Families Of Elliptic Curves With The Same Mod 8 Representations

Abstract

Let $E:y^2=x^3+ax+b$ be an elliptic curve defined over $\mathbb{Q}$. We compute certain twists of the classical modular curves $X(8)$. Searching for rational points on these twists enables us to find non-trivial pairs of $8$-congruent elliptic curves over $\mathbb{Q}$, i.e. pairs of non-isogenous elliptic curves over $\mathbb{Q}$ whose $8$-torsion subgroups are isomorphic as Galois modules. We also show that there are infinitely many examples over $\mathbb{Q}$.