Endomorphism Rings of Supersingular Elliptic Curves and Quadratic Forms

Given a supersingular elliptic curve, the supersingular endomorphism ring problem is to compute all of its endomorphisms. We use the correspondence between maximal orders in quaternion algebra $B_{p,\infty}$ and positive ternary quadratic forms with discriminant $p$ to solve the supersingular endomorphism ring problem. Let $c<3p/16$ be a prime or $c=1$. Let $E$ be a $\mathbb{Z}[\sqrt{-cp}]$-oriented supersingular elliptic curve defined over $\mathbb{F}_{p^2}$. There exists a subgroup $G$ of order $c$, and $\text{End}(E,G)$ is isomorphic to an Eichler order in $B_{p,\infty}$ of level $c$. If the endomorphism ring $\text{End}(E,G)$ is known, then we can compute $\text{End}(E)$ by solving two square roots in $\mathbb{F}_c$. In particular, let $D