Complex Multiplication Tests for Elliptic Curves

TL;DR

This paper presents two polynomial-time algorithms, one randomized and one deterministic, for testing whether an elliptic curve over a number field has complex multiplication, with the randomized version also determining the endomorphism ring's discriminant.

Contribution

It introduces efficient algorithms for complex multiplication testing of elliptic curves, improving computational methods in algebraic number theory.

Findings

Both algorithms run in polynomial time.

The randomized algorithm can compute the discriminant of the endomorphism ring.

The deterministic algorithm provides a reliable test for complex multiplication.

Abstract

We consider the problem of checking whether an elliptic curve defined over a given number field has complex multiplication. We study two polynomial time algorithms for this problem, one randomized and the other deterministic. The randomized algorithm can be adapted to yield the discriminant of the endomorphism ring of the curve.