On families of 9-congruent elliptic curves

TL;DR

This paper computes explicit equations for families of elliptic curves that are 9-congruent to a given curve, demonstrating the existence of infinitely many such pairs over Q with isomorphic 9-torsion subgroups.

Contribution

It provides explicit equations for 9-congruent elliptic curves and proves the existence of infinitely many non-trivial pairs over Q.

Findings

Explicit equations for families of 9-congruent elliptic curves

Existence of infinitely many non-isogenous 9-congruent pairs over Q

Identification of isomorphic 9-torsion subgroups as Galois modules

Abstract

We compute equations for the families of elliptic curves 9-congruent to a given elliptic curve. We use these to find infinitely many non-trivial pairs of 9-congruent elliptic curves over Q, i.e. pairs of non-isogenous elliptic curves over Q whose 9-torsion subgroups are isomorphic as Galois modules.