# Endomorphism ring of a generic elliptic curves in positive characteristic

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:
1. $$E$$ cannot be defined over a finite field if and only if $$\operatorname{End}(E)\otimes\mathbb{Q}=\mathbb{Q}$$.
2. Suppose that $$E$$ is defined over a finite field. Then $$\operatorname{End}(E)\otimes\mathbb{Q}$$ is imaginary quadratic over $$\mathbb{Q}$$ if and only if $$E[p]\cong\mathbb Z/p\mathbb Z$$, i.e., if and only if $$X$$ is ordinary.
3. If $$E$$ is supersingular, i.e., if $$E[p]=0$$, and thus necessarily defined over a finite field, then $$\operatorname{End}(E)\otimes\mathbb{Q}$$ is a quaterion algebra over $$\mathbb Q$$.