## Abstract

We describe theorems and computational methods for verifying the Birch and Swinnerton-Dyer conjectural formula for specic elliptic curves over Q of analytic ranks 0 and 1. We apply our techniques to show that if E is a non-CM elliptic curve over Q of conductor ≤ 1000 and rank 0 or 1, then the Birch and Swinnerton-Dyer conjectural formula for the leading coefficient of the E-series is true for E, up to odd primes that divide either Tamagawa numbers of E or the degree of some rational cyclic isogeny with domain E. Since the rank part of the Birch and Swinnerton-Dyer conjecture is a theorem for curves of analytic rank 0 or 1, this completely veries the full conjecture for these curves up to the primes excluded above.

Original language | American English |
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Pages (from-to) | 2397-2425 |

Number of pages | 29 |

Journal | Mathematics of Computation |

Volume | 78 |

Issue number | 268 |

DOIs | |

State | Published - Oct 2009 |

## ASJC Scopus subject areas

- Algebra and Number Theory
- Computational Mathematics
- Applied Mathematics