  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 when
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 small-degree isogenies 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.
