - 4  Kodaira  (2''p+i)*  p  2  T* • T* 27-4' 2  2,,-8 2  i  p  (2" "p+l) 2<'-' p +  1  T*-T* V 1  2  (p_2^-") 22(2,-8)  7  p  J  • T*  T*  Mc;-12i M  (p + 2"+")2  2«+V  1  a  p  2  4  j-invariant  2"+V _  215  2  2  2  2  a  2,+4 8  2  2  2  2  P  A  \T \  a-2  2p  )  4. p — 2 fs « square for m = 2 or m > 4, and E fs Q-isomorphic to one of the elliptic curves: m  a  Cl  -pVp ~  C2  2p^p -  IT2I is 2 for  A  0,4  2  -2 - p m  2  m  2  P  2  m  j-invariant  2 p  2  2m  -2  3  (p-2">-*) 2,„-l2p ".+2_ .  7  m + 6  Kodaira  3  p  ( 2  >l2m-8i^l  II  2  8  p )  2"-bp  IS) I m - 2 i I2  2  both curves.  5. p + 32 is a square and E is Q-isomorphic to one of the elliptic curves: a  04  2  DI D2  |T |  A  2  2 p  2  -Ps/p + 32 2 p 2p^p + 32 P 3  2  10  2  3  2 p n  4(p+8f  7  El E2  m\  -Pv/p +  2  m  2  m-2p2  2PA/P + 2  m  P  m + 6  p  Fl F2  PsJ2™ -2psJ2  m  -  2  p -  p  m - 2  -p  8. there exists m > 7 SMC/J f/jflf to one of the elliptic curves:  2  p  2  2 2  3  -2 2  2  2 m  m + 6  1 p  Kodaira  2 „ - l 2  7"  m+J  P  2"—Hp2  j-invariant  p p  7  (p-2"- ) 2  2  8  • T*  m-2i  l  1  2  — p is a square and E is  2(«.-6)  3  m-6 2 p  Kodaira  ;i  • T*  T*  p  (2"'+ -p) 2  2  2 m  1  1  ( +2 y  8  A  |T |  <Z4  3' 2  is a square and E is  2  7. £/zere exists an odd integer m > 7 such that 2 Q-isomorphic to one of the elliptic curves:  2  1  (p+2"- )-  7  m  a  1  j-invariant  2 p 2  2  3  A  2> l  T*-T*  J  2  2m  2  1  2(p+2') P  8  rn  0,4  T*. T t  P  6. there exists an odd integer m > 7 such that p + 2 Q-isomorphic to one of the elliptic curves: 02  Kodaira  j-invariant  2m-8'  1  T* 1  m-2i  l  x  • 1  1*  2  fs a square integer and E is Q-isomorphic  Chapter 6. Classification of Elliptic Curves of conductor  GI  2  p  3  m + 6  p  3  -2  ~P  2  2  2  -2  G2 HI  2 m  (1_2"'- )  2  2  m-2 3 p  H2  2  -P  3  9. f/rere exists m > 5 swc/i  2 m  + 1 p  2  216  2  j-invariant  2  2 ~p m  p ^ ¥  a  A  |T |  a.4  a-2  2p  2 r r  2  2  y  m+6 9 p  2m-12  m +  ( 2  Kodaira  3  2"-  2_  3  ] )  i - ;ni  6  m  (1-2'"- ) 2  22m-12 (2"'+ -l) 2  2  3  12m-8 i m  3  Im-2J  m-6  2  m  fs a square integer and E is Q-isomorphic  to one of the elliptic curves: a 11  2  2 " p m  12  2  P  j-invariant (2"- + 2  l) 2 2 T H - 12  2  2 >  2  m+6 3  (2"' + l )  9  9  2  2  3  p  +2  2 "p  2  2 p  p  2  2 p  (2"' + l )  2  3  3  J2  Theorem 6.6 TTze elliptic curves E/Q  Kodaira 3  I5 _ ;1II W  : !  2m  m+6  T  2  i m-8; 2  +2  2  2  3  2m-12  2  8  i - ;ni m  (2 "- + l )  m  JI  A  \T \  04  2  m-0  3  m  im- ;nr 2  of conductor 32p with a rational point of 2  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 Appendix C. 2. p > 5 and E is Q-isomorphic to one of the elliptic curves: (i) p = 1 (mod 4), a  04  \T \  A  Al  0  A2  0  -P 2p  4 2  2 p -2 p  A3  6p  p  2  2V  A4  —6p  P  BI  0  B2  2  Cl C2  2  2  2  2  2  j-invariant  6  1 2  2  2 p  6  2  2 p  3  0  ~P 2p  0  -p  2  2 p  0  2p  2  -2 p  (ii) p = 3 (mod 4),  2  3  2  3  2  6  -2  1 2  6  1 2  6  p 9  9  III;IS  2 3 11  3  T* . 7*  2 3 11  3  7*. 7*  3  3  Kodaira  1728 3  9  2  1728  6  3  3  1728  III;III  1728  13;III  1728  III;III*  1728  15; HI"  Chapter 6. Classification of Elliptic Curves of conductor  a  a  1221  DI D2  0  -P 2 p  4  0  D3 D4  6p  p  —6p  El E2  0 0  p P -2 p  FI  0  P  F2  0  4  2  2  2  2  2  2  2  3  -2V  A  217  2p a  2  j-invariant  Kodaira  1728 1728  III;15  2  2 p -2 p  6  2  2 p 9  6  2 3 11  3  T* • T*  2  2 p  6  2 3 11  3  T * . T*  2 2  -2V  2 2  -2 p 2 p  6  12  6  9  2 p 12  3  3  3  6  1 2  3  9  9  3  15; 15  1728 1728  III;III  1728  nijiir  1728  15; i n *  15; n i  3. p - 1 zs a square and E is Q-isomorphic to one of the elliptic curves:  GI  2 y/ - 1  -p  G2  -2'py/p - 1  2 p  GI'  -2p^P " 1  -p  G2'  2'py/p - 1  2 p  P  A  j-invariant  Kodaira  2  2<y  64(4p-l)  III;I*  2  -2 p  2  2 p  2  -2 p  04  «2 P  2  2  3  2  2  3  1 2  6  8  J  I5;i2  64(4p-l)  7  1 2  P 64(4-p) p*  8  P 64(4-p) P  J  J  III;I*  J  I5;i2  2  4. p — 8 z's a square and E is Q-isomorphic to one of the elliptic curves:  HI H2 HI' H2'  A  j-invariant  Kodaira  2  2<y  64(p-2)  III;I*  2  -2 p  2  2p -2 p  (24  0-2  pVp-8  -2p  -2pyJp-%  p  3  -P-y/P ~ 8  -2p  2 /P - 8  p  PX  2  2  3  2  9  6  8  P -8(p-32) P 64(p-2)  J  15; 15  2  7  9  J  J  III;IJ  V 8  -8(p-32) P  J  T*- T*  2  5. p + 8 is a square and E is Q-isomorphic to one of the elliptic curves: a  a.4  Py/p + 8  2p  2  11 12  -2  P v  /p + 8  2  P  3  11'  -pVP + 8  2p  12'  2 /P + 8  p  P n  2  3  Theorem 6.7 The elliptic curves E/Q  m\  A  j-invariant 64(p+2)  2  6  2 p  7  2  9  2 p  8  p 8(p+32) P  2  6  2 p  7  64(p+2f  2  2 p  8  8(p+32)  9  J  15; 15 III;IJ  V 2  III;I*  J  2  P  Kodaira  J  T*. 1  T*  0' 2 1  of conductor 64p with a rational point of 2  order 2 are the ones such that one of the following conditions is satisfied:  Chapter 6. Classification of Elliptic Curves of conductor  218  2p a  2  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  \T \  A  2  Al A2 A3 A4  0 0 12p -12p  -2 p  P 2p 2p  4 2 2 2  2 p -2 p 2 p 2 p  BI B2  0 0  P -2 p  2 2  -2 p 2 p  Cl C2  0 0  P  2 2  -2V  2  2  2  2  2  2  2  2  3  -2V  12  6  6  6  15  6  15  6  6  1 2  3 3  2'V  j-invariant  Kodaira  1728 1728 2 3 11 2 3 11  15; lo  i i ; 15  3  3  3  is;i$  3  3  3  T* • T*  1728 1728  II;III I ;III  1728 1728  11; III* I*,; III*  2  (ii) p = 3 (mod 4), 0,2  DI D2 D3 D4  0 0 12p -12p  El E2  0 0  Fl F2  0 0  04  -2 p p 2p 2p  A  2 2 2  -P 2p  2 2  2 p -2 p  2p  2p  2 2  2  2  2  2  2  2  2  2  3  3. p = 2 + 1, where k > 4, and k  m\ 4  2 -2 2 2  2  12  G  15  15  p p p p  6  6  1728 1728 2 3 11 2 3 11  6  6  6  3  ] 2  2  9  -2 p 1 2  j-invariant  6  9  3  3  3  3  3  3  1728 1728 1728 1728  Kodaira J * . T*  i i ; 15 is;iS 1* •  T*  II;III 1*; i n n iir ;  i5;iir  is Q-isomorphic to one of the elliptic curves:  Chapter 6.  Classification of Elliptic Curves of conductor  a-2 2p(2p-  GI  2 (p-l)p 2  -2 p(2p-l)  G2  2 p  2  G3  2  2  p(p+l)  2 p  2 p  GI'  -2p(2p-l)  2 (p-l)p  G2'  2 p(2p-l)  2 p  G3'  -p(p+l)  2  2  G4'  4  2  2  3  2  - 2 p ( p - 2)  2 p  2  2  3  (2 +"p+l)  t  V°  -2  fc+1  2  fe  2 . T*  fc+4' 1 1  3  • T*  -Mfc-8' 1 1  T*  1  • T*  3  22fc-8p2  1  t  p  2  l  x  T*  fc+14 7  fc+1  4fc  4  1  p  * - - )  22(2fc-8)p  2 + p  p  2  0>  T*  22fc+10p8 2  2fc-2 2  x  ;J  (2 +' -p ) 2 -"p' (2*p+lJ _)l  -2  2 p(p-2)  7  T * . T*  2  p  2*--i ( p + 2  2 p  G4  2 f c - 8  (2 V+l)  7  4k+2  Kodaira  3  fc+  k+14  p  2  fc  2  2  219  2  j-invariant (2 p+l)  22fc+10p8  3  2fc-2 2  2  a  A  04  1)  2p  (p+2  7  2 f e  -")  :i  3  2(2fc-8)p  2  (2 + " - p ) 2  fc  V°  2  0'  2  1  T*  • T*  T*  • T*  T*  • T*  3  fr-4p.l  1  fc+4'  4  |T | = 4/or G2 and GI', and \T \ = 2/or a// offers. 2  2  4. p = 2 — 1, where q > 3 is a prime, and E is Q-isomorphic to one of the elliptic curves: q  a  0-2  A  4  HI  2p(2p+l)  2 (p+ l)p  H2  - 2 p ( 2 p + 1)  2 p  H3  p(p - 1)  H4  2 p(p+2)  2 p  HI'  -2p(2p+l)  2 (p+ l)p  H2'  2 p(2p+ 1)  2 p  H3'  -p(p-l)  2 ''~ p  H4'  - 2 p ( p + 2)  2 p  2  2  2  2  3  7  2  2  2  2  +2  2 , , - 8  3  2  2 "+ p 2"+ p  2  14  -2 '  2  4 r  + 2  (  +  2  ,,+  14  .,  2  )  a  2  + 4  q+4 •i T* 4 1  x  3  !  T* T*  • T* • T*  p  3  T*  • T*  2  \T \ = 4 for HI and HI', and \T \ = 2 for all others. 2  2  5. p — 1 is a square and E is Q-isomorphic to one of the elliptic curves: a  11 12 11' 12'  a.4  2  2p p~ T /  ::  v  p  3  -2 p 2  -2 /p - 1  p  2Wp - 1  -2 p  Pv  2  3  2  2  A  j-invariant  Kodaira  2  -2 p  -64(p-4)  II; I*  2  2 p  2  -2 p  2  2 p  6  12  6  12  8  P  7  3  2  64(4p-l)  J  P 8  -64(p-4)  3  P*  7  64(4p-l) P  J  1  J^O' 2  (p + 2 " ) <y-4p-l 2  • T*  L  2 , , - 8 p 2  2 ( 2 , , - 8 )  T*  1  3  2  10  • T*  <J+4I  T*  pi  (2''+"p+l) 2''--ip (p-2 ''-")'  7  T* 1  2(2,,-8)p  2  p  (2"p+l)  8  7  p  2 '+ p f  4  1  )  2  p  10  J-O' 2  2  p  (p-i "-")-'  7  , + 14 10 2  J  1  2-1-1  2  2  (2"+' p+I)2"->p  p  -2 <> p  2  2  9+14 7  p  4  4  Kodaira  (2''p+l)  9  2  p  2  j-invariant  2 +10 8  2  2  2 -2 2  2  2  15; i t II; 15 15; I I  Chapter 6.  Classification of Elliptic Curves of conductor  6. there exists m > 2 such that p — 2 of the elliptic curves: 0 4  JI J2  2p^p -  2  -2 p  m  -2'py/p  -  m  2  2 p 2  m  -2p^/p  -  2  -2 p  J2'  2 pyJp -  2  2p  \T2\  A  2  2 +V.  m  -2  2  2  2  j-invariant  2  m + 1 2  2 m + 6  p  2 r , . - 1 2  2  2 " ' -  m  2  -2  2  3  m + 1  04  KI  2 y/ P  + 2  m  P  V  ( 2 " ' + - p f 2 « . - 6 2  2  2  2 m - 1 2  + p  2  2  2m+12p8  2  2  6  K2  -2'pyf +2  2 p  KI'  -2 y/p+2  2 p  2 pv/p + 2  2 p  m  2  P  m  P  K2'  z  m  2  3  2  2  3  p  CI4  (12  LI  2psJ2  L2  -2 p^/2  LI' L2'  m  z  - p - p  m  ~2p^2  - p  m  2 p^2 z  m  - p  2  3  m  2  2  3  9. there exists m > 3 such that "' 2  1 p  the elliptic curves:  2  -2 +V r  n  + 12 8  2  2  2 m +  V  m+12 8 p  • T*  2  (  » + ^_  2  m  2  )  3  2  - p ) m - 6 2 2  p  T* • T* m+2> 2 A  J  1  T*  2 , „ - 1 2 p  2  (2'" +  II  ^2m-4>  p  p )  - b p 2  ( p - 2 ™ -  Kodaira  3  p  2  -2  T*  — p is a square and E is  2  2  • T* 1  ( p - 2 — ) 2rn 1 2  2m  T*  *-m+2i 2  j-invariant  2  -2 p  p  J  A  2  2p  2  m  a  2 „ . - 1 2  6  1  1  (p+2"' + ) 2 " —  • T*  T* • T* m+2> 2  2  p  2  2  x  T*  3  \T \ 2  -2 p 2  m - B  (p+2'"~ )  2»n+12p8  2  2p m  • T*  m + 2 i 2  Kodaira  2 , „ - 1 2 p 2  7  8. zTzere exz'sfs an odd integer m > 3 swc/z £/za£ 2 Q-isomorphic to one of the elliptic curves:  T*  is a square and E is  (P+2-+ )2  2 m + 6  • T*  1  (p+2"'-'r 2  m  1  T*  p  7  m+2> 2  p  j-invariant  2  2 m  • T*  2  A  \T \  2p m  T* 1  {r-r"- r  m  2  • T*  2  2  7. f/zere exists an odd integer m > 2 such that p + 2 Q-isomorphic to one of the elliptic curves: a  p  6  7  p  T* p  (2"^-pf  8  2  A  Kodaira  2m  2  3  JI'  m  2  a  is a square and E is Q-isomorphic to one  m  «2  220  2p  • T*  •••2771-45 3  T* I  1  1  • T*  77l + 2 !  2  X  is a square and E-is Q-isomorphic to one of  Chapter 6.  Classifica tion of Elliptic Curves of conductor  Ml  2p  _ 2m+6 3  -2 p  m+12 3  m  \J  V  Zp  M2  2  2p  M2'  -2 p  _  (  2  p  (  p  )  3  1  )  3  2 _  +2  -(2"'- -l)  2  (2"'+ -l) m-6 2  2 +  1 2  p  3  3  l2m-4) HI  2 m - 1 2  2  m  i™ ;ni  m - G 2  _ 2m+6p3  2  Kodaira  1  2 , n - 1 2  r „ +  2  2  m  2  m - 2 _  2  2  2  Ml'  a  j-invariant  A  04  a-2  221  2p  3  Im+2> H I  2  o„2  NI  N2  /2"'-l  P V  2  - 2  o„2  NI'  2  ^  m  3  -2 p 2  /2'"-l  N2'  -2  2p  V  p  2  -(2"'~ -l) 2  2  -2 p 2  3  2  2  m  p  +  2 _  1  3  )  i™+2 ;nr  m - 6  2  +V  -(2"- -l)  m+12 9  (2"'+ _l)  -2  3  (  2 m  2  2  2  2  3  ^2m-4>  - 1 2  m  2  p  l2m.-4; H I  2 r „ - 1 2  2  m+129  3  2p m  +V  2 m  3  HI  3  Im+2; H I  m - «  |T 1 = 2/or a/Z f/tese curves. 2  10. there exists rn > 2 swc/z i/zaf  2 m  + 1 p  zs A square and E is Q-isomorphic to one of  the elliptic curves: 0,4  02  2p m  0 1  02  2p 2  or 02'  2  p  ^  PI  - 2 p  P2'  2  2  p  ^ 2  ^  2p  3  2p 2  3  Kodaira  3  ^2m-4'' HI  2,r,-12  277i+6 3  2  m  )  +  2  ,..-(5  l)  3  I 7 W 2 ; in  2  2  2p  1  (2"— -l)  2"'  3  m - 2 _  2  2  p  2  3  (  m+12 3  2p 2  2  PI'  2  p  3  ( 2 " ' +  2  2p  2  2  2  2 m + 6  2  m  P2  2  j-invariant  A  2  2p m  2  |T |  p  2  2  + 1  V  27fi+6 9 p  3  I2777.-4; in  2 m - 1 2  ( 2 ' " + l) „,-o  3  + 2  17*77+2; in  2  (2"- -l) 2  2  3  I2777.-4; in  2 , „ - 1 2  ( 2 " ' + l) 2"'-  3  2  277i+6p9  3  2  (2"-  2  2  (2"'  2•  2  m  +  1  V  + 2  2  m + 1 2  p  T* TTT* 77l+2'  e  9  2  -l)  1  2 m - 1 2  2  + 2  + l)  m - 6  1 1 1  T* TTT* 2777.-4 1  I  1  3  1^+2;  1  1  nr  Theorem 6.8 The elliptic curves E/Q of conductor 128p with a rational point of 2  order 2 are the ones such that one of the following 1. p = 13 and E is Q-isomorphic  conditions is satisfied:  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  Classification of Elliptic Curves of conductor  6.  a  0,4  2  Al A2  A -2 p 2 p  6  -2 p 2 p  6  2  2p  2p  2  -2 p  -2 p  2  -2p  2p  2p  -2 p  2  Al' A2'  2  2  2  2p  2p  -2p  -P 2p  2  3  7  2  1 4  2  2  III; 15  5 3  15; 15  7  2 27 7  6  2p -2 p  2  Kodaira  6  1 4  2  7  2  2p -2 p 7  2  2  2  13  a  j-invariant 6  8  2  3  2p  13  2  2  -2 P  BI' B2'  2  -p  2  8  2  2  2  BI B2  \T \  2  222  2p  3  T * . T*  27 2  3  III;15  5  6  n i ; 15  5  7  6  2  ?  2  6  15; 15 n i ; 15  5 3 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: a  04  2  C2  2p 2pV2p - 1 - 2 W 2 p " 1 -2 p  Cl'  -2 /2p - 1  2p  C2'  2W2p -1  -2 p  Cl  2  DI D2  3  P v  2 /2p  -  Py  1  o2 /o -\ -2'pyJ2p - 1  DI'  -2psJ2p  D2'  2'pV2p - 1  - 1  \T \  A  2  2  .  2  3  2  -P 2p  2  2  3  -p  3  2  j-invariant  -2 p  8  -128(p-2)  2  2 p  7  P» 32(8p-l)  2  -2 p  2  2 p  2  2p  2  -2 p  2  2p  2  -2 p  8  13  13  -256(p-2)  8  P  2p  3  n i ; 15  J  15; II  P  32(8p-l)  J  i i ; II  P  1 4  8  -128(p-2) P  1 4  8  in*; 15  J  -128(p-2) P  J  2  32(8p-l)  7  E  3  15; II J  2  32(8p-l)'  7  7  7  Hi; 15  J  P  8  7  Kodaira J  !J —  2  II; II i n * ; 15  4. 2p — lis a square, p = 1 (mod 4), and E is Q-isomorphic to one of the elliptic curves: 2  a El  \T \  04  2  2 /2p  -  2  Py  1  2  2p  4  E2  -2W2p - 1  -2 p  El''  -2 /2p - 1  2p  E2'  2 p /2p - 1  -2 p  Fl  2p^2p - 1  -p  F2  -2 p 2p - 1  2p  Fl'  -2PVV -1  -P  F2'  2 pv/2p - 1  2p  2  2  P x  2  2  x  2  2  /  2  V  2  2  2  4  2  2  3  3  4  2  4  2  j-invariant  2 2  -2 p  2  2 p  2  2p  2  -2 p  2  2p  2  -2 p  8  13  8  13  7  1 0  1 4  2  8  Kodaira J  ± , ± 2  -128(p*-2) p" 32(8p -l) 2  8  P P  :  2  P 1 0  4  m*; I* i i ; 15  2  2  P  J  J  -128(p -2)  ill;I* i i ; 15  2  32(8p -l)  8  J  -128(p' -2) p"  2  15; 15  2  2  1 0  a  J  32(8p -l)  8  n i ; I*  : i  P* 1 0  1 4  7  -128(p -2) p* 32(8p -l) 2  -2 p 2 p  2 2  A  J  IH*; I*  Chapter  Classification of Elliptic Curves of conductor  6.  5. p — 2 is a square and E is Q-isomorphic 04  «2  2py/p - 2  GI  G2  -2'p^p  -2 p  - 2  -2py/p- 2 2W P - 2  GI'  G2' HI  2 /p  H2  -2'psjp  2  P  3  -2 p 3  -2p  - 2  Ps  2  3  2  2  2p 2  - 2  3  HI'  -2 /p-2  -2p  H2'  2W P - 2  2p  2  Py  2  3  2  A  j-invariant  -2V  -32(p-8)  2 p  128(2p-l)  2  3  a  to one of the elliptic curves:  \T \  P  223  2p  2 2 2  14  P  7  2 p  128(2p-l)  P  7  2 2  III*;I*  -2 p  2 2  2p -2 p  13  J  II;  2  J  J  III;II  P  -32(p-8)'  8  P  )  T*- T*  2  128(2p-l)  7  J  III;II  P  -32(p-8)'  8  P  i  T*- T* 2i *2  2  x  6. p" + 2 fs a square for some integer n > 1 and E is Q-isomorphic elliptic curves: 04  0.2  11  2py/p  + 2  n  12  -2'pyjp  11'  -2p^/p  12'  2 pv/p" + 2  JI  J2 JI'  J2'  + 2  n  z  2p\Jp  + 2  n  -2 ?p 2  + 2  n  Pv  -2 /p  n  Py  2W  P  2p 3  n  p  2p 3  2p  2 p  7  2  2 p"+  2  6  128(2p" + l ) p™  6  32(p '+8) p " 128(2p" + l ) p™  Kodaira  3  n;i n  P " 2  T  2  3  128(2p ' + l ) p" 32(p"+8)' T  6  n;i2„ J  m*;i; J  IILI;  J  T*. T*  P " 2  2 p"+ 2  A  128(2p" + l ) p" 32(p"+8) p "  6  13p2n+6  m*;i;  3  2  p  8  2  32(p"+8)  213^271+6  2  2  6  14 n+6  2 p"+  n + 2  +2  2n  8  2  2p 2  n  2p +  2  2  + 2  2  14  2  n+2  2  2 p "+ 2 p+ 7  2  p  + 2  n  2  to one of the  j-invariant  A  \T \  n+2  n  III*;II  P  128(2p-l)  2V  13  J  P  -2V  8  II; 15  2  -32(p-8)  14  Kodaira  a  J  J  2  2i 2n 1  in  r  T*- T* 2i 2n x  I  Theorem 6.9 T/ie elliptic curves E/Q of conductor 256p with a rational point of 2  order 2 are the ones such that one of the following 1. p — 23 and E is Q-isomorphic  conditions is satisfied:  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  0,4  a-2  0 0  Bl B2  0 0  -2p  CI C2  0 0  2p  DI D2  0 0  -2p  El E2  0 0  2p  FI F2  0 0  -2p  GI G2  2p  2p  -2 p  2p  -2 p  2p  2p  2p  GI' G2'  -2 p 3  2p 3  2  -2 p 3  2  2  2p 3  2  -2 p  3  3  2p 3  3  2  2  3  3  2  2  2  3  3  2 2  -2V  2 2  2 p  15  9  -2  2  1 5  3  3  p  3  2 2  -2V  2 2  2p  2  1 5  9  -2  1 5  / p  6  2 p 2 p 15  9  2 2  -2 p  2 2  2 p 2 p  2 2  2 p 2 p  9  1 5  9  9  9  6  15  15  2  Kodaira  1728 1728  III;III III*;III  1728 1728  III;III III*;III  1728 1728  HI*; 15  III;15  1728 1728  6  -2V 9  a  j-invariant  2 p  2 2  3  3  A  2  2p  Al A2  \T \  224  2p  6  6  6  III;15  1728 1728  IU*; 15 III;III* III*;III*  1728 1728  III; III* III*;III*  2 5 2 5 6  3  III;15  6  3  III*;15  2 5 2 5 6  3  III;15  6  3  III*;15  3. -^- is a square and E is Q-isomorphic to one of the elliptic curves: p  (X4  0,2  X?2\  2p  3  HI  2  A  j-invariant  Kodaira  -2V  -64(p-4)  III;I  P  -2V^  -2V  -aV*?  2p -2V  2  -2 p  2  2 p  11  -2p  2  2V  12  2p  2  -2 p  11' 12'  -2p  H2 HI' H2'  2V^  3  2  3  3  2  - 2 V ^  23V T 2  1  2p 3  3  2  2 2  2 y 9  15  8  -64(p-4)  7  64(4p-l)  P  8  P -64(p-4) p  8  3  3  2  64(4p-l)  7  -2 p  3  P  2p 15  in*; it 3  2  64(4p-l)  P -64(p-4) P  2  2  3  P  15  9  2  64(4p-l)  i j  3  III;15 III*;It III;It III*;15  3  III;It 3  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 2 p a  a-2  a  2p  JI  2  4  2/  JI'  J2'  2  -2V  2 V ^  -2p  KI  K2  2 p  KI'  -2p  3  K2'  -2V° 2  2  2  2V  1 5  p  8  -2V°  P-I  2  1 5  p  8  2  64(p -4)  9  _  2  i  5  8  i o  p  9  8  _ 15 10 2  p  III;12  P  4  64(4p -l)  3  64(4p -l)  3  P  2  III*; III; 15  P  2  64(p -4) 2  3  III*;II  P  4  64(4p -l) 2  2 p  in*; 15  3  2  2 p  3  P  2  2  2  III;IS  3  64(4p -l) 2  2  4  Kodaira  64(p -4) 2  2  2  j-invariant 2  2  -2V  J2  A  1^21  4  225  2  3  III;15  P  2  64(p -4) 2  P  4  3  in*; 13  5. Y~ is a square and E is Q-isomorphic to one of the elliptic curves: PJ  a  2  LI L2  2p  - 2  p ^  3  LI'  3  2p  M2'  3  - 2  2 P  v  / £ ± i  2 Px/^ 3  2p  2  2 p  2  2 p  2 2  3  2 2  2  2 p 3  2  2  2  2 p  M2  2  3  2 p  Ml  Ml'  3  2p  L2'  3  2 p  A  \Ti I  04  3  2  9  9  2  1 5  2 p 9  2  1 5  9  2  1 5  III;15  64(4p+l) 3  64(4p+l)  3  64(4p+l)  3  P P  7  8  64(p+4)  3  2  P  7  8  64(p+4)  P  2  3  III*;I* III;I* HI*; 15  P  64(4p+l)  in*; i* III;15  pi  7  p  3  P  64(p+4)  p  2 p  64(p+4) 2  7  8  p  Kodaira  3  P  8  15  2 p  j-invariant  3  ni; I*  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  a  4  122 | A 2 2V°  NI  2p  N2  2V  2  NI'  2p  2  2V°  N2'  2V  2  2 p  Ol  2p  2  02  2V  2  or  2p  2  2V  2  4  4  2  2  02'  aW ^ 1  2p a  226  2  j-invariant 64(p +4) p" 2  6 4 ( 4 p + l) 2  2 p 15  8  P  Kodaira  3  III;12 3  2  3  III;13  6 4 ( 4 p + l) 2  1 5  8  P  3  9  6 4 ( 4 p + l)  8  P 64(p +4)  1 5  9  III;I?  1 0  P  ;  4  6 4 ( 4 p + l)  8  P 64(p +4) p-i  1 5  3  III;15  2  2  2 p  nr i;  3  2  2 p  3  2  2  2 p  III*;IJ  2  2  2 p  in*; 1*2  2  64(p +4) p-i  1 0  3  III*; I*  In Chapter 8 we w i l l be interested i n knowing, up to isogeny, the elliptic curves with conductor of the form 32p or 256p , and their j'-invariants. We 2  2  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 32p . Then E is isogenous over Q to a curve 2  of the form y — x 4- a x 2  3  2  2  with coefficients given in the following p  0-2  + C14X  table. j-invariant  0,4 2  any  0  any  0  —p (_l)(p+l)/2  any  0  (_l)(P+l)/2 3  7  ±7  2-7  7  ±7  2• 7  -2  7  ±7  2 • 7  s + i, s e z  2ps  -p  -2 64(4p-l)  s + 8, s e Z  ps  -2p  s - 8 , s <E Z  ps  2p  2  2  2  p  2  1728 1728 8000/7  2  2  1728 p  3  6  P  2  2  6  64(p-2) P  a  64(p+2) P  a  a  Chapter 6. Classification of Elliptic Curves of conductor  227  2p a  2  Corollary 6.11 Suppose p > 5 is prime and that E/Q is an elliptic curve with a rational 2-torsion point and conductor 256p . Then E is isogenous over Qtoa 2  curve  of the form y = x + ax 2  3  2  2  + ax 4  with coefficients given in the follozuing table. «2  any  ±2p  any  0 0 0  any  ±4p  2p  any  1728 1728 1728 2 5  ±2p 2  ±2p  3  6  2  23  ±2  3  -23-39  2 • 23  23  ±2  4  -23-39  2 • 23 3  2s + 1, s G Z  ±4ps  2p  2s + 1, s e z  ±4ps  -2p  V 2 s + 1, s £ Z  ±4ps  2p  s/2s + 1, s G Z  ±4ps  -2p  2s - 1, s G Z  ±4ps  2p  2s — 1, s G Z  ±4ps  2p  V 2 s — 1, s G Z  ±4ps  2p  \/2s - 1, s G Z  ±4ps  2p  2  2  2  2  2  2  2  2  6.2  j-invariant  0,4  p  5  5  3  3  2 3 4057 23« 2 3 4057 23« -64(p-4) b  a  a  b  3  ii  P  3  4  64(4p-l)-  2  s  P  64(p -4)2  A  "  i  P^  64(4p -l) 2  2  p  64(p+4)  3  a  2  P  a  2  2  A  _  ?  64(p +4) 2  P  64(4p + I)1!  2  J  4  5  P*  T h e Proof  We w i 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 i n 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 conductor N = 2p 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: 2  1) d = 2 p 2  2) d = 2  m  2  rn  n  + 1,  + p, n  Chapter 6. Classification of Elliptic Curves of conductor  228  2p a  2  3) d = 2 - p , 4) d = p 2, 5) pd = 2 + 1, 6) pd = 2 - l , with n > 0 and m > 7. A p p l y i n g the Diophantine lemmata from Chapter 4, the solutions of these are respectively as follows: 1) (p, d, n) = ( 2 - + 1, 2p - 1,1) and (p, d, n) = {2 ~ - 1, 2p + 1,1), 2) (p,d,m,n) = (17,71,7,3), (p,d,n) = ( 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 + l , p - 2,2), and solutions with n = 1. 5) ^ is a square, 2  m  n  2  n  m  2  m  2  m  m  2  m  m  2  2  m _ 2  6) is a square. Thus, p must satisfy one of these conditions. Suppose p satisfies the first condition in 1, that is p = 2 "-~ +1, d = 2p — 1, and n = 1. Then £ is Q-isomorphic to one of 7  2 y  2  = x + p(2p - l ) x + 2 " p x , 3  2  m  3  2  y = x - 2p(2p - l ) x + p x, 2  3  2  2  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  y + xy = x 2  2  3  ^  y +xy  = x  3  +  p ( 2 p - 1) - 1 4  , p(2p - 1) - 1  +  p ( p - 1) 16 ' 3  2  X  +  , - p ( p - 1) 3  2  x  +  , -p (p-l)(2p-l) 4  x+  —  .  Thus E is isomorphic to either A l or A 2 in Theorem 6.2. Suppose that p satisfies the first condition in 4, that is, p = 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. m  2  Chapter 7 O n  the  Classification  Q with  2-torsion  a n d  of Elliptic C u r v e s c o n d u c t o r  over  23p a  2  A more appropriate title for this chapter w o u l d be "Classification of primes for which there exist elliptic curves over Q with 2-torsion and conductor 2 3 p with a G {1,2,3}", since it is the collection of primes we will be studying, not the curves themselves. The tables i n Section 3.2 provide a classification of elliptic curves of conductor 2 3 p in which the prime p must satisfy one of a list of Diophantine equations. In this chapter, we use the lemmata 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 w i l l be interested in the primes for which there are no elliptic curves of conductor 2 3 p, with a G {1,2,3}. Our main focus here w i l l be to "determine properties of this set of primes. We w i l l show that for all primes p = 317 or 1757 (mod 2040) there are no elliptic curves with 2-torsion and conductor 2 3 p with a G {1, 2, 3}. a  a  a  Q  7.1  2  2  2  2  Statement of Results  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 3. p=  2  m _ 2  ' " "  2  3 ' ± 1 with +  1  with  m>7,e>0; m>7J>l;  229  Chapter 7.  On the Classification of Elliptic Curves of conductor  4. P = e 3  +  2  m-2  w  5. P = £ _ m-2 3  2  6. P = m - 2  _  2  3  i  w  t  i  <? w  U  m  h  h  m  > 7 ^ > 0;  m  > j  f  £  m  e  2  0;  e  2  with rn>7,£>  1  11. P = 3d + 2 2  2  1;  with m > 7;  m  12. P = 3d - 2 13. P = 2  and £ > 0;  - d with m>7and£>  10. P = + " d2  ;  0;  m  9. P = 2 3  > Q  with m>7and£>  2  230  7  m  8. P = d - 2 3  2  /  7. P = d + 2 3* with m>7 2  Q  >7 £>0;  h  t  2 3 p  with m > 7;  m  - 3d with m > 7;  m  2  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. 2.  {5,13}; n p  = d + 4 • 3 with £>0 2  3. p  e  even, n = lor P {n) min  = d - 4 • 3* with £ > 1 odd, n = 1 or P (n)  n  2  mm  4. p  > 7; > 7;  = 4 • 3 - d with £ > 1 odd, n = 1 or P „ ( n ) > 7;  n  e  2  m i  5. p" = 4_±3i  £ > l odd, n = 1 or P {n)  6. 4p = 3d + 1 with n G {1,2} andp  mm  n  2  n  > 7, p = - 1 (mod 4);  = 1 (mod 4);  7. p = 3d - 4; 2  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 3 p a  3. p  = a? + 4 • 3 with £>1 odd, n = 1 or P  4.  = d _ 4 • 3^ with l>0  n  e  n  2  p  m i n  even, n = lor P  £  231  ( n ) > 7;  m i n  5. p™ = 4 • 3 - d with £>0 even, n = 1 or P  2  ( n ) > 7;  2  m i n  ( n ) > 7;  6. p = 2 " 3 ± 1 with m e { 4 , 5 } , * > 0; m  2  f  7. p = d + 2 n  2  m  8. p = d - 2 n  9.  2  p = 2 n  rn  m  • 3 with m <E { 4 , 5 } , £ > 0, n = 1 or P e  • 3 with m e { 4 , 5 } , l > 0 , n = l or P e  • 3 - d with m G e  10. p =  m i n  2  {4,5},  m i n  ( n ) > 7;  ( n ) > 7;  £ > 0, n = 1 or P {n) min  > 7;  wif/t £ odd, p = 1 (mod 12), n = 1 or P „ ( n ) > 7;  n  m i  22. p = 3 ±4with£>  0;  e  22. p = 3' ± 8 with ^ > 0; 13. p =  with rn <E { 4 , 5 } and £ odd;  14. p =  with £ odd, n=lor  n  15. p =  P  mill  ( n ) > 7;  to;  W. p =  to;  2  27. p = 3d - 2  with m e { 4 , 5};  28. p = 3d  with m € { 2 , 4 , 5 } .  2  2  Corollary7.4  rn  + 2  rn  Letp>5beaprime.  2. If there exists an elliptic curve over Q with 2-torsion and conductor 18p then one of the following 5 (mod 8), or p  must hold: p = 5, p ^ 2 (mod 3), p ^ 2 (mod 5), 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 2 3 p a  2  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 2 9p, where a £ {1,2,3}, for p satisfying p = 317 or 1757 (mod 2040) (i.e. p = a  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 2 9p, 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 Cremona's tables of elliptic curves up to conductor 130000 reveals the following list of the first few primes: a  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 p = Also, let S'(X) = {p £ S' : p < X}. We prove the following. n  Lemma 7.5  Chapter 7. On the Classification of Elliptic Curves of conductor 2 3 p a  7.2  2  233  The Proofs  7.2.1 Proof of Theorem 7.1 We proceed through the cases of Theorem 7.1 (with b = 2) and use the lemmata of Chapter 4 to resolve the Diophantine equations that arise. Notice that in all cases we are only concerned with solutions to the Diophantine equations 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 d = 2 3 p is of one of the following forms 2  m  e  + 1 is solvable then the prime p  n  om-2  p = 2 ~3 m  2  e  ± 1, p =  ¥  I  1  — ,  orp=  17.  2) If d = 2 3 + p is solvable then the prime p is of one of the following forms (see Lemma 4.9): 2  m  e  n  d - 2 3, 2  m  2 '3  e  m 2  - 1, 3 - 2 ~ ,  e  e  rn  2 "- - 3 , 5, 7, 17, or 73.  2  1  2  e  3) If d? = 2 "3 — p is solvable then the prime p is of one of the following forms (see Lemma 4.9): 7, 23, or 2 " ^ - ^ . T  f  n  4) If d = 2 "p + 3 is solvable then the prime p is of one of the following forms (see Lemma 4.8): 2  r  n  e  2  m-2  ±  e/2_  3  o  _  r 5  5) If d — 2™ + 3 p is solvable then the prime p is of one of the following forms (see Lemma 4.10): 2  e  y—^,  n  3 ± 2 ' (  m  2 +  \  d - 2, 2  m  7, 2 ~ m  2  - 1, or 17.  3^ 6) If d = 2 — 3 V is solvable then the prime p is of one of the following forms (see Lemma 4.10): 2  m  2  r „ / 2 + l _ 3^  m _ ^2^  2  ^  Q  r  ?  Chapter 7. On the Classification of Elliptic Curves of conductor 2 3 p a  7) If d? = 3 p — 2 e  n  234  2  is solvable then the prime p is of one of the following  rn  forms (see lemma 4.10): 2  m+l  +  2  1  e/2  3  d  » > n  >  17  1 9  '  m  +  2  —#—'  or  8) If d = 3 — 2 p is solvable then the prime p is of one of the following forms (see Lemma 4.8): 3 ^ - 2 " , or 7. 2  e  m  n  2  m  2  9) If d = p - 2 3 is solvable then the prime p is of one of the following forms (see Lemma 4.9): 2  n  rn  e  2 -3 m  2  + l,  e  3 + 2 ~, e  m  17, or 2 3  2  m  e  + d. 2  10) If 3d = 2 + p is solvable then the prime p is of one of the following forms (see Lemma 4.11): 11, or 3d - 2 . 2  rn  n  2  m  11) If 3d = 2 - p is solvable then the prime p is of one of the following forms (see Lemma 4.11): 5,or2 -3d . 2  rn  n  m  2  12) If 3d = p - 2 is solvable then the prime p is of the form 3d + 2 (see Lemma 4.11). This proves Theorem 7.1. 2  n  rn  2  m  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 d = 4 • 3 + p is solvable then the prime p is of one of the following forms (see Lemma 4.9): 2  e  n  13,  orp  n  d -4-3 , 2  e  = d -4-3 with P ( n ) > 7. 2) If d = 4 • 3 - p is solvable then n = 1 or P 2  e  m i n  2  e  n  m i n  (n) > 7 (see Lemma 4.9).  Chapter 7. On the Classification of Elliptic Curves of conductor  23p a  2  235  3) If d = 4p - 3^ is solvable then n = 1 or P ( n ) > 7 (see Lemma 4.8) and p = — 1 (mod 4). 4) If d = p — 4 • 3 is solvable then either p = 5, n = 1 or P j (n) > 7 (see Lemma 4.9). 5) If 3d = 4p - 1 is solvable then n 6 {1,2} (see Lemma 4.11). 6) If 3d = p + 4 is solvable then n = 1 so p = 3d — 4 (see Lemma 4.11). This proves Theorem 7.2. 2  n  m i n  2  n  e  m  2  n  2  7.2.3  n  n  2  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 d = 2 3 p + 1 is solvable then the prime p is of one of the following forms (see Lemma 4.7): 2  rn  e  n  2 - 3'±l, m  or 5.  2  2) If d = 4 • 3^ + p is solvable with £ even then the prime p is of the form p = d -4- 3 with n = 1 or P „ ( n ) > 7 (see Lemma 4.9). 3) If d = 2 3 + p is solvable then the prime p is of one of the following forms (see Lemma 4.9): 2  n  n  2  e  mi  2  rn  e  n  ^2 _ 2 3^ m  2 ~ 3^ rn  1  2  3^  <2 ~ rn  2, ~  2  m  2  — 3^  or p = d - 2 3 with P ( n ) > 7. 4) If d = 4 • 3^ - p is solvable with £ even then n = 1 or P j (?i) > 7 (see Lemma 4.9). 5) If d = 2™3 —p is solvable then one of the following must hold: p = 47, n = 1 or P i ( n ) > 7 (see Lemma 4.9). 6) If d = 2 p + 3' is solvable then the prime p is of one of the following forms (see Lemma 4.9): n  2  m  e  rain  2  n  m  2  i  m  2  n  n  m  n  3^+1  f  2  m  - 2  +  3  //  2  j  5  )  o  r  7  .  n  Chapter 7. On the Classification of Elliptic Curves of conductor  236  23p a  2  7) If d = Ap — 3 with p = 1 (mod 4) is solvable then the prime p is of the form or n = 1 or P i ( n ) > 7 (see Lemma 4.8). 2  n  e  m  n  8) If d = 4 + 3 V is solvable then the prime p is of one of the following forms (see Lemma 4.10): 3<±1 5 or — - — . 4 2  1  9) If d = 2 2  + 3p  m  i  is solvable then the prime p is of one of the following  n  forms (see Lemma 4.10): d - 32, 3 ± 8, or 7. 2  e  10) If d = 2 — 3 p is solvable then the prime p is of one of 5, 7, 23, or 31. 11) If d = 3 p - 2 is solvable then the prime p is of one of the following forms (see Lemma 4.10): 2  m  2  e  e  n  n  m  2  m-i  +  1  e/2  > > 5  3  >  67  or n = 1 or P i „ ( n ) > 7. 12) If d = 3 - 2 p is solvable then the prime p is of the form 3 l - 2 ~ (see Lemma 4.8). 13) If d — Ap - 3 , with t odd, is solvable then p = 13 or n = 1 or Pminfa) > 7 (see Lemma 4.8). 14) If d = p — 2 3 is solvable then the prime p is of one of the following forms (see Lemma 4.9): m  2  e  rn  2  n  ( 2  n  2  n  2  e  Tn  2  m  m _ 2  e  3 ' + l , 3^ + 2 " , 72, 193, 1153, 5, m  2  or n = 1 or P (n) > 7. 15) If 3ti = 2 +p is solvable then n - 1 and so p = 3d - 2 mm  2  m  n  2  (see Lemma  m  4.11). 16) If 3d = 2 +p is solvable p is either 5,13 or 29 (see Lemma 4.11). 17) If 3d = Ap - 1 is solvable then n 6 {1, 2} so p = ^<£±1 2 Mjtl 2  m  2  n  n  o  r  p  =  (see Lemma 4.11). 18) If 3d = p - A is solvable then n = 1 so p = 3d + A (see Lemma 4.11). 19) If 3d =p - 2 is solvable then n = 1 so p = 3d + 16 or p = 3d + 32 (see Lemma 4.11). This proves Theorem 7.3. 2  2  n  n  2  rn  2  2  Chapter 7. On the Classification of Elliptic Curves of conductor 2 3 p a  7.2.4  2  237  Proof of Corollary 7.4  We show that all the primes appearing i n 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 will 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 = 2 ~ — 3 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: m  2  e  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 have  m  — 3d with- m > 7. Modulo 3 we 2  1 (mod 3)  if m is even,  2 (mod 3)  if m is odd.  Chapter 7. On the Classification of Elliptic Curves of conductor 2 3 p a  2  238  So, assume m is odd. In this case we have  P =  3 (mod 5)  if m  3 (mod 4) and d  2  0 (mod 5),  0 (mod 5)  if m  3 (mod 4) and d  2  1 (mod 5),  0 (mod 5)  if m  3 (mod 4) and d  2  4 (mod 5),  2 (mod 5)  if m  1 (mod 4) and d  4 (mod 5)  if rn  4 (mod 5)  if m  2  0 (mod 5),  1 (mod 4) and d  2  1 (mod 5),  1 (mod 4) and d  2  4 (mod 5),  Thus, the only trouble seems to occur when m = 1 (mod 4) and 5 | d, In this case the prime is of the form P = 2  m  - 75fc  2  (7.2)  with m = 1 (mod 4).  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 10 however we chose this collection of primes since their reductions hit every congruence class modulo 7, 11 and 13. This means, to characterize such primes locally, we have to go as far as 17. We w i l l show for p of the form (7.2) thatp ^ 6 (mod 17). 7  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 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),  Thus, for p = 2 — 3d 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. m  2  Chapter 7. On the Classification of Elliptic Curves of conductor 2 3 p a  239  2  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) = 3d to obtain 4p = 3d\ + d?, = 1 (mod 3), where d — d\d - Finally, if p is of the form 2  2  (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 i n (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 3 p satisfy one of the following p = 5, p = 29, p = 1 (mod 3), or p = ± 1 or 3 (mod 8). 3  2  7.2.5 Proof of Lemma 7.5 We list the primes appearing in (7.1), (7.2), and (7.3) (except p the following table. Unless otherwise stated £ > 0.  n  = +f ) d2  m  in  Chapter 7.  On the Classification of Elliptic Curves of conductor 2 3 p Q  conditions  p 2 -3 m  2  e  m  m.-2  -4-3^  £odd  p" = ± ( d - 4 • 3 ) 2  £  n = 1 or P {n)  > 7,  mm  ( even  +  = d + 4 • 3*  ra = 4,5 or > 7, £ >4  - 3<)  2  2  ±(d  2  ra = 5 or > 7  3 ±(2 -  conditions  m = 4, 5 or > 7  ±1  e  240  2  3*  TO =  2  n  4, 5 or > 7  Ci  2  =  P  +  3^  4  n = 1 or P  min  (n) > 7  n = 1 or P (n)  > 7,  mm  £odd ±(d  -  2  2 3) m  d + 23 2  m  ±(3d -  TO =  m  3d + 2 3« + l 4 3d -4 2  P  p  ra > 7  i  2)  2  n  m >7  e  n  r  f  "+  = 4 =d ±2 2  n  4, 5, or > 7  3  p  =  me  _  3  2  n = 1 or 2  1  m  2  d  • 3*  n = 1 or P  min  (n) > 7  771 = 4,5 n = 1 or P „(n) m = 4,5  > 7  mi  m = 2,4,5, or > 7  rn  £odd  2  We are interested in counting the number of primes of each of these forms up to X. First we observe that for the forms i n the second column there are only finitely many primes satisfying the conditions with P m(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 w i l l just bound the number of integers of each form listed in the table. This w i l l then bound the number of primes as well. If rj(X) is an upper bound on the number of integers up to X satisfying one of the forms i n 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 limA^oo ^ p f j = 0. m  1) If 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 log X = O(TT(X)) such integers. 2) If ' " ^ is an integer then 2 ~" = - 1 (mod 3^)- It follows that the order of 2 modulo 3 , which is 2 • 3 , must divide 2(m — 2). Thus, 3 ^ \m — 2, hence £ < c log m for a fixed constant c. N o w if " \ < X then m < c\ log X for m  2  £  2  2  2 + 1  m  e  2  2  £ _ 1  _1  2  + 1  t  2  Chapter 7. On the Classification of Elliptic Curves of conductor 2 2> p a  2  241  some constant c\. The number of integers of this form is bounded by rj2(X) = (ci l o g X ) ( c l o g (  Cl  \ogX))=  o(ir(X)).  3) If ± ( 2 " - 3*) is an integer such that | 2 - 3 \ < X, it follows from a result of Ellison that m < c log X for some fixed constant c. Thus, m  2  m _ 2  r e<l  +  t o g  i  io 3 [c logX  f  2  _ _ ,  m  2  if 2 ~ m  3  <  0  , - 3'> 0  g  2  e  2  for some fixed constant c . Therefore, the number of primes of this form up to X is bounded above by 2  773 W  < < [c log X  log3  &  if2 " -3^>0,  2  m  3  2  where Cz is some fixed constant. Thus, n-i(X) = o(n(X)). 4) If 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 log X = O(TT(X)). 5) Consider the set of primes of the form p = \d — 2 3 \ up to X. If m and £ are even then factor to obtain m  2  2  2  P  =  \d +  m  2  e  2 l 3 ' \-\d-2 l 3 ' \. m 2  i 2  m 2  e 2  It follows thatp = d + 2 / 3 <" and 1 = \d - 2 l 3 > \. Eliminate d to obtain p = 2 / 3 ^ ± 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 m 2  m  2 r f J  e  2  m 2  t  2  2  p = |d - 2 3 ^ ° v 2 | | d + 2 ° 3 ° ^ | < X. mo  Let e = I A/2 -  /  m  £  1 and F = |d + 2 ° 3 ° v 2 | . The equation can be written as rn  2 3 °eF mo  e  f  /  < X.  According to Ridout [62], e cannot be too small, e =  V2-  d 2 o3 ° rn  e  > (2™o3*o)l+<5 '  (7.3)  Chapter 7. On the Classification of Elliptic Curves of conductor  2 3 p a  2  242  for any b > 0, where > means "except for finitely many mo and no" (independent of X). From (7.3) it follows that F (mo3<?o)<5  <X,  2  that is, +  V2{2 3 °) ' mu  e  1  < X.  This implies and so, where 5\ satisfies (1 - 5)(1 + 5\) = 1. Therefore, 23 m  <  e  X  2+&2  where S = 25\, whence 2  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) 2 3* < X ~ \ (ii) X ~ * < 2 3 < X +*\ In the first case, it follows directly from \d - 2 3 \ < X that m  2  2  6  2  m  s  Tn  e  2  e  d <  cX ~^l . x  2  In the second case, if d is large, say d > do := [\/2 3 ] + 1, write d = do + k. Then \d - 2 3^| < X becomes m  2  £  m  | ^ - 2 3 ^ + 2d fc + fc | < X . m  2  0  from which it follows that 2d k + k  2  0  < X.  But 2d > X ^ / so /c < A " / . Thus, the number of primes of the form \d — 2 3 \ up to X is bounded above by 1  2  2  5 2  2  0  2  m  e  max{X ^ 6  2  log X, X ~ ^ 2  l  s  2  log X = 2  o(n(X)).  Chapter 7. On the Classification of Elliptic Curves of conductor 2 3 p a  2  243  A similar argument works in the case when m is even, £ is odd, and i n 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 3 < X then m,i < clogX and d < \fX thus the number of primes of this form, 776 PO, satisfies 2  m  e  (X)  < c^/Xlog  X =  2  V6  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 , up to X, is of order O(TT(X)). 8) If 3d + 2 < X then m < c \ogX and d < c2VX, for some fixed constants c\ and c thus the number of primes of this form, 77s(X), satisfies 2  2  m  m  l  2l  77 pO < c V x log X = 2  8  o{ir{X)).  3 1 e  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 — 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  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  the  x  n  equation  y  n  +  2 pz a  =  2  In this chapter, we show, if p is prime, that the equation x + y — 2pz has no solutions in coprime integers x and y with \xy\ > 1 and n > p v , and if p ^ 7, the equation x™ + y = pz has no solutions in coprime integers x and y with jrcj/l > 1, z even, and n > p . A modified version of the contents of this chapter has been published [4]. n  n  2  vi2  n  2  l2p2  8.1  Introduction  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 representations to Diophantine equations of the form Ax  + By  p  q  (8.1)  = Cz , r  for p, q and r positive integers with 1/p + 1/q + 1/r < 1. We briefly outlined i n 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 x  + y  n  n  = 2 pz , a  2  (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 rational 2-torsion and conductor 2 p . In the case when p = 2 or 3 it is shown i n [BS04] that the only solution in nonzero pairwise coprime integers (x, y, z) is a  2  244  Chapter 8. On the equation x  n  +y  n  - 2 pz a  245  2  (p, a, x, y, z, n) = ( 2 , 0 , 3 , - 1 , ± 1 1 , 5). Thus, i n this chapter, we may take p to be a prime > 5. O u r main results are as follows: Theorem 8.1 If n an odd prime and p > 5 a prime (p ^ 7), then the Diophantine equation x  n  + y  n  = pz  2  has no solutions in coprime integers x, y and z with \xy\ > 1, z even, and n >  p . 12p2  Theorem 8.2 Ifn an odd prime and p > 5 a prime then the Diophantine equation X  n  + y = 2Z n  P  2  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 x  n  + y  n  = pz  2  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 x + y = 2pz n  at most finitely  n  2  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 x + y n  = 2 pz  n  a  246  2  that c = 0 (mod 2). A s i n [5] we associate to the solution (a, b, c) an elliptic curve •  E (a,  b, c):Y  = X  2  a  3  + 2 cpX a+1  +  2  2 pb X. a  n  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 = E (a, A{E)  (b) The conductor N(E)  b, c) is given by  a  =  2 p (ab ) . 3a+G  3  2  n  of the curve E = E (a, 6, c) is given by Q  N(E)  = 2 p 3a+5  2  ]Jq. q\ab  In particular, E has multiplicative reduction at each odd prime p dividing ab. (c) The curve E (a,  b, c) has a Q-rational point of order 2, namely (0,0).  a  (d) The curve E (a,b, a  c) obtains good reduction over Q(\/2 p) a  dividing p. Over any quadratic field K, the curve E (a, a  at all primes ideals  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 of the curve E = E (a, a  ord (j(E)) q  In particular, if ab ^ ± 1 then E (a, a  8.3  j(E)  b, c) satisfies < 0.  b, c) does not have complex  multiplication.  Outline of the Proof of the main theorems  To the elliptic curve E (a, b, c) we w i l l associate a weight 2 cuspidal newform / of level 32p (if a = 0) or 256p (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 w i l l a  2  2  refer to [Kf : Q] as the dimension of / .  If / has dimension > 2 then eg £ Q for some £. We w i l l see that n must divide Norm /Q(ce - af), for some a ^ e Z such that \af\ < £+ 1 (Proposition Kf  Chapter 8. On the equation x  n  +y  247  = 2 pz  n  a  2  8.6). This gives a bound on n in terms of £. The question then arises: H o w 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. N o w suppose / is of dimension 1, that is c\, e Z for all i. We again have that n must divide Norm /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 rational 2-torsion and conductor 32p or 256p . By Lemma 8.4 (d) E (a,b, c) has potentially good reduction at p, we w i l l see (Proposition 8.7) that this implies F has potentially good reduction at p, i.e. p does not divide the denominator of j(F). Also, by Lemma 8.4 (e) E (a, b. c) does not have C M , we w i 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 32p or 256p , potentially good reduction at p and without C M we w i 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. Kf  2  2  a  a  2  2  8.4 Galois Representations and Modular Forms In this section we describe how to associate to the elliptic curve E (a, b, c) a weight 2 modular form. Let E-= E (a,b,c) for some primitive solution (a,b,c) to (8.2). We associate to the elliptic curves E a Galois representation a  a  pi : Gal(Q/Q) -> G L ( F „ ) , 2  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.  248  On the equation x + y = 2 pz n  n  a  2  Let F„ be an algebraic closure of the finite field F 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 n  p  JiV  : Gal(Q/Q) -  GL (F„) 2  unramified outside of Nn and satisfying, if f(z) = Yln°=i " 9 ^ c  n  o r  Q  :  ^  =  tracep^j,(Frob ) = c (mod u) p  p  for all p coprime to Nn. Here, Frob is a Frobenius element at the prime p. If the representation p%, after extending scalars to F , is equivalent to pj_, for some newform / , then we say that pf is modular, arising from / . The next lemma follows from [5] Lemma 3.3. p  n  h  Lemma 8.5 Suppose that n > 7 fs a prime and that ptf is associated to a primitive h  solution (a, b, c) to (8.2) with ab ^ ± 1 . Put  The representation pf arises from a cuspidal newform of weight 2, level N (E), t  and  n  trivial Nebentypus character.  This lemma says that we can associate to the elliptic curve E = E (a b, c) a weight 2 modular form of level 32p (if a = 0) or 256p (if a = 1). a  2  8.5  :  2  Useful Propositions  In this section we collect together some results concerning the newforms of level N (E) from which our representation pf can arise. The proofs of these propositions can be found i n [5]. The first proposition gives a relationship between n and the coefficients of the newform. We w i l l use this result to obtain the bounds on n in the main theorem. n  t  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 oo  f =YJ  rn C  M  Q  (q := e ™ ) 2  Chapter 8.  On the equation x +y n  = 2 pz  n  a  is a newform of weight 2 and level N (E)  249  2  giving rise to pf and that Kj is a number  n  field containing the Fourier coefficients of f. If q is a prime, coprime to 2pn, then n divides one of either Norm  (c ± {q + 1))  Kf/Q  q  or Norm (c Kf/Q  q  ± 2r),  for some integer r < y/q.  In the case when the space of cuspforms of level N (E) contains newforms associated to elliptic curves with rational 2-torsion we will find the following result useful. n  Proposition 8.7 Suppose n ^ p is an odd prime and E = E (a, 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(E')  j-invariant  is not divisible by p.  Finally, i n the case when the space of cuspforms of level N (E) contains newforms associated to elliptic curves with rational 2-torsion and C M we w i l l need the following result. n  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 = ±2 , r > 0, 2 /ABC  and 2 splits in K.  r  (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 3 with s > 0 and 3 ramifies in K. r  8.6  s  Elliptic curves with rational 2-torsion  It is possible that the modular form associated to E = E (a, b, c) has rational integer coefficients in which case the results of the previous section w i l l not help i n 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: 32p a  2  Chapter 8.  250  On the equation x + y = 2°pz n  n  2  or 256p . We use the classification of such elliptic curves found i n Chapter 6. 2  We restate the relevant results (Corollaries 6.10 and 6.11) of Chapter 6 i n 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 S2p . Then E is isogenous over Q to a curve 2  of the form y = x +ax 2  3  + ax  2  2  4  with coefficients given in the following table. P  0-2  any  0  any  0  -P +l)/2  (  _  1 ) ( p  p  any  0  (_1)(p+l)/2p3  7  ±7  2 • 7  7  ±7  2-7  7  ±7  2 • 7  2  2ps  -P  + 8, s £ Z  ps  -2p  s - 8, s £ Z  ps  2p  2  2  1728  2  s + l, s e Z s  j-invariant  0,4  2  2  3  2  2  2  1728 1728 8000/7 -2  6  -2  6  64(4p-iy p 64(p-2)' P 64(p+2) P  i  a  Proposition 8.10 Suppose p > 5 is prime and that E/Q is an elliptic curve with a rational 2-torsion point and conductor 256p . Then E is isogenous over Qtoa 2  of the form y  2  = x  3  + 1J2X + a 4 X  with coefficients given in the following table.  2  curve  Chapter 8.  On the equation x  n  +y  = 2 pz  n  a  251  2  V  «2  a\  any any  0 0  ±2p  any  0  any  ±Ap  23  ±2  3  23  ±2  4  j-invariant  1728 1728  ±2p 2  ±2p 2p  1728 2 5  3  2  -23-39  2 - 23  6  5  - 23 - 39 2 • 23 3  3  2 3 4057 23 2 3 4057 236 -64(p-4) b  ::)  ;5  6  b  5  a  is  a  2s + 1, s e Z  ±4ps  2p  2s + 1, s <E Z  ±4ps  -2p  v"2s + 1, s e Z  ±Aps  2p  \/2s + 1, s € Z  ±Aps  -2p  2s - 1, s € Z  ±Aps  2p  2s - 1, s e Z  ±Aps  2p  V 2 s - 1, s € Z  ±Aps  2p  64(p' +4)  \/2s - 1, s e  ±Aps  2p  64(4p* + l)' p*  2  2  2  2  2  2  2  2  z  3  P  2  64(4p-l)  2  a  P  64(p*-4)  A  P  3  4  64(4p*-l)  2  64(p+4) >> p  3  3  a  z  64(4p+l)  2  PJ  4  P  3  4  2  2  ;i  ;1  1  The main feature of these propositions we w i l l use is that an elliptic curve E/Q with rational 2-torsion and conductor 32p or 256p either has C M or p dividing the denominator of j{E), with one exception: there are curves of conductor 32p when p = 7 without C M and potentially good reduction at p, namely y = x ± 7x + 14x and y = x ± A9x + 686x. 2  2  2  2  3  2  2  3  2  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 w i 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 l\N  + y), ^  '  •  Chapter 8. On the equation x +y n  252  = 2 pz  n  a  2  where the product is over prime I. Proposition 8.11 (Kraus) Let N be a positive integer and f = 2^2 >i nQ c  n  n  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 c  e Z. Then we may conclude that  p  c £ Z / o r all n > 1. n  Proposition 8.12 (Kraus and OesterU) Let k be a positive integer, x « Dirichlet character of conductor N and f = Y^ >o " 9 c  n  n  acter xfo  r  be a modular form of weight k, char-  To(N), with C,,. £ Z. Let pbe a rational prime. If Cn = 0 (mod p) for all  n < p.{N)k/V2, then c = 0 (mod p) for all n. n  We now proceed with the proofs of Theorems 8.1 and 8.2; i n each case, from Lemma 8.5, we may assume the existence of a weight 2, level N cuspidal newform / (with trivial character), where N £ {32jD ,256p } . 2  2  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 ,  flWp+l)  lfN =  [64p(p+l)  ifJV = 256p .  3V, 2  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 coefficients of the form / . Next, we note that [K : Q] < g^N) f  where  g^N)  denotes the dimension (as a C-vector space) of the space of cuspidal, weight 2, level N newforms. A p p l y i n g Theorem 2 of Martin [49] we have 5o (32/) < +  < 3p , 2  Chapter 8. On the equation x + y n  and  253  = 2 pz  n  a  1  ,, 256p + 1 o g+(256p ) < — < 22p 2  9 2s  P  2  Combining these with inequalities (8.3) and (8.4), we may therefore conclude that f (y8p(p+l) + l ) n  ^ \ )  ,  i f / V = 32p ,  6 P  2  4 V  N  I f >/64p(p + 1) + l j  (  8  -  5  )  i f / ^ = 256p . 2  It follows, after routine calculation, that ,12p jf /V = 39r) 12p ifAr 32p , 22  2  =  p  l i32 p  ifAT = 256p^  P  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 c for all n > 1. In such a situation, / corresponds to an isogeny class of elliptic curves over Q with conductor N. Define Tl  r =  E n>l,(n,2p) = l  a  n  d  9*=  E n>l,(n,2p)=l  i( )^ '  CT  re  n  whereCTI(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 512p . A p p l y i n g 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: 3  (i) There exists a prime /, coprime to 2p, satisfying I < 128p (p + 1) and ci = 1 (mod 2). 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<  128p (p + 1) + 1 + 16JVP + 1 < p , 2  2p  (8.6)  where the last inequality is valid for p > 5. In the latter situation, there necessarily exists a curve, say F, in the given isogeny class, with a rational 2-torsion  Chapter 8. On the equation x  n  +y  n  =  254  2°pz  2  point. Propositions 8.9 and 8.10 then immediately imply Theorems 8.1 and 8.2. Regarding Theorem 8.1, where N = 32p , 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 i n this case since we are assuming c = 0 (mod 2) and a, b, c pairwise coprime). Regarding Theorem 8.2, where TV = 256p , we apply Proposition 8.10 to conclude 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. 2  2  /  Corollary 8.3 is an easy consequence of Theorems 8.1 and 8.2, after applying a result of Darmon and Granville [DG95] (which implies, for fixed values of n > 4 and a, that the equation x + y = 2 pz has at most finitely many solutions in coprime, nonzero integers x, y and z. n  8.8  n  a  2  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 x + y = 7z n  n  2  with x, y and z coprime nonzero integers, may, as in e.g. Kraus [38], be treated tor fixed values for n. We w i l l not undertake this here.  Chapter 9 O n  the  equation  x  + y  3  =  3  ± p  m  z  In this chapter we restrict our attention to determining primes p for which x + y = ±p z can be shown to be unsolvable in integers [x,y, z) for all suitable large primes n. 3  9.1  3  rn  n  Introduction  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 i n 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 x + y i  3  = ±p z m  n  has no solutions in coprime nonzero integers x, y and z, and prime n n >p  8p  and n \ rn.  255  (9.1) satisfying  Chapter 9. On the equation x + y = 3  ±p z  3  256  m  We remark that in the case that n \ m the equation can be written as x + y = z . Kraus has treated these equations in [44]. So, i n what follows, we w i l l assume n\m. As an almost immediate consequence of this theorem, we have: 3  3  n  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  9.2  n\m.  Frey Curve  Let p be a prime number > 5, n a prime > 7 and m a positive integer. We consider a proper, nontrivial solution (a, b, c) of the equation a + b = ±p c , i.e. pgcd(a, b,pc) = 1 . We suppose, without loss of generality, that the following conditions are satisfied: 3  3  rn  n  1  {  — 1 (mod 4) 1 (mod 4)  if c is even, if c is odd.  (9.2)  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 E ,b, with equation a  = x  2  3  y  + 3abx + b - a , 3  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: 'c (a,b) 4  =  -2 3 ab 4  2  c {a,b) = 2 3 ( a - 6 ) 5  3  3  3  6  A(a,b) = - 2 3 V c 4  m  (9.4)  2 n  We determine the conductor Ns 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. a  J  pgcd denotes the pairwise gcd.  b  Chapter 9. On the equation x + y = 3  3  ±p z  257  m  Lemma 9.3 We have (under conditions (9.2) on a, b, and c) 2 • 3 pTZ  if c even, b = — 1 (mod 4),  2  Ns ,  a b  = { 2 3 pTl 3  ifc odd, v (a)  2  = 1 and b = 1 (mod 4),  2  2 3V&  ifc odd, v (a) > 2 and b = 1 (mod 4).  2  2  Proof. We w i l l use Theorems 2.1, 2.3, and 2.4 to compute v e ( N E ) for all primes I. To do this we first need to move the point of order 2 to (0,0). Applying the change of variables A  B  x = X + (a - b), y = Y,  the curve E ^ is Q-isomorphic to a  E.  a b  : Y  = X  2  + 3(a - b)X  3  + 3(a  2  2  (9.5)  - ab + b )X. 2  The invariants of this model are still as i n (9.4). 1) Let I be a prime number > 5 which divides pc. A s the integers a, b and pc are prime to each other we have I \ a — b and equation (9.5) is minimal at /. O n 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 vi(N )  [1  if £ divides pc,  [0  if £ does not divide pc.  = {  EAB  (9.6)  2) We determine the exponent of 2 in N E 2.1) Suppose that c is even. In this case ab is odd, because pgcd(a, b,pc) = 1. We have ±p c = (a + b)(a — ab + b ) and the number a — ab + b is odd. As n is > 5, we have UB  rn  n  2  2  2  2  a + b = 0 (mod 32).  As a and b are odd, it follows that 4 does not divide a — b. Therefore iv (3(a-b))  = v (a-b)  2  \v (3(a  2  2  2  - ab + b ))  = l,  =0.  2  Thus, from Theorem 2.1, the value of v (N ^ ^) depends on the congruence class of 3(a - 6) modulo 8 (note v (A) > 4 + 2n > 14). It follows from a + b = 2  2  E  A  Chapter 9. On the equation x + y = 3  ±p z  3  258  m  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(NE 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 0  (v (3(a-b))  = 0,  \v (3(a  + b )) = 0,  2  -ab  2  2  2  thus, from Theorem 2.1 the value of v 2 { N ( ^ ) depends on the congruence classes of 3(a - b) and 3(a - ab + b ) modulo 4. Since 6 = 1 (mod 4) then E  2  A  2  f 1 (mod 4) 3(a - b) = < 1-1 (mod 4)  if a EE 0 (mod 4), i f a = 2(mod4),  and , -l(mod4) 3 a - a6 + b ) = { 1 1 (mod 4) 0 2  0  if a = 0 (mod 4), '  2  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 • 3.1) Suppose that 3 divides c. Under this hypothesis 3 does not divide ab. From the equality a + 6 = p c we have a = -b (mod 3) and 3 does not divide a — b or a — b . It follows that 3 divides a — ab + b . Therefore vs(3(a — b)) = 1 and v (3(a -ab + b )) > 2. The Neron type of E at 3 is then i ; with v = 2nv^(c) - 3 and v ^ ( N E ,,) = 2 by Theorem 2.3. 3.2) Suppose that 3 divides ab. We have in this case 3 does not divide a - b or a — ab + b since gcd(a, b) = 1. Therefore v-s(3(a — b)) = 1 and v^(3(a — ab + b )) = 1. The Neron type of E ^ at 3 is thus III and again v (NE ) = 2 by Theorem 2.3. B  3  3  3  m  3  B  n  2  2  2  2  3  a}b  A  2  2  2  2  a  3  a  b  Chapter 9. On the equation x + y = 3  3  259  ±p z m  n  3.3) Suppose that 3 does not divide abc. A s 3 does not divide pc, we have from the equality a? + b = p c that a = b (mod 3) and so a - b = 0 (mod 3) and 3 does not divide a - ab + b . Thus v (3(a - b)) > 2 and v (3(a - ab + b )) = 1. The Neron type of E ^ at 3 is thus III, and we have again vs(Ns J = 2. This completes the proof of Lemma 9.3. • 3  m  n  2  2  2  3  2  3  a  a  T h e M o d u l a r G a l o i s Representation p%  9.3  b  Let Q be an algebraic closure of Q and E ^[n] the subgroup of n-torsion points of E b(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 E fi{n] by a  ai  a  : Gal(Q/Q) -  GL (F„). 2  Let k and N(pn' ) denote the weight and conductor of pn' respectively, which are defined by Serre i n [64]. b  Lemma 9.4  b  1. k = 2.  {  18p  if c even, b = — 1 (mod 4).  36p  if c odd, v (a)  >2andb=l  (mod 4),  72p  ifcodd, v {a)  = 1 and 6 = 1  (mod 4).  2  2  3. The representation pn is irreducible. b  Proof. 1) Recall n ^ p by assumption. If n \ c then E b has good reduction at p. Otherwise, E ^ 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 E has multiplicative reduction at q (Lemma 9.3) and the exponent of q in the minimal discriminant of E fi 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). a>  a  a<b  a  Chapter 9. On the equation x + y = 3  260  ±p z  3  m  n  3) Suppose pn is reducible. Since E ^ has a point of order 2 there exists a subgroup of E {Q) of order 2 stable under Galois G ( Q / Q ) . b  a  atb  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 —15 and 255 , 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 E ^ has j-invariant 3  3  a  . _ 6912(a6)  3  the lemma follows.  •  Given an integer N > 1 we let S2(TQ(N)) denote the C-vector space of cuspidal 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'n is irreducible of weight 2 and E ^ is modular (by the extraordinary work of Breuil, Conrad, Diamond, Taylor, and Wiles: [80], [77], [8]) there exists a newform / e 5 {N(pn' )) whose Taylor expansion is b  a  +  b  2  t^g  +  Yl  where  n(f)Q  a  n  q  = e" 2  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,, one has H  ai{f) = ai{E ) atb  (mod  B).  It follows that n | Norm  X / / Q  ( a ( ( / ) - a,(£•„,(,)),  where Kf denotes the field of definition of the coefficients.  (9.7)  Chapter 9. On the equation x + y = 3  3  ±p z m  261  n  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 i n 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 > p (in case N = 18p or 36p) or n > p (in case iV = 72p). This follows from 9.7 and applying Theorem 1 of [49] to obtain 4p  8p  {5p/4  ii N = 72p.  To finish the proof of Theorem 9.1 we w i 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 a (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 3 p . A p p l y i n g the Proposition of Kraus and Oesterle to / * — g*, and using a(l) = 1 + 1, for all primes I one of the following necessarily occurs: m  4  3  2  (i) There exists a prime I, coprime to 6p, satisfying I < 144p(p + 1) and ai(/) = l ( m o d 2 ) . (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{E ) obtain the inequality ab  n <l + l + 2Vl<  144p(p + 1) + 1 + 24yfp{p+  we  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 x + y = 3  ±p z  3  m  n  262  n < p (if N = 18p or 36p) or n < p (if N = 72p). This completes the proof of Theorem 9.1. 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Acad. Sci. Paris Ser. A-B, 273:A238-A241,1971.  Bibliography  269  [80] A . Wiles. Modular elliptic curves and Fermat's last theorem. Ann. of Math. (2), 141(3):443-551,1995.  Appendix A O n  the Q-lsomorphism  C u r v e s w i t h 2-Torsion  C l a s s e s of Elliptic a n d C o n d u c t o r  2 3V a  5  In this appendix, we provide the proof of the main lemmata used in our classification of elliptic curves. In particular, we w i l l give a list of elliptic curves which contains a representative for each Q-isomorphism class of curves containing 2 torsion and having conductor of the form 2 3 p . Let E be an elliptic curve over Q of conductor 2 3 p , 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 a  M  2 y  l3  L  s  N  — x + ax + bx, 2  3  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 , !  b (a - 46) = ± 2 3 V 2  2  Q  (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 integers 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}. y = x +ax 2  3  + (14X  2  2  Diophantine Equation 1  d - 2 3p  2  d - 23  = ±p  3  d - 2p  = ±3  4  \2 _  5  pd - 2 3  6  pd - 2  7  3d - 2 p  8  3d - 2  9  3pd - 2 ' = ± 1  2  m  2  e  m  2  e  m  2  ±3  2  2  m  2  n  10  ,d - 3 p  11  d - 3 =  ±2 p  12  \d - p =  ±2 3  13  d - 1= ± 2  14  , pd - 3 = ±2  15  pd - 1 =  16  3d - p = ±2  17  3d - 1 =  18 19  e  2  m  f  2  m  n  2  m  ±2 3  2  m  n  e  ±2 p  3pd - 1 = ± 2  V=  21  2d - p = ±3  22  2d - 1 =  e  2  2pd -l  25  6d  26  6d -  27  6pd - 1 = ± 1  2  2  = ±3  1 =  2  n+1  T  r2  r2+1  +1  2  3+1  r 2 +  ± 1  ±p  n  r  r2  2  d 2  r3+1  3  2  r i + 2  3  r 2 + 1  p  r 3 + 1  d  p  d  r 3  2* 3 p d 2  r i + 2  3  2  p  3  2r +l 3  2  2  p d  2  r2+1  2r + l 2r  22ri+l3«+2r p2r  r3  r 2 + 1  2  p  d  +  3  2 p  22n+l3«+2T22r'i + l 3 2 r 2  2  2 p  3  3  2r +l 3  2r +l 3  2 n +1 3 2 r + l ^ n + 2 r 2  2  r 3 + 1  1  3  3  2n+l32r pn+2r 2r l 2r 2r 2  2  r 2  +2  3  2r,+l3^+2r pn+2r  r3  r i + 2  r  3  lpn+2r  2  r3  r  +  2  +l  3  +\  3  2  2 3 p d 2 »+ 3  2  2  22r!32r +l  r3  r2  3 p  22r £2r  2 3 p rf ri+2  3  2r3  T3  2 3 p d r2  2  2r2  1  2r,  3  £2r pn+2r  2r 3f+2r p2r  2  3 + 1  3  2p  l  V  T2  2  x  d  +\  3  22r <i2r p2r  r3+1  2^+ 3 p d  3  ^e+2r 2r  2  3d  r2+1 r3  3  r i + 1  3  + lp2r  ^e+2r pn+2r  a  2n  2ri  T3  P  2r  r  r2  n+2r  2  22T\ p  +l  2  2  2  +l  3  m + 2 r i - 2 £ 2 r + lp2r  3 p' d 3 p d  ri+1  ri+2  f  n  2  r3  2> 3  n  = ±l  - p =  d  3  3  r  ±3 p  e  r2  ri+2  24  2  2 2  1  ; 2pd -3 2  1  +l  3  2  2-m.+2r -2^2r  r3  ''l+l r  2  2  23  2  ri+1  e  e  2  2 3 p d  n  n  2  r2  r2  2  2d - 3 ± 2d - 3 = ±p 2  n  m  20  2  3  2m+2n-2g2r  2 3 p d  ri+1  m  2  p  3  r i  2 3 p d  m  2  3  ri+1  3 V m  r 2 +  3  m+2 -2^2r p2r  2  r i  2  2  2 3 V ^ 2 3 V d n 3^2+l r +l ri+1  e  e  p d 3  3  ^2r pn+2r  1  r3+1  r 2 +  2  3  2m+2ri-23f+2r p2r  T3+1  r i  2  2  2m.+2r -2^2r2p2r  pd  r2  2  3  m+27'i —2  2 3 p'' (i  n  m  2  2  2  = ±2  n  2  2 ^3 p d  n  2  2  r3  r3  2 *3  = ± 1  2  1  pd  r2  r  m+2ri- ^e+ r pn+ r 2m+2r -2 e+2r p2r  p d  r2  r  2  T3  2 '3  p  = ±p  m  r3  r2  r  e  n  pd  2  T2  ri  = ±3  m  r  23  = ± 1  l  3  i  2 3  en  =  r  Ti  e  m  a.4  2  2  n  n  m  2  d  a  = ± 1  n  £  2r +l32r 1  2  + lp2r  3  3  22ri +l32r +lp2r + l 2  3  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 righthand side of the Diophantine equation, otherwise they must be > 1. A warning to the reader. The following proof is tedious and very repetitive. We have included all the details only for the purpose of completeness.  272  Appendix A. Q-Isomorphism Classes  For those interested i n getting an idea of the flavor of the proof, we suggest only reading a few cases. P r o o f . 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 = 23p , i  j  and 0 < 2i < a, 0 < 2j < (3, 0 < 2k < 5.  k  We obtain from ( A . l ) 2 _ i+ y 2  a  2  k p  =  ±  2  ^ - ^ - ^  6 p  -  2  k  In what follows we consider the following twenty-seven cases: l ) z + 2 > a-r 2i,j>(32j, k> 5- 2k, k<5- 2k, 2)z + 2 > a - 2z, j > 0 - 2j, 3)i + 2 > a - 2i, >j32j, k = S- 2k, 2j, k> 5- 2k, 4)z + 2 > a 2i,3<[32j, k<6- 2k, 5)i + 2 > a 2i,j</36) 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, - 2j, k > 5 - 2k, 13) z + 2 < a -2i,j<(3 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, 3  L  7  17) i + 2 < a- -2i,j = f3 18) i + 2 < a -2i,j = 0 19) i + 2 = a - 2%, j > >0 20) z + 2 = a -2i,j 21) i + 2 = a -2i,j> 0 22) z + 2 = a -2i,j<0 < (3 23) z + 2 = a -2i,j 24) z + 2 = a -2i,j 25) i + 2 = a -2i,j  - 2j, k<5 - 2k, - 2j, k = 5 - 2k, 0 - 2j, k>5 - 2k, -2j,k<  6- 2j, k = 6 - 2j, k> 6 - 2j, k <S < 0 - 2j, k = 5 = p - 2j, k>5 -  2k, 2k, 2k, 2k, 2k, 2k,  (A.2)  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 v (a?) = a - 2i, V2,(a ) = (3 — 2j and v (a?) — 5 — 2k so a, (3, and S are even. Therefore, v (a) = § - i, V3(a) = f — j , and v (a) = | - k. Let 2 a u = K 22 *32 -?r)2 P 2  2  p  2  p  so (A.2) becomes 2  _  u  3i-a+2 3j-P 3k-6  2  3  p  =  ±  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 d - 2 3p 2  m  e  = ±1,  n  with m, £, n > 1. The model for E can be written y = x + 2? _ i 32 ~ p2- dx 2  3  j  k  + 2 3V^-  2  J  There exist six integers r i , q\, r , q ,7- , and g such that 2  - - « = 2gi+ri,  2  - - J =  3  3  2<7  2  + r  - - fc = 2g + r ,  2 )  3  with r i , r , r € {0,1}. There are two cases to consider: l . i ) W e h a v e ( m , r i ) = (1,0). Putting 2  3  •7"  2  QI  2(«-l)32g  2j)2g  3 '  2 («- )3 «73 '23 ' 3  1  3  3(  we obtain the new model for E Y  2  = X + 2 3 p dX 3  2  r2  T3  2  +  23 p 3  e+2r2  which is the curve i n case 1 of the lemma with r\ = 2.  X  n+2r3  3  Appendix A.  Q-Isomorphism  274  Classes  l . i i ) W e h a v e ( m , r i ) > (1,0) . Putting 1  x  x  =  Y  2<?i 3 <?2^2(73 2  2  -  V  2 'i 3 3f  '  3 l  ?2p3g ' 3  we obtain the new model for E Y  2  = X  + 2 3 p dX  s  ri  r2  rs  +  2  m 2  +  2 r  i- 3 2  £ + 2 r 2  p"  + 2 r 3  X  which is the curve i n 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; (a ) = a — 2i, ^3(a ) = (3 — 2j and t> (a ) = A; so a, (3, and fc are even. Therefore, v {o) = f - h V3(a) = f - j, and u (a) — | . Let 2  2  2  2  p  p  2  a u= — 22- 32-lp2  :  %  so (A.2) becomes 2  u  _  2  3z-a+2 3j-,a 3  -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 + 2'- 3%- p?dx 3  i  j  + 2*3? p x.  2  k  There exist six integers v\,q\, r , q , r , and q such that 2  a'  2  3  3  /?  -'-t  = 2q +.n,  - - J  1  k - = 2q +  = 2q +r , 2  2  3  r, 3  with r\,r ,r e {0,1}. There are, again, two cases to consider: 2.i)Wehave ( m , n ) = (1,0). Putting 2  3  ' 1  Y=  Lexicographic order.  X  22(c?l-1)3292p2g3 '  Y=  2 ^ i - )3 <?2j 3 3 ' 3  1  3  9  (?  I  Appendix A.  Q-Isomorphism  Classes  275  we obtain the new model for E Y  = X  2  + 2 3 p dX  3  2  r2  r3  +  2  23 p X 3  e+2r2  2r3  which is the curve i n case 2 of the lemma with r\ = 2. 2.ii) We have ( m , r ) > (1,0). Putting a  y -  x Y  =  V  we obtain the new model for E Y  2  = X  3  + 2 ¥ p dX ri  2  ri  +  2  2  m + 2 r  ^H  p  i + 2 r 2  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(a ) = ct — 2i,v (a ) = (3 — 2j so a and /3 are even. Therefore, v (a) = ^ — i,v (a) = § — j. Also, v (a ) > k = 8 — 2k so v (a?) > where £3 denotes the residue of k modulo 2. Let 2  2  3  2  3  2  p  p  22-'-32"ipso (A.l) becomes - 2 ~ 3 J-  pu t3  2  3i  a+2  3  = ±1,  0  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 y = x + 22^32 -ip^dx 2  3  +  2  2 3 p x. i  j  k  There exist six integers r\, q\, r , q , r , and q such that 2  a - - i = 2q +r , 1  1  2  3  3  (3 - - j = 2q + r , 2  2  k - = 2q + r , 3  3  Appendix  A.  Q-Isomorphism  276  Classes  with r\, r , r G {0,1}. We have two cases to consider: 3.1) Suppose £3 = 0. 3.1.i) If (m,n) = (1,0), then putting 2  3  X=  * „ 22(9i-l)32q p2 2  ,Y'  g3  V  2 ('?i- )3 ?2p3q ' 3  1  3<  3  we obtain the new model for E Y  = X  2  + 2 3 p dX  3  2  T2  T3  +  2  23 p X 3  e+2r2  2r3  which is the curve in case 2 of the lemma with n = 0 and r\ — 2. 3.1.ii) If ( m , r ) > (1,0), then putting x  X=  * ,Y= 2<?1 3 92p2q ' 2  V  2 ? 3 « p « '  n  2  31  3  1  3  2  3  we obtain the new model for E Y  = X  2  + 2 S p dX  3  ri  r2  Ti  +  2  2  m + 2 r  ^- 3 2  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 =  x  —  -  —  - — • ,  —  22(91-1)3292^2(93-1+1-3) '  v  Y -  y  2 (9l- )3 92p3(9.3-l+r ) ' 3  1  3  3  we obtain the new model for E Y  = X  2  + 2 3 p ~ dX  3  2  r2  2  r3  +•  2  23 p- X 3  e+2r2  3  2r3  which is the curve in case 5 of the lemma with r\ = 2 and r = 1 — r . 3  3  3.2.ii) We have (rn, n ) > (1,0) and r = 0. Putting 3  X  = 2  „ 29i  „  !,  , ,  ,,  3292^2(93-l+r-i)  '  Y = ~  2 9l3 92p (<?3-l+r3) ' 3  3  3  we obtain the new model for E Y  2  = X  3  + 2 ¥ p - dX ri  2  2  r3  2  +  2  m + 2 r x  - 3 2  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.  277  Q -Isom orphism Cla sses  4. We have i + 2 > a - 2i, j < (3 — 2j and k > 5 — 2k. In this case V2{o?) = a — 2i, t> (a ) = j and f ( a ) = 5 — 2k so a, j, and 5 are even. Therefore, 2  2  3  v (a) 2  — § ~~  p  ^ 3 ( ) = o' a  a  n  d  V  P(°)  f 2 ^- Let —  =  a 22" 35p2l  so (A.2) becomes 2 _ 23i-<*+2p3fc- 5 _ -J-3/3-3J 1  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 - 2p 2  rn  =  n  ±3 , e  with m, £, n > 1. The the model for £ can be written y = x 2  3  + 22- 3ip*~ dx i  k  +  2  2 3 p x. i  j  k  There exist six integers v\, q\, r , q , r^, and 93 such that 2  a. --i  with  = 2q .+ n, 1  2  j - = 2q + r , 2  5 - - k = 2q + r ,  2  3  3  r , r3 6 {0,1}. There are, again, two cases to consider: 2  4.i) We have (m, n) = (1,0). Putting 22( i-l)32 ,2f/3 ' 9  2 (<?i-l)3392p3 3 ' 3  92?  9  we obtain the new model for E Y  =X  2  3  + 2 3 p dX 2  r2  r3  2  + 2 3 3  2 r 2  p  n + 2 r 3  which is the curve in case 3 of the lemma with 7*1 = 2. 4.ii) We have (m, rj) > (1,0). Putting  x  = 2291 ,5, „, , y = 3292p2g ' 3  2 <?l 3392p3Q3 ' 3  X  Appendix A.  Q-Isomorphism  278  Classes  we obtain the new model for E Y  = X  2  + 2 3 p dX  3  ri  r2  r3  + 2  2  ^- 3  m + 2 r  2  2 r 2  p  n + 2 r 3  X  which is the curve i n 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 v (a ) = ct — 2i, v (a ) = j and v (a?) = k so a, j , and k are even. Therefore, v (a) = f - i, v (a) = 5 , and v (a) = | . Let 2  2  2  3  p  3  2  p  a  u  :—77  =  2?-*3M  so (A.2) becomes u  - 2 *~  2  3  =  a+2  ±3< ~ ip ~ , 3  3  s  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 d - 2 2  =  rn  ±3 p , e  n  with m, £, n > 1. The the model for E can be written y = x 2  3  + 22-^p^dx  +  2  2 3 p x. i  3  k  There exist six integers r\,q\, r , q , T , and 173 such that 2  '^-i  = 2q +r , 1  2  3  ^ = 2q + r ,  1  2  ^ = 2g + r ,  2  3  3  with ri, r , r e {0,1}. There are, again, two cases to consider: 5.i) We have (m, r ) = (1,0). Putting 2  3  x  22(<y-i-l)32<, 2g ' 2p  2 (9i- )3 92p393 ' 3  3  1  3  + 2 3  2 r 2  we obtain the new model for E Y  2  = X  3  + 23 2  T2  p dX T3  2  3  p  2 r 3  X  Appendix A. Q-Isomorphism  279  Classes  which is the curve in case 4 of the lemma with r\ = 2. 5.ii)Wehave (m,r\) > (1,0). Putting  2 ?l 2 (  3 92p2q  2 '' 3 ?2p^l* '  '  2  3  3  1  3l  we obtain the new model for E Y  =X  2  3  + 2 3 p dX ri  r2  r3  +  2  2 ^- 3 p X m+2  2  2r2  2rs  which is the curve i n 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 ) = a — 2i and vz(a ) — j so a and j are even. Therefore, v (a) = § — i arid W3(a) = | . Also, v (a ) > k = 6 — 2k so u (a) > where £3 denotes the residue of k modulo 2. Let 2  2  2  2  p  p  a 1* =  :  ZZ  22 32p 2 J  so (A.2) becomes p« 2_ u  2  3t-a+2  =  ±  3  P - 3 i  )  with 3i - a + 2 > 1 and P - 3j > 1. Let d = u, m. = 3i — a + 2,  then (d, m,  £ =ft— 3j,  is a solution to p  t 3  d -2 2  = ±3^  m  with m,£> 1, and the model for E can be written 1  y  2  = x + 2^~ 3ip ^ dx 3  l  t  L  +  2  2 3 p x. z  j  k  There exist six integers r\,q\, r , qi, r , and 53 such that 2  a --i  = 2qi+n,  3  j' - = 2q + r , 2  2  k + 63  =2q  with r\, r , r e {0,1}. We have two cases to consider: 6.1) Suppose 63 = 0. 2  3  3  + r, 3  Appendix AJ Q-Isomorphism  6.1.i) If (m,n)  280  Classes  = (1,0), then putting a;  X =  Y  22(gi-l)32 2p2 ' 9  =  g3  y 2 (9i-I)33q2p3c73 ' 3  we obtain the new model for E Y  = X  2  + 2 3 p dX  s  2  r2  r3  +  2  23 p X 3  2r2  2ri  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  2 91 3292p2q  '  2  3  Y  =  y  2 ?l 3 92p <33 3(  3  '  3  we obtain the new model for E Y  = X  2  + 2 3 p dX  3  ri  r2  r3  +  2  2 + ^- 3 p X m  2r  2  2r2  2r3  which is the, curve in case 4 of the lemma with n = 0 and r\ = 0 or 1. 6.2) Suppose 63 = 1. 6.2.i) If (m, ri) = (1,0) and r = 0, then putting 3  x  X = 2  y  -. Y  2(9i-l)329 2(93-l) '  2 < ?l-l)3 92p3(93-l)' 3  2 p  3  <  we obtain the new model for E !  Y  = X  2  + 2 T p dX  3  2  2  2  +  2  23 pX 3  2r2  3  which is the curve in case 6 of the lemma with r\ = 2 and r = 1. 6.2.ii) If (m, n) = (1,0) and 7-3 = 1, then putting 3  2 ('/l-l)32<72p293 2  y =  23(91- l ) 3 3 9 2 p 3 9 3 '  we obtain the new model for E Y  = X  2  3  + 2 3 pdX 2  T2  2  +  2 3 pX 3  2r2  which is the curve in case 6 of the lemma with r\ = 2, r = 0. 6.2.iii) If ;(m, r\) > (1,0) and r = 0, then putting ;  3  3  !  ,Y =  X  2 <n 3292^,2(93-1)' 2  Y=  V  2 'i3 92p ( «-i)' 3 (  3  3  (  Appendix A:  Q-Isomorphism  Classes  281  we obtain the new model for E Y  = X  + 2 3 p dX  3  2  n  r2  2  +  2  2  -3 pX  rn+2ri  2  2r2  3  which is the curve i n case 6 of the lemma with r\ = 0 or 1 and r = 1. 6.2.iv) If \(m,r\) > (1,0) and r = 0, then putting 3  3  X—  1  x 2 <?i 3 92p2q ' 2  2  Y=  3  v 2 ?l 3 ?2p3<y ' 3<  3<  3  we obtain the new model for E Y  = X  + 2 ¥ pdX  3  2  ri  2  + 2  2  m + 2 r i  - 3 2  2 r 2 P  X  which is the curve in case 6 of the lemma with r\ = 0 or 1 and r = 0. 3  7. We have i + 2 > a - 2i, j = (3 - 2j and /c > 8 - 2k. In this case u (a ) = a — 2i, and v (a ) = 8 — 2k so a and 8 are even. Therefore, u (a) = § — i and •Up(a) = I — k. Also, ^ ( a ) > j = (3 - 2j so v (a) > where e denotes the residue of j modulo 2. Let 2  2  2  p  2  2  3  2  a  u = —  2 2 " ' 3 2 p5"  so (A.2) becomes 3 -a - 2 - p £2  2  3r  a+2  3k  = ±1,  s  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 p" = ±l, 2  m  with m, n > 1, and the model for E can be written 1  y = x + 2 ? - ^ p ^ d a ? 2  3  +  2 ffp x. i  k  There exist six integers r\, q\, r , g , r , and <? such that. 2  -~ i i  = 2qi+ri,  2  2  3  3  = 2g +r , 2  2  - - A; = 2g + r , 3  3  Appendix A. Q-Isomorphism  282  Classes  with r\, r , r € {0,1}. We have two cases to consider: 7.1) Suppose e = 0. 7.1.i) If (rn,n) = (1,0), then putting 2  3  2  i i  f  Y = 2 ('»- )3 "2p3g  X = ^-, . „ p2< „ 3 ,' 22(qi-l)32<; 2  3  ?  1  3  3  '  we obtain the new model for E Y  =X  2  + 2 3 p dX  3  2  T2  r3  +  2  23 p 3  2r2  X  n+2r3  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  2 1l 32<72p2<7 '  V  2 ? i 3 </2p3(j3 ' 3 ,  2  3  3  we obtain the new model for E Y  2  =X  3  +2 3 y' dX n  r  3  +  2  m  +  2  2  r  i- 3 2  2  r  p  2  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 e = 1. 7.2.i) If (m, n) = (1,0) and r = 0, then putting 2  2  ii  X  Y  22(91-1)32(92-1)^293 X = — —— — — ,'  2 («- )3 ( '2-l)p 93 ' 3  1  3  f  3  we obtain the new model for E Y  =X  2  + 2 3 p dX  3  2  2  r3  +  2  23p 3  3  X  n+2r3  which is the curve i n case 7 of the lemma with r\ = 2 and r = 1. 7.2.ii) If (m, n) = (1, 0) and r = 1, then putting 2  2  ^ =22(91-1)3292^293 _ _—_' Y =2 (<?i-l)3392p3g3 ' 3  we obtain the new model for E Y  2  =X  3  + 2 3p dX 2  r3  2  +  2 3p 3  X  n+2r3  which is the curve i n case 7 of the lemma with r\ = 2, r = 0. 2  Appendix A. Q-Isomorphism  283  Classes  7.2.iii) If (TO, n) > (1,0) and r = 0, then putting 2  v  x =  Y  2qi32(r -l)p2q '  2  =  V  2 « 3 (<?2- l)p3<j ' 3(  3  /2  3  3  we obtain the new model for i? .  Y  = X  2  + 2 3 p dX  3  Ti  2  T3  +  2  i- 3 p X  m+2r 2  2  3  n+2r3  which is the curve i n case 7 of the lemma with r\ = 0 or 1 and r = 1. 7.2.iv) If (TO, r i ) > (1,0) and r = 1, then putting 2  2  _ 2 n 32?2p2<7 'Y= x  V  2 '/l 3 « p 3 q '  2c  Y  3  3  3  3  we obtain the new model for E Y  = X  2  3  + 2 3p W + r i  r  2  m 2  + i- 3p 2  2 r  n + 2 r 3  X  which is the curve i n case 7 of the lemma with r\ = 0 or 1 and r = 0. 8. We have i + 2 > a — 2i, j = (i - 2j and k < 5 - 2k. In this case u (a 2^ a — 2i, and v (a ) — so a and fc are even. Therefore, t> (a) = f — i and t; (o) = | . Also, v (a ) > j = 8 — 2j so i>3(a ) > where e denotes the residue of j modulo 2. Let 2  :  2  2  p  2  2  p  2  3  2  u • J + t-2 k 22~ 3 2 p2 l  so (A.2) becomes 3 u - 2 t 2  2  3 i  "  a + 2  =  ±p - , s 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 d - 2 2  m  = ±p , n  with rn, n > 1 and the model for E can be written y = x + 2? 3 2 p t d x + 2*3 p z2  3  _ i  2  J  fc  Appendix A.  Q-Isomorphism  284  Classes  There exist six integers n , q\, r , qi, r%, and q% such that 2  3 - = 2q + r ,  a  --i  = 2q +n, 1  2  k - = 2q + r ,  2  s  3  with r\,r ,r G {0,1}. We have two cases to consider: 8.1) Suppose £2 = 0. 8.1-i) If (m,n) = (1,0), then putting 2  3  2 (9i- )3 '?2p393 '  22(91-1)3292^293 '  3  3  1  we obtain the new model for E Y  = X + 2 y p dX  2  3  2  2  Ti  +  2  23 p X 3  2r2  2rs  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  y v  y_  22913292^293 '  2 9 3 ? p '? ' 3  1  3<  2  3  3  we obtain the new model for E y  2  = X +2 3 V dX + 3  r i  T  3  2 -3 p X  2  m+2ri 2  2r2  2r3  which is the curve in case 4 of the lemma with ~£ = 0 and ry = 0 or 1. 8.2) Suppose £2 = 18.2.1) If (m, n ) = (1, 0) and r = 0, then putting 2  X  =  —  ,  —  7 7 - ^ - ,  ^  2(91-l)3 (92-l)p293 2  2  '  Y  =  2 (9l- )3 fe- )p ' ' 3  1  3  1  3<  3  we obtain the new model for E Y  2  =X  3  + 2H p dX 2  ri  +  2  2Sp X 3  3  2r3  which is the curve in case 8 of the lemma with r\ = 2 and r = 1. 2  8.2.ii) If (m, ri) = (1,0) and r = 1, then putting 2  x  =  — T T — - r - r — - — , y  22(91-l)32g p293 ' 2  =  2  7  2 (<> ~ )3 'J2p <» ' 3  1  1  3  3  Appendix A.  Q-Isomorphism  Classes  285  we obtain the new model for E Y  = X  2  + 2 3p dX  3  2  T3  +  2  2 3p X 3  2r3  i  which is the curve i n case 8 of the lemma with r\ = 2 , r = 0. 8.2.iii) If ( m , r i ) > (1,0) and r = 0, then putting 2  2  x  x  =  Y  =  V  2 ' 3 ( ? - )p '?3  2 9l3 (92-l)jr,2( 3 ' 2  3f  2  ?  3  1  <  2  1  3  '  we obtain the new model for E Y  = X  2  + 2 3 p dX  3  ri  2  r3  + 2  2  m + 2 r i  - 3 p 2  3  2 r 3  X  which is the curve i n case 8 of the lemma with r i = 0 or 1 and r = 1. 8.2.iv) If ( m , r i ) > (1,0) and r = l,then putting 2  2  y=  x x  =  V  2 <n 3292pigs'  2 " 3 '' p i3 '  2  3 f  3  2  3(  we obtain the new model for E Y  2  = X  + 2 3p dX  3  Ti  r3  + 2  2  m + 2 r  '- 3p 2  2 r 3  X  which is the curve in case 8 of the lemma with r\ = 0 or 1 and r = 0. 9. We have i + 2 > a — 2i, j = [3 — 2j and k = 5 — 2k. In this case v (a ) = a — 2i, so a is even. Therefore t> (a) = | - i. Also, v (a ) > j = (3 — 2j and v (a ) > k = d - 2k so v-i(a) > and v (a) > where e denotes the residue of j modulo 2 and 63 denotes the residue of k modulo 2 . Let 2  2  2  2  2  3  2  p  p  2  a 22~ 3  2 p 2  l  so (A.2) becomes 3 V tx £  3  2  2  3  i  -  a  +  2  ±1,  =  w i t h 3 i - a + 2 > 1. Let d = u, m = 3i — a  +  then (d, m) is a solution to 3 p d - 2 e 2  £ 3  2  rn  = ±1,  2,  Appendix A. Q-Isomorphism Classes  286  withTO,n > 1, and the model for E can be written y = x+ 2  2^- 3 ^ p ^dx +  3  i  2  Z  t  2 3 p x.  2  i  j  k  There exist six integers r\, q\, r , q2, r , and q such that 2  a  j'  - - ' i = 2gi+ri,  3  +  3  k  £2  — - — = 2<j + r , 2  + £3  — - — = 2g + r ,  2  3  3  with n , r , r £ {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 2  3  11  X  : x = 22(gi-l)32 ^—A 2p2„3 ', y 9  2 (9i- )3 92p3g ' 3  1  3  3  9  we obtain the new model for E Y  = X  2  3  + 2 ¥ p dX 2  2  r3  +  2  23 p X 3  2r2  2r3  which is the curve i n case 4 of the lemma w i t h £ = 0, n = 0 and r\ = 2. 9.1.ii) If (TO, n) > (1,0), then putting x  v  2 91 32g2p2q ' Y = 2 'l 3 ' « p 9 3 ' 2  ;  Y  =  3(  3  3  3  we obtain the new model for E Y  = X  2  + 2 3 p dX  3  ri  r2  T3  +  2  2 ^H p X m+2T  2r2  2r3  which is the curve i n case 4 of the lemma with t = 0, n = 0 and r i = 0 or 1. 9.2) Suppose £ = 0 and £3 = 1. 9.2.i) If (TO, r i ) = (1,0), then putting 2  x = -, ' „ 22(g -l) 1  3  2  *p 2 ( g  ( ; 2 :  „ y =  3  -l+r ) ' 3  3 2  ( « - ^ ^ t e - H - r s ) '  we obtain the new model for E '  y  2  = X + 2 3 V~ dX + 2 3 3  2  r  r 3  2  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  287  Classes  9.2.ii) If (m, n) > (1, 0), then putting  x = . „2 2 3_ „* „y= 2g p2(g -l+r- ) ' ~ g i  3  2  2 9l 3 9 2 p 3 ( 3  3  3  9 3  -l+r ) ' 3  we obtain the new model for E Y  + 2 ~ 3 VdX 2  =X  2  3  r i  r  +  2  r n + 2 2  -  3  2 r i  2 r 2  p 3  2 r 3  X  which is the curve in case 6 of the lemma w i t h £ = 0, r\ = 0 or 1 and r = l—r . 9.3) Suppose e = l and e = 0. 9.3.i) If (m, r ) = (1,0), then putting 3  2  3  3  a  x=,„,2 2 ( i ^,l ) 3 2 (*q - l + r ,) 2„ ,' y 9  2  2  p  3 2  g 3  («-l)3 ('/2-l+r )p3r/ ' 3  2  3  we obtain the new model for E Y  =X  2  3  + 2 3 - p dX 2  2  r2  rs  +  2  23- p X 3  3  2r2  2r3  which is the curve in case 8 of the lemma with n = 0, r\ = 2 and r = 1 —