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.
