Consider the case of curves of genus $2$. If $\mathrm{A}$ is an abelian variety and $\mathrm{C}$ a smooth curve in $\mathrm{A}$ of genus $2$, then $\mathrm{A}\simeq\mathrm{J}(\mathrm{C})$ and $\mathrm{C}$ is the theta divisor of $\mathrm{J}(\mathrm{C})$. The special case $\mathrm{A}=\mathrm{E}\times\mathrm{E}$ (where $\mathrm{E}$ is an elliptic curve) was studied in this paper by Hayashida. It is known that for a given abelian variety $\mathrm{A}$ there are only finitely many curves with Jacobian $\mathrm{A}$ (see this paper by Narasimhan and Nori; for surfaces this was proven much earlier by Hayashida and Nishi). The first paper I mentioned gives in fact formulae (depending on the nature of $\mathrm{End(E)}$) for the number of curves $\mathrm{C}$ with Jacobian $\mathrm{A}=\mathrm{E}\times\mathrm{E}$.

More explicit examples (in the sense that the equations for the curves $\mathrm{C}$ can be written down) were constructed by Howe.

Consider the case of curves of genus $2$. If $\mathrm{A}$ is an abelian surface and $\mathrm{C}$ a smooth curve in $\mathrm{A}$ of genus $2$, then $\mathrm{A}\simeq\mathrm{J}(\mathrm{C})$ and $\mathrm{C}$ is the theta divisor of $\mathrm{J}(\mathrm{C})$. The special case $\mathrm{A}=\mathrm{E}\times\mathrm{E}$ (where $\mathrm{E}$ is an elliptic curve) was studied in this paper by Hayashida. It is known that for a given abelian variety $\mathrm{A}$ there are only finitely many curves with Jacobian $\mathrm{A}$ (see this paper by Narasimhan and Nori; for surfaces this was proven much earlier by Hayashida and Nishi). The first paper I mentioned gives in fact formulae (depending on the nature of $\mathrm{End(E)}$) for the number of curves $\mathrm{C}$ with Jacobian $\mathrm{A}=\mathrm{E}\times\mathrm{E}$.

More explicit examples (in the sense that the equations for the curves $\mathrm{C}$ can be written down) were constructed by Howe.