We describe theorems and computational methods for verifying the Birch and Swinnerton-Dyer conjectural formula for specific elliptic curves over\u00a0\n\n \n \n Q<\/mml:mi>\n <\/mml:mrow>\n \\mathbb {Q}<\/mml:annotation>\n <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula> of analytic ranks \n\n \n 0<\/mml:mn>\n 0<\/mml:annotation>\n <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula> and \n\n \n 1<\/mml:mn>\n 1<\/mml:annotation>\n <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula>. We apply our techniques to show that if \n\n \n E<\/mml:mi>\n E<\/mml:annotation>\n <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula> is a non-CM elliptic curve over\u00a0\n\n \n \n Q<\/mml:mi>\n <\/mml:mrow>\n \\mathbb {Q}<\/mml:annotation>\n <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula> of conductor \n\n \n \n \u2264<\/mml:mo>\n 1000<\/mml:mn>\n <\/mml:mrow>\n \\leq 1000<\/mml:annotation>\n <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula> and rank \n\n \n 0<\/mml:mn>\n 0<\/mml:annotation>\n <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula> or \n\n \n 1<\/mml:mn>\n 1<\/mml:annotation>\n <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula>, then the Birch and Swinnerton-Dyer conjectural formula for the leading coefficient of the \n\n \n L<\/mml:mi>\n L<\/mml:annotation>\n <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula>-series is true for\u00a0\n\n \n E<\/mml:mi>\n E<\/mml:annotation>\n <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula>, up to odd primes that divide either Tamagawa numbers of\u00a0\n\n \n E<\/mml:mi>\n E<\/mml:annotation>\n <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula> or the degree of some rational cyclic isogeny with domain\u00a0\n\n \n E<\/mml:mi>\n E<\/mml:annotation>\n <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula>. Since the rank part of the Birch and Swinnerton-Dyer conjecture is a theorem for curves of analytic rank \n\n \n 0<\/mml:mn>\n 0<\/mml:annotation>\n <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula> or \n\n \n 1<\/mml:mn>\n 1<\/mml:annotation>\n <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula>, this completely verifies the full conjecture for these curves up to the primes excluded above.<\/p>","DOI":"10.1090\/s0025-5718-09-02253-4","type":"journal-article","created":{"date-parts":[[2009,6,30]],"date-time":"2009-06-30T14:39:02Z","timestamp":1246372742000},"page":"2397-2425","source":"Crossref","is-referenced-by-count":10,"title":["Computational verification of the Birch and Swinnerton-Dyer conjecture for individual elliptic curves"],"prefix":"10.1090","volume":"78","author":[{"given":"Grigor","family":"Grigorov","sequence":"first","affiliation":[]},{"given":"Andrei","family":"Jorza","sequence":"additional","affiliation":[]},{"given":"Stefan","family":"Patrikis","sequence":"additional","affiliation":[]},{"given":"William","family":"Stein","sequence":"additional","affiliation":[]},{"given":"Corina","family":"Tarni\u0163\u01ce","sequence":"additional","affiliation":[]}],"member":"14","published-online":{"date-parts":[[2009,6,8]]},"reference":[{"key":"1","unstructured":"[ABC] B. 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