Computing Isogenies at Singular Points of the Modular Polynomial

In this paper we present a method which, given a singular point $(j_1,j_2)$ on $Y_0(\mathrm{\ell })$ with $j_1,j_2\mathrm{\ne }0,1728$ and an elliptic curve $E$ with $j$-invariant $j_1$, returns an elliptic curve $\tilde{E}$ with $j$-invariant $j_2$ that admits a normalized $\mathrm{\ell }$-isogeny $\phi \text{\hspace{0.17em}⁣}:E\to \tilde{E}$. The isogeny $\phi$ can then be efficiently computed from $E$ and $\tilde{E}$ using an algorithm of Bostan, Morain, Salvy, and Schost.