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Consider the elliptic curve $E\!: y^2 = x^3 + 1$ of $j$-invariant $0$ over an algebraically closed field $k$ of characteristics $p$. Let me remind that $E$ is ordinary (i.e., non-supersingular) iff $p \equiv 1$ ($\mathrm{mod}$ $3$).

Also consider the direct square $E^2$. It is known that for $p \equiv 2$ $(\mathrm{mod}$ $3$) it is the unique superspecial abelian surface, i.e., $E^2 \cong E_1 \!\times\! E_2$ for any two supersingular elliptic curves $E_1$, $E_2$ over $k$.

What is known for $p \equiv 1$ $(\mathrm{mod}$ $3$)? More precisely, are there two ordinary elliptic curves $E_1$, $E_2$ over $k$ (at least one of them with $j \neq 0$) such that $E^2 \cong E_1 \!\times\! E_2$?

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