In 1972, Serre proved that the Galois representations arising from the $p$-power torsion points of non-CM elliptic curves over $\mathbb{Q}$ have open image in $\operatorname{GL}_2(\mathbb{Z}_p)$, and Mazur later initiated a vast programme to determine all such possible images explicitly -- for fixed $p$, it is known that there are only finitely many possibilities. Much progress has been made for small primes $p$, but a complete classification remains open beyond $p \in \{2,3,13,17\}$. In this talk, I will describe recent progress on this problem for $p = 7$, based on a surprising correspondence between rational points on modular curves and primitive integer solutions to certain generalised Fermat equations of signature $(2,3,7)$, such as $a^2 + 28b^3 = 27c^7$. We show that these Diophantine equations can be reduced to determining the rational points on a finite collection of genus-3 curves. As a consequence, we are able for example to determine the rational points on a modular curve of genus 69 and establish that the $7$-adic Galois images of elliptic curves over $\mathbb{Q}$ are determined by their reduction modulo $7^2$.