On families of n-congruent elliptic curves

Details On families of n-congruent elliptic curves On families of n-congruent elliptic curves

2011-05-09

arxiv.org/abs/1105.1706v1

Mathematics

Number Theory

57 pages

Scientific paper

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.

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