A bound for the conductor of an open subgroup of GL2 associated to an elliptic curve

Abstract

Given an elliptic curve $E$ without complex multiplication defined over a number field $K$, consider the image of the Galois representation defined by letting Galois act on the torsion of $E$. Serre's open image theorem implies that there is a positive integer $m$ for which the Galois image is completely determined by its reduction modulo $m$. In this note, we prove a bound on the smallest such $m$ in terms of standard invariants associated with $E$. The bound is sharp and improves upon previous results.

This replacement contains minor revisions and includes MAGMA scripts in an appendix for all computations referenced in the paper

References in corpus (1)

• Modular curves of prime-power level with infinitely many rational points