explict form of the equation of elliptic curve

Question

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?

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 $k = 2$.