On families of n-congruent elliptic curves

Abstract

We use an invariant-theoretic method to compute certain twists of the modular curves X(n) for n=7,9,11. Searching for rational points on these twists enables us to find non-trivial pairs of n-congruent elliptic curves over Q, i.e. pairs of non-isogenous elliptic curves over Q whose n-torsion subgroups are isomorphic as Galois modules. We also show by giving explicit non-trivial examples over Q(T) that there are infinitely many examples over Q in the cases n=9 and n=11.

arXiv comment57 pages

References in corpus (2)

• Minimisation and reduction of 2-, 3- and 4-coverings of elliptic curves
• The Hessian of a genus one curve

Cited by in corpus (5)

• Families Of Elliptic Curves With The Same Mod 8 Representations
• On pairs of 17-congruent elliptic curves
• P-torsion monodromy representations of elliptic curves over geometric function fields
• Congruences of elliptic curves arising from non-surjective mod $N$ Galois representations
• Explicit Families Of Elliptic Curves With The Same Mod 6 Representation