Trace of Frobenius of elliptic curve is integer

Question

I recently started to read the book "Arithmetic of Elliptic Curves" by Silverman. And I can't solve an exercise 5.10. Let $E/\mathbb F_q$ be an elliptic curve and $\phi$ is Frobenius endomorphism, then $tr(\phi) \in \mathbb Z$ and $E$ is supersingular iff $tr(\phi) = 0 \mod p$. I would be glad of any help.

Answer 1

Since these are meant to be exercises, it would not be helpful to give you a complete solution. But here is a hint for your first question. To show that $\text{tr}(\phi)$ is an integer, try to express it in terms of $\deg(\phi)$ and $\deg(\phi-1)$, using the fact that if $\psi$ is any endomorphism, then $\deg(\psi)=\det(\psi_\ell)$. Here $\psi_\ell$ is the map that $\psi$ induces on the Tate module $T_\ell(E)$. (This actually contains a hint for your second question, too, namely look at what's happening to the Tate module.)

In future, you should post questions at this level to math stackexchange, not math overflow, because they're really not research-level questions. So don't be discouraged if the question is closed here and migrated there.