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arXiv · 2301.11169

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

John E. Cremona·Andrew V. Sutherland
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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.

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John E. Cremona, Andrew V. Sutherland. 2023-12-15. Computing the endomorphism ring of an elliptic curve over a number field. https://doi.org/10.1090/conm%2F796%2F15998

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