Consider a finite field $\mathbb{F}_p$ such that $p \equiv 1 \ (\mathrm{mod} \ 3)$, $p \equiv 3 \ (\mathrm{mod} \ 4)$, $\mathbb{F}_{p^2}$-isomorphic elliptic curves (of $j$-invariant $0$) $$ E\!:y_1^2 = x_1^3 + b, \qquad E^\prime\!: y_2^2 + x_2^3 + b = 0, $$ where $b \in \mathbb{F}_p^* \setminus (\mathbb{F}_p^*)^3.$

Also, for $i \in \mathbb{Z}/3$ let $$P_i := (\zeta^i\sqrt[3]{-b}, 0) \in E[2] \cap E^\prime[2],$$ where $\zeta^3 = 1$, $\zeta \neq 1$. Note that $\zeta \in \mathbb{F}_p$.

Finally, denote by $G \subset E[2] \!\times\! E^\prime[2]$ the $\mathbb{F}_p$-invariant subgroup $(\simeq \mathbb{Z}/2 \!\times\! \mathbb{Z}/2)$ generated by the point $(P_0, P_0)$ (or, equivalently, by any point $(P_i, P_i)$).

What is the quotient $E \!\times\! E^\prime / G$? Is this Jacobian of genus 2 curve, the direct product of elliptic curves, or the Weil restriction (with respect to $\mathbb{F}_{p^2}/\mathbb{F}_p$) of an elliptic curve?