  In this paper, we add the information of level structure to supersingular
elliptic curves and study these objects with the motivation of isogeny-based
cryptography. Supersingular elliptic curves with level structure map to Eichler
orders in a quaternion algebra, just as supersingular elliptic curves map to
maximal orders in a quaternion algebra via the classical Deuring
correspondence. We study this map and the Eichler orders themselves. We also
look at isogeny graphs of supersingular elliptic curves with level structure,
and how they relate to graphs of Eichler orders.
