A formula for counting points on a elliptic curve over a finite field

Question

Let $E(\mathbb{F}_p)$ is an elliptic curve, #$E(\mathbb{F}_p)=p+1-a$ and $x^2-ax+p=(x-\alpha)(x-\beta)$.

Now I found the following formula here https://joeylitalien.github.io/assets/elliptic-curves.pdf (Theorem 3.3)

# $ E(\mathbb{F}_{p^2})=p^2+a-(\alpha^2+\beta^2)$ but it doesn't work in problems.

What is wrong?

Answer 1

If $|E(\Bbb F_p)|=p+1-\alpha-\beta$ with $\alpha\beta=p$ then $$|E(\Bbb F_{p^n})|=p^n+1-\alpha^n-\beta^n.$$ Here, $$|E(\Bbb F_{p^2})|=p^2+1-\alpha^2-\beta^2=p^2+1-a^2+2p$$