Let E be a generic elliptic curve over an algebraically closed field $k$ of characteristic $p>0$ (i.e. an elliptic curve corresponding to a geometric point over the generic point of $M_{1,1}$).
What is $End_{k}(E)$?
The following will give you $\operatorname{End}(E)\otimes\mathbb{Q}$. One can probably then figure out which orders appear as $\operatorname{End}(E)$. (As noted by Will Sawin, the generic case that you've asked about is Case 1, so in your situation the endomorphism ring is $\operatorname{End}(E)=\mathbb Z$.)
Theorem (Deuring, see Mumford's Abelian Varieties Section 22, page 217) Let $E$ be an elliptic curve in characteristic $p>0$. We have the following equivalences: