On families of 13-congruent elliptic curves

TL;DR

This paper investigates families of elliptic curves that are 13-congruent, computes related modular curve twists, and demonstrates the existence of infinitely many such non-isogenous pairs over rationals.

Contribution

It explicitly constructs and analyzes the surfaces parametrizing 13-congruent elliptic curves, revealing infinite families of non-trivial pairs over ${f Q}$.

Findings

Identifies equations for the surfaces parametrising 13-congruent elliptic curves.

Shows there are infinitely many non-trivial 13-congruent pairs over ${f Q}$.

Distinguishes between 13-congruences that do and do not respect the Weil pairing.

Abstract

We compute twists of the modular curve $X(13)$ that parametrise the elliptic curves 13-congruent to a given elliptic curve. Searching for rational points on these twists enables us to find non-trivial pairs of 13-congruent elliptic curves over ${\mathbb Q}$, i.e. pairs of non-isogenous elliptic curves over ${\mathbb Q}$ whose 13-torsion subgroups are isomorphic as Galois modules. We also find equations for the surfaces parametrising pairs of 13-congruent elliptic curves. There are two such surfaces, corresponding to 13-congruences that do, or do not, respect the Weil pairing. We write each as a double cover of the projective plane ramified over a highly singular model for Baran's modular curve of level 13. By finding suitable rational curves on these surfaces, we show that there are infinitely many non-trivial pairs of 13-congruent elliptic curves over ${\mathbb Q}$.