# Trace of Frobenius of elliptic curve is integer

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.

## migrated from mathoverflow.netApr 5 '14 at 14:46

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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.)