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A finite field is a field with a finite number of elements. For each prime power $q^k$, there is a unique (up to isomorphism) finite field with $q^k$ elements. Up to isomorphism, these are the only finite fields.

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The product of two supersingular elliptic curves is independent of which ones we pick

See Theorem 3.5 in "Supersingular K3 surfaces" by TetsuJi Shioda, or a recent paper "Abelian varieties isogenous to a power of an elliptic curve" at https://arxiv.org/abs/1602.06237. Let $C_0$ be a …
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