\frametitle {Examples applying Kato's theorem}
{\dblue
\begin{verbatim}
sage: E = EllipticCurve('37b')
sage: E.analytic_rank()
0
sage: E.non_surjective()
[(3, '3-torsion')]
sage: E.shabound_kato()
[2, 3]
sage: E.three_selmer_rank() # calls magma
Traceback (most recent call last):
...
NotImplementedError: Currently, only the case with irreducible phi3 is implemented.
sage: E.two_selmer_shabound()
0
\end{verbatim}
}
Thus Kato and $2$-descent
implies that only $3$ could divide $\#\Sha(E/\Q)$.
Moreover, Wuthrich's generalization of Kato's theorem
implies that moreover $3\nmid\#\Sha(E/\Q)$, so the
BSD conjecture is true for $E$.
\vfill
{\bf Thus BSD is true for the modular abelian variety $J_0(37)$.}