Equations of elliptic curves

Question

First part of question I have asked on mathoverflow already: https://math.stackexchange.com/questions/467088/explict-form-of-the-equation-of-elliptic-curve

1) 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 (something curve with same number points) in the explicit form? It will be interesting to get answer for infinetely family of $q$.

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

Answer 1

For (2), you'll need the coefficients of your equation to depend on $n$. More generally, if $E$ is defined over $q$ for any prime power $q$, for example $q=2^k$, and if $\#E(F_{q^n})=q^n+1$, then $\#E(F_{q^{2n}})=q^{2n}+2q^n+1$. So your formula in (2) can't hold for both $n$ and $2n$. For (1), the example you give in your comment has CM and you're choosing the inert (hence supersingular) primes. So for (1) you might try taking CM curves defined over some quadratic (or larger) extension of $\mathbf{Q}$ and reducing modulo ss primes of norm $q^2$. I don't know if that will work, just a thought.

Answer 2

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 $k = 2$.