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

## TL;DR

This paper establishes a sharp bound on the minimal integer m that determines the Galois image of an elliptic curve without complex multiplication over a number field, improving previous results.

## Contribution

It provides a new, sharp bound on the conductor of an open subgroup of GL2 linked to an elliptic curve, based on standard invariants.

## Key findings

- Bound on the smallest m is explicitly given
- The bound is proven to be sharp
- Improves upon previous bounds

## 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.