I've read a few books and papers about isogeny-based cryptography and its mathematic but didn't get the idea how to find the endomorphism ring of a supersingular elliptic curve. I know how to do it for an ordinary curve but not for a supersingular one.
For example, I have a supersingular elliptic curve $E: y^2 = x^3 + x$ over $\mathbb{F}_{5^2}$. We know that $End(E) \cong \mathbb{Q}_{p, \infty}$ (maximal order of quaternion algebra ramified in $p$ and $\infty$). Can someone give the algorithm for this toy example? Is it a hard problem in general?
Any answer would be useful, especially with some literature or logical reasoning what should be done to solve this problem.