nt.number theory - The current status of the Birch & Swinnerton-Dyer Conjecture

Reading Olivier's comment reminded me that it is always possible to verify rigorously, for a given elliptic curve $E$ over $mathbb Q$ for which the rank of $E({mathbb Q})$ is at most $3$, that one has an equality of algebraic and analytic ranks (i.e. in the notation of the question, whether or not $r = d$), assuming that it is indeed true.

The point is the following: one can compute the sign of the functional equation, and hence
the parity of the order of vanishing of $L(E,s)$ at $s = 1$ (i.e. the parity of $r$,
in the notation of the question).

If this parity is even, one proceeds as follows: the computation of $L(E,1)/Omega$ (where
$Omega$ is the real period of the curve) is exact
(one does it via modular sybmols), and in particular one can determine whether or not $L(E,1)$ vanishes. If the rank of $E({mathbb Q})$ vanishes, then one expects that in fact
$L(E,1)$ is non-zero; if not, one has a counterexample to BSD.

Now if the rank of $E({mathbb Q})$ equals 2, then one knows that necessarily
$L(E,1)$ vanishes, and that in fact it must vanish to order at least 2. On the other hand,
one can approximate the 2nd derivative of $L(E,s)$ at $s = 1$ as accurately as one wants,
and so in particular, can verify that it doesn't equal zero (again, as must be the case if BSD is true).

If the rank of $E({mathbb Q})$ is 1 or 3, then one expects that $L(E,1)$ vanishes with odd order, and this can be verified by computing the sign in the functional equation (and if it doesn't hold, again one has a counterexample to BSD). The Gross-Zagier formula lets one compute the derivative of $L(E,s)$ at $s = 1$, by using an explicit modular parameterization of $E$ and seeing if the Heegner point is torsion or not;
if it doesn't vanish, then one know that the
rank of $E({mathbb Q})$ must be 1. If it does vanish, then the rank of $E({mathbb Q})$ will have to be 3 (or else BSD fails), and one knows that $L(E,s)$ vanishes at $s = 1$ to
an odd order that is greater than 1. Again, one can verify that the third derivative
of $L(E,s)$ at $s = 1$ is non-zero, and so show that the order of vanishing is exactly 3,
verifying BSD. (Otherwise, BSD would fail.)

I suppose in practice it could happen that in the case of rank 2 or 3, the non-zero 2nd or 3rd derivative that you have to compute would be so small that it was hard to distinguish from 0. On the other hand, one can use the conjectural formula for the leading term (coming from the full BSD conjecture) to determine an expected lower bound for this term, and again this should be an actual lower bound unless BSD fails. (Here it is useful to note that the regulator and the order of Sha appear in the numerator of the BSD formula, so one doesn't need to know about Sha, or the precise generators of $E({mathbb Q})$, to determine this lower bound.)

I once (back in 1993) read a masters thesis from Macquarie University (by a student named Chris Daniels, if I remember correctly) which verified a rank 3 example via the above scheme. I don't know how many other examples have been verified.