Constructing Cycles in Isogeny Graphs of Supersingular Elliptic Curves

Loops and cycles play an important role in computing endomorphism rings of supersingular elliptic curves and related cryptosystems. For a supersingular elliptic curve $E$ defined over $\mathbb{F}_{p^2}$, if an imaginary quadratic order $O$ can be embedded in $\text{End}(E)$ and a prime $L$ splits into two principal ideals in $O$, we construct loops or cycles in the supersingular $L$-isogeny graph at the vertices which are next to $j(E)$ in the supersingular $\ell$-isogeny graph where $\ell$ is a prime different from $L$. Next, we discuss the lengths of these cycles especially for $j(E)=1728$ and $0$. Finally, we also determine an upper bound on primes $p$ for which there are unexpected $2$-cycles if $\ell$ doesn't split in $O$.