TITLE: Supersingular elliptic curves
QUESTION [14 upvotes]: I've read that an elliptic curve is supersingular if and only if its endomorphism ring is an order in a quaternion algebra.  Does anyone have a simple explanation of this (or a good reference)?

REPLY [4 votes]: I'd like to add a question to this discussion: the endomorphism ring of a supersingular elliptic curve is not just an order in a quaternion algebra, it is a maximal order in such an algebra.  Is there a simple explanation for this?