For p=2: Cremona verified this several years ago via a 2-descent.

For all pÕ3 with p==7: Kolyvagin's bound implies that p=j#Sha(E).

For p=7: we computed the 7-adic regulator, found that it was nonzero. Then using
that the mod-7 representation is surjective, the bounds coming from Iwasawa
theory, and thoerems of Kato, Perrin-Riou, Schneider, and others imply that ord7(Sha(E))=0.

We conclude that the full BSD conjecture is true for the elliptic curve 141a.

Key Point

Applying the general theory is now fairly straightforward. Combining together
the wide range of theorems people have proved toward BSD gives quite a lot, and illustrates
and motives work to fill in the remaining gaps.