First part of question I have asked on mathoverflow already: http://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?