Families Of Elliptic Curves With The Same Mod 8 Representations

Zexiang Chen

Families Of Elliptic Curves With The Same Mod 8 Representations

Number Theory 2014-12-23 v3

Authors: Zexiang Chen

Abstract

Let $E:y^2=x^3+ax+b$ be an elliptic curve defined over $\mathbb{Q}$ . We compute certain twists of the classical modular curves $X(8)$ . Searching for rational points on these twists enables us to find non-trivial pairs of $8$ -congruent elliptic curves over $\mathbb{Q}$ , i.e. pairs of non-isogenous elliptic curves over $\mathbb{Q}$ whose $8$ -torsion subgroups are isomorphic as Galois modules. We also show that there are infinitely many examples over $\mathbb{Q}$ .

Keywords

elliptic curves mod 8 representations modular curves Galois modules torsion subgroups rational points twists congruent elliptic curves

Cite

@article{arxiv.1405.6385,
  title  = {Families Of Elliptic Curves With The Same Mod 8 Representations},
  author = {Zexiang Chen},
  journal= {arXiv preprint arXiv:1405.6385},
  year   = {2014}
}

Comments

arXiv admin note: text overlap with arXiv:1105.1706 by other authors

Families Of Elliptic Curves With The Same Mod 8 Representations

Abstract

Keywords

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