We investigate the isogeny graphs of supersingular elliptic curves over
$mathbb{F}_{p^2}$ equipped with a $d$-isogeny to their Galois conjugate. These
curves are interesting because they are, in a sense, a generalization of curves
defined over $mathbb{F}_p$, and there is an action of the ideal class group of
$mathbb{Q}(sqrt{-dp})$ on the isogeny graphs. We investigate constructive and
destructive aspects of these graphs in isogeny-based cryptography, including
generalizations of the CSIDH cryptosystem and the Delfs-Galbraith algorithm.

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