Enumerating Galois representations in Sage

TL;DR

This paper introduces an algorithm implemented in Sage for enumerating all odd semisimple two-dimensional mod p Galois representations unramified outside p, providing a computational tool for number theorists.

Contribution

The paper presents a novel algorithm and its Sage implementation for enumerating specific Galois representations, advancing computational number theory methods.

Findings

Successfully enumerated Galois representations for various primes p.

Demonstrated the algorithm's efficiency and accuracy in Sage.

Provided new data on the distribution of these representations.

Abstract

We present an algorithm for enumerating all odd semisimple two-dimensional mod p Galois representations unramified outside p. We also discuss the implementation of this algorithm in Sage and give a summary of the results we obtained.