Computing the endomorphism ring of an elliptic curve over a number field

TL;DR

This paper introduces new algorithms to efficiently compute the endomorphism ring of elliptic curves over number fields, aiding in identifying complex multiplication properties and classifying elliptic curves.

Contribution

It presents deterministic and probabilistic algorithms that improve speed and simplicity in computing elliptic curve endomorphism rings over number fields.

Findings

Algorithms are faster than existing methods.

They can determine CM properties of elliptic curves.

Implementation is straightforward and practical.

Abstract

We describe deterministic and probabilistic algorithms to determine whether or not a given monic irreducible polynomial H in Z[X] is a Hilbert class polynomial, and if so, which one. These algorithms can be used to determine whether a given algebraic integer is the j-invariant of an elliptic curve with complex multiplication (CM), and if so, the associated CM discriminant. More generally, given an elliptic curve E over a number field, one can use them to compute the endomorphism ring of E. Our algorithms admit simple implementations that are asymptotically and practically faster than existing approaches.