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Let $E(\mathbb{F}_{q^2})$ is elliptic curve with #$E(\mathbb{F}_{q^2}) =q^2 + q + 1$. Can we write equation of this curve in the explicit form?

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Is $q$ a fixed number? Or you mean that for any prime (or power of prime) $q$? –  asatzhh Aug 14 '13 at 8:47
It will be interesting to get answer for infinetely family of $q$ –  Alexey Aug 14 '13 at 9:35
For this to happen the zeros of the $L$-function of the curve $E(\mathbb{F}_{q^2})$ must be $$\omega_{1,2}=qe^{\pm2\pi i/3}.$$ So if a curve works for some $q$, the same curve will work for $Q=q^n, 3\nmid n$. –  Jyrki Lahtonen Aug 15 '13 at 12:02

1 Answer 1

Curve $E$ with equation $y^2 = x^3 + 1$ has $p+1$ rational points over $\mathbb{F}_p$ when $p = 5$ (mod $6$). Let $q = p^n$. $|E(\mathbb{F}_{q^2})| = q^2 \pm 2q +1$.

Let $\zeta_6$ is generator of $\mathbb{F}_{q^2}^*/\mathbb{F}_{q^2}^{*6}$.

Consider curve $E'$ with equation $y^2 = x^3 + \zeta^k$.

As it write in "Constructing supersingular elliptic curves" of Reinier Broker(http://www.math.brown.edu/~reinier/supersingular.pdf) $|E'(\mathbb{F}_{q^2})| = q^2 \pm q +1 $ when $k = 1$ and $|E'(\mathbb{F}_{q^2})| = q^2 \mp q +1 $ if $k = 2$.

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