Computing Congruences Between an Elliptic Curve with Integer Coefficients and the Old Space

TL;DR

This paper presents a fast algorithm that leverages theoretical results and computational tools to determine congruence primes linking elliptic curves with integer coefficients to the old space at the same level.

Contribution

It introduces a novel, efficient algorithm combining theoretical insights and computational methods for identifying congruence primes in elliptic curve contexts.

Findings

Algorithm effectively computes congruence primes

Utilizes theoretical results and SageMath for efficiency

Applicable to elliptic curves with integer coefficients

Abstract

We put together some known theoretical results and the fact that certain computations can be done efficiently in SAGE to come up with a fast algorithm for calculating congruence primes linking a newform with integer coefficients (i.e. a newform associated to an elliptic curve) with the old space at the same level.