Computing Isogenies at Singular Points of the Modular Polynomial

Abstract

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