  Let $E$ be an optimal elliptic curve over $\Q$ of prime conductor $N$. We
show that if for an odd prime $p$, the mod $p$ representation associated to $E$
is reducible (in particular, if $p$ divides the order of the torsion subgroup
of $E(\Q)$), then the $p$-primary component of the Shafarevich-Tate group of
$E$ is trivial. We also state a related result for more general abelian
subvarieties of $J_0(N)$ and mention what to expect if $N$ is not prime.
