 Prime-square point-count formula (source code)

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= Prime-square point-count formula
{title2=$\#E(\mathbb F_{p^2})=N(2p+2-N)$}

For an <elliptic curve> over $\mathbb F_p$, put $N=\#E(\mathbb F_p)$ and $a=p+1-N$. The <trace of the square of an elliptic-curve endomorphism> gives $\operatorname{tr}(\pi^2)=a^2-2p$ for the <Frobenius isogeny>. Therefore $\#E(\mathbb F_{p^2})=p^2+1-a^2+2p=N(2p+2-N)$.

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