$E/\mathbb{F}_p$ supersingular elliptic curve, what's $\# E(\mathbb{F}_{p^n})$?

Question

Let $E/\mathbb{F}_p$ be a supersingular elliptic curve with $p \ge 5$ prime, and let $n \ge 1$ be an integer. How do I see that$$\# E(\mathbb{F}_{p^n}) = \begin{cases} p^n + 1 & \text{if }n\text{ is odd,} \\ (p^{n/2} - (-1)^{n/2})^2 & \text{if }n\text{ is even?}\end{cases}$$

Answer 1

The characteristic polynomial of the Frobenius acting on the $l$-adic Tate module, where $l$ is a prime different from $p$, is given by $x^2+a_px-p=(x-\alpha)(x-\beta)$ for some $\alpha,\beta\in \overline{\mathbb Q}$. If $a_p=0$, you get $\alpha=\sqrt{p}$ and $\beta=-\sqrt{p}$. Now just use the fact that $a_{p^n}=\alpha^n+\beta^n$ together with the definition of $a_{p^n}=p^n+1-|E(\mathbb F_{p^n})|$.