To prove the BSD conjecture, one has to know about 'the finiteness of the Shafarevich Tate group'. But, an example of an elliptic curve of rank 2 (whose Sha group $Ш(E/\mathbb{Q})$ is finite) is not yet known.

 Is there any example of an elliptic curve of rank 2 such that $p$-primary components of Ш are trivial for $p$ outside a finite set of primes?.
In particular, $Ш(E/\mathbb{Q})[p]$ is trivial for $p$ $\neq$ 2, 3, 5, 7.