  We give an algorithm for computing inseparable endomorphisms of a
supersingular elliptic curve $E$ defined over $\mathbb F_{p^2}$, which,
conditional on GRH, runs in expected $O(\sqrt{p}(\log p)^2(\log\log p)^3)$
time. With two calls to this algorithm, we compute a Bass suborder of
$\text{End}(E)$, improving on the results of Eisentr\"{a}ger, Hallgren,
Leonardi, Morrison, and Park (ANTSXIV) who only gave a heuristic algorithm for
computing a Bass suborder. We further improve on the results of Eisentr\"{a}ger
et al. by removing the heuristics involved in an algorithm for recovering
$\text{End}(E)$ from a Bass suborder. We conclude with an argument that $O(1)$
endomorphisms generated by our algorithm along with negligible overhead suffice
to compute $\text{End}(E)$, conditional on a heuristic assumption about the
distribution of the discriminants of these endomorphisms.
