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  $\mathbb{Q}$ of analytic ranks 0 and $1$. We apply our techniques to show that if $E$ is a non-CM elliptic curve over  $\mathbb{Q}$ of conductor $\leq 1000$ 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.