Supersingular Curves With Small Non-integer Endomorphisms
We introduce a special class of supersingular curves over $\mathbb{F}_{p^2}$, characterized by the existence of non-integer endomorphisms of small degree. A number of properties of this set is proved. Most notably, we show that this set partitions into subsets in such a way that curves within each subset have small-degree isogenies between them, but curves in distinct subsets have no isogenies of degree $O(p^{1/4-\varepsilon})$ between them. Despite this, we show that isogenies between these curves can be computed efficiently, giving a technique for computing isogenies between certain prescribed curves that cannot be reasonably connected by searching on $\ell$-isogeny graphs.