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?



What is the quotient $E \!\times\! E^\prime / G$?
I claim that $E\times E'/G$ is the Weil restriction of $E\_{\mathbb{F}\_{p^2}}$ w.r.t. $\mathbb{F}\_{p^2}/\mathbb{F}\_{p}$. (I don't know about the product question, or the Jacobian question; the answers might depend on $p$).


Let $A$ be the Weil restriction in question (which is an abelian surface). For each $\mathbb{F}\_{p}$-algebra $R$, write $R\left[\sqrt{-1}\right]:=\mathbb{F}\_{p^2}\otimes\_{\mathbb{F}\_{p}}R=R[t]/(t^2+1)$.


By definition, $A(R)=E\left(R\left[\sqrt{-1}\right]\right)$. We have an obvious inclusion $E\to A$: in terms of $R$-points this comes from $R\subset R\left[\sqrt{-1}\right]$, so we may desribe it as $(x\_1,y\_1)\mapsto(x\_1,y\_1)$. There is also a morphism $E'\to A$ given by $(x\_2,y\_2)\mapsto \left(x\_2,\sqrt{-1}y\_2\right)$. Putting these together yields
$$f:E\times E'\to A, \qquad((x\_1,y\_1),(x\_2,y\_2))\mapsto (x\_1,y\_1)\oplus\left(x\_2,\sqrt{-1}y\_2\right)$$
where $\oplus$ stands for the group law on $E\left(R\left[\sqrt{-1}\right]\right)$.  

Let me show that $\ker(f)=G$. (It will follow that $f$ is surjective and $E\times E'/G\cong A$).
If the RHS is zero, we must have $x\_1=x\_2$ and $y\_1=-\sqrt{-1}y\_2$ but the second relation forces $y\_1=y\_2=0$ because $R\cap\sqrt{-1}R=\{0\}$. In other words, $((x\_1,y\_1),(x\_2,y\_2))\in G$. QED


Of course you can do the same thing with any ellitic curve over a field of char. $≠2$ and its twist by a quadratic extension. 


