Computing the endomorphism ring of an elliptic curve over a number field
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866908322783821824 |
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| author | Cremona, John E. Sutherland, Andrew V. |
| author_facet | Cremona, John E. Sutherland, Andrew V. |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2301_11169 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Computing the endomorphism ring of an elliptic curve over a number field Cremona, John E. Sutherland, Andrew V. Number Theory 11G05 (Primary) 11G15, 11Y16, 11Y40 (Secondary) 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. |
| title | Computing the endomorphism ring of an elliptic curve over a number field |
| topic | Number Theory 11G05 (Primary) 11G15, 11Y16, 11Y40 (Secondary) |
| url | https://arxiv.org/abs/2301.11169 |