Consider the elliptic curve defined by $\ y^2 = x^3 + 1\ $ over $\ \mathbb{Z}_p,\ $ where $\ p \equiv 2 \pmod{3}\ $ is prime. Prove that the number of points on the curve is exactly $\ p + 1.\ $
Hint: for $\ y \in \mathbb{Z}_p,\ $ prove that there is exactly one $\ x \in \mathbb{Z}_p\ $ satisfying the equation.
Prove without showing that $\ x \mapsto x^3\ $ is a bijection