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We introduce a special class of supersingular curves over $\mathbb{F}_{p^2}$,characterized by the existence of non-integer endomorphisms of small degree. Anumber of properties of this set is proved. Most notably, we show that whenthis set partitions into subsets in such a way that curves within each subsethave small-degree isogenies between them, but curves in distinct subsets haveno small-degree isogenies between them. Despite this, we show that isogeniesbetween these curves can be computed efficiently, giving a technique forcomputing isogenies between certain prescribed curves that cannot be reasonablyconnected by searching on $\ell$-isogeny graphs.

Supersingular Curves With Small Non-integer Endomorphisms