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Let C be a curve defined over a number field $k$, and suppose $C(k)$ is nonempty. We say a closed point $x$ on $C$ of degree $d$ is isolated if it does not belong to an infinite family of degree d points parametrized by the projective line or a positive rank abelian subvariety of the curve's Jacobian. In this talk we will identify the non-CM elliptic curves with rational $j$-invariant which give rise to an isolated point of odd degree on $X_1(N)$ for some positive integer $N$. This is joint work with David Gill, Jeremy Rouse, and Lori D. Watson.