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Let $E$ be the elliptic curve $y^2=x^3-x$ defined over $\mathbb F_5.$ It is ordinary with $j=-2$ and $End_{\mathbb F_5}(E)=\mathbb Z[i],$ where $i:(x,y)\mapsto(-x,2y).$ So $Aut_{\mathbb F_5}(E\times E)=GL_2(\mathbb Z[i]).$ Which matrices (and in particular, how many) preserve the principal polarization given by the divisor $E\times0+0\times E?$

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    $\begingroup$ Rosati involution on $E$ (via the unique principal pol.) is complex conjugation, so Rosati involution on $E \times E$ is conjugate-transpose on ${\rm{Mat}}_2(\mathbf{Z}[i])$. Hence, want $M \in {\rm{GL}}_2(\mathbf{Z}[i])$ such that conjugate-transpose of $M$ is inverse to $M$ (which is equivalent to preservation of the Weil self-pairing on Tate modules induced by the polarization). You can take it from there, I presume. $\endgroup$
    – BCnrd
    Commented Nov 14, 2010 at 15:13

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