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Let $E_1$, $E_2$, $E_3$, $E_4$ be supersingular elliptic curves over a finite field $\mathbb{F}_{p^2}$, where $p$ is an odd prime. There is a well known theorem stating that over the algebraic closure $\overline{\mathbb{F}_{p^2}}$ direct products $E_1 \times E_2$ and $E_3 \times E_4$ are isomorphic. Under what conditions are they isomorphic over $\mathbb{F}_{p^2}$? And how can this isomorphism be explicitly constructed?

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