Counting elements of $y^2 - y = x^3$ in finite fields The problem I have to solve is the following:

Let $p$ be a prime number with $p \equiv 2$ mod $3$. Let $E$ be the elliptic curve given by $y^2 - y = x^3$. Show that $\#E(\mathbb{F}_p) = p+1$ and $\#E(\mathbb{F}_{p^2}) = (p+1)^2$.

I have solved the first part in the following way: $\mathbb{F}_p^*$ has order $p-1$ which is not divisible by $3$, and so every element in $\mathbb{F}_p$ has a unique cube root. So the elements of $E$ in $\mathbb{F}_p$ are the $p$ elements $(\sqrt[3]{y^2-y},y)$ and the point at infinity, which gives a total of $p+1$ elements.
In the second part this trick no longer works, as cube roots are no longer unique. I don't know where to start to show that $\#E(\mathbb{F}_{p^2}) = (p+1)^2$. Any help is appreciated.

Use Lemma 4.2 of the lecture notes.
