Given the elliptic curve $y^2 + y = x^3$ over $\mathbb{F}_q$ ($q=p^r$, where $p$ is prime), I want to prove that if $q \equiv 2 \bmod 3$, then the elliptic curve has $q + 1$ points.
My exercise says that $q$ can be even, but I thought this kind of curve assumed that $q = 2^r$. And I don't know how to apply the Legendre symbol to this case because the left hand side contains $+ y$.
It is actually easier than you thought.
It is known that the multiplicative group $\Bbb F_q^\times$ is a cyclic group with $q - 1$ elements. If $q \equiv 2\mod 3$, then its order is prime to $3$.
This means that the map $z \mapsto z^3$ is an automorphism of $\Bbb F_q^\times$.
Therefore, for any $y \in \Bbb F_q$, there exists a unique $x$ such that $x^3 = y^2 + y$.
Together with the infinite point, we get $q + 1$ points in total.