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

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.

arXiv commentminor corrections; 21 pages

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