#### Vol. 308, No. 2, 2020

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A bound for the conductor of an open subgroup of $\mathrm{GL_2}$ associated to an elliptic curve

### Nathan Jones

Vol. 308 (2020), No. 2, 307–331
##### 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 $m$. 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.

##### Keywords
elliptic curves, Galois representations
##### Mathematical Subject Classification 2010
Primary: 11F80, 11G05