Computational verification of the Birch and Swinnerton-Dyer conjecture for individual elliptic curves

We describe theorems and computational methods for verifying the Birch and Swinnerton-Dyer conjectural formula for specific 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 L-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 verifies the full conjecture for these curves up to the primes excluded above.