Torelli's theorem states that:

(Torelli's Theorem). Let $R$, $R'$ be compact
Riemann surfaces of genus $g$, $J(R)$, $J(R')$ their Jacobian varieties, $\Theta$, $\Theta'$ their respective theta divisors. The Riemann surfaces $R$ and $R'$ are isomorphic if and only if $(J(R), \Theta)$ and $(J(R'), \Theta')$ are isomorphic as principally polarized abelian varieties.

In this theorem, $J(R)$, $J(R')$ are required to be
isomorphic not only as abelian varieties but also as principally polarized abelian varieties. It turns out that the condition for $J(R)$ and $J(R')$ to be isomorphic as abelian varieties alone need not imply that $R$ and $R'$ are isomorphic.

Does anyone know where I can find an example that shows that being $ J (R) $ and $ J (R ') $ isomorphic just as Abelian variety, does not imply that $ R $ and $ R' $ are isomorphic?