To put a little color on the BSD conjecture, it states that the rank (0, 1, 2, 3, etc.) of rational points on an elliptic curve is related to the residue (coefficient of 1/q) of the L-function for the curve. There are some additional multiplicative factors, in particular the size of the Tate-Shafarevich group.
No one knows how to compute the size of that group in general (in fact no one has proved that it's finite!). Computing the rank of a curve via non-analytic means is more akin to a bespoke proof than a straightforward computation (see Noam Elkies' work).
So saying you're going to disprove BSD with blind computation is rather naive unless you're sitting on several career-defining proofs and not sharing them.
If the BSD rank conjecture were false, then the simplest counterexample might be an elliptic curve with algebraic rank 4 and analytic rank 2. This could be established for a specific curve by rigorously numerically computing the second derivative of the L-series at 1 to some number of digits and getting something nonzero (which is possible because elliptic curves are modular - see work of Dikchitser). This is a straightforward thing to do computations about and there are large tables of rank 4 curves. This is also exactly the problem I suggested to the OP in grad school. :-)
In number theory doing these sorts of “obvious computational investigations” is well worth doing and led to many of the papers I have written. I remember doing one in grad school and being shocked when we found a really interesting example in minutes, which led to a paper.
No one knows how to compute the size of that group in general (in fact no one has proved that it's finite!). Computing the rank of a curve via non-analytic means is more akin to a bespoke proof than a straightforward computation (see Noam Elkies' work).
So saying you're going to disprove BSD with blind computation is rather naive unless you're sitting on several career-defining proofs and not sharing them.