\begin{theorem}
Suppose that $E$ is an elliptic curve over~$\Q$ and
that~$n$ is an integer such that
\begin{enumerate}
\item $\ds\gcd(n,2N_E\prod_{p}\#\Phi_E(\Fpbar)) = 1$, and

\item there is a prime $\ell\con 1\pmod{n}$ such that
 for all nontrivial $\chi:(\Z/\ell\Z)^*\ra \C^*$ of order
dividing~$n$, 
$$
        L(E,\chi,1)\neq 0,
$$
and $a_{\ell} \not \con \ell+1\pmod{p}$ for all $p\mid n$.
\end{enumerate}
Then there exists a rank~$0$ abelian variety~$A$ over~$\Q$ of
dimension $n-1$ such that
$$
  E(\Q)/n E(\Q) \subset \Sha(A)
$$
and $\ds{}L(A,s) = \prod_{\chi} L(E,\chi,s)$, where
the product is over~$\chi$ as above.
\end{theorem}


\begin{theorem}
Let~$E$ be an elliptic curve over~$\Q$,
and suppose $\chi:(\Z/\ell\Z)^*\ra \C^*$
is a Dirichlet character of prime conductor
$\ell\nmid N_E$ and order~$n$ such that
\begin{itemize}
\item $L(E,\chi^a,1)\neq 0$ for $a=1,\ldots, n-1$,
\item $\ds\gcd\left(n,\,\,2\ell{}N_E\prod_{p\mid N_E}\#\Phi_E(\Fpbar)\right	) = 1$, and
\item $a_\ell \not\con \ell+1 \pmod{p}$ for any $p\mid n$.
\end{itemize}
Then there exists an abelian variety~$A$ over~$\Q$ of rank~$0$ 
and dimension $n-1$ such that $L(A,s) = \prod_{a=1}^{n-1} L(E,\chi^a,s)$
and
$$
  E(\Q)/n E(\Q) \subset \ker(\Sha(A).
$$
\end{theorem}