On families of n-congruent elliptic curves

2011/05/09 by Tom Fisher, Fisher, Tom · 1 citation
Computer Science · Mathematics · #11F80 #11G05 #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #Cryptography and Residue Arithmetic #FOS: Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.1105.1706

openalex publication_date 2011/05/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

Cited by

• Congruences of elliptic curves arising from non-surjective mod N Galois representations

Related

• On families of 9-congruent elliptic curves
• Families Of Elliptic Curves With The Same Mod 8 Representations
• On families of 13-congruent elliptic curves
• Congruences of elliptic curves arising from non-surjective mod N Galois representations
• On 12-congruences of elliptic curves