On elliptic curves with an isogeny of degree 7

TL;DR

This paper investigates elliptic curves over Q with a rational 7-isogeny, showing that their Galois representation is as large as possible except for CM curves, and classifies exceptions using advanced number-theoretic methods.

Contribution

It extends previous results to the case p=7, classifies exceptions as CM curves, and provides explicit parameterizations for elliptic curves with a 7-isogeny over arbitrary fields.

Findings

The Galois image is maximal except for CM curves.

Exceptions correspond to rational points on a genus 12 curve.

Explicit parameterizations for elliptic curves with a 7-isogeny.

Abstract

We show that if $E$ is an elliptic curve over $\mathbf{Q}$ with a $\mathbf{Q}$-rational isogeny of degree 7, then the image of the 7-adic Galois representation attached to $E$ is as large as allowed by the isogeny, except for the curves with complex multiplication by $\mathbf{Q}(\sqrt{-7})$. The analogous result with 7 replaced by a prime $p > 7$ was proved by the first author in [7]. The present case $p = 7$ has additional interesting complications. We show that any exceptions correspond to the rational points on a certain curve of genus 12. We then use the method of Chabauty to show that the exceptions are exactly the curves with complex multiplication. As a by-product of one of the key steps in our proof, we determine exactly when there exist elliptic curves over an arbitrary field $k$ of characteristic not 7 with a $k$-rational isogeny of degree 7 and a specified Galois action on the kernel of the isogeny, and we give a parametric description of such curves.