Isolated $j$-invariants arising from the modular curve $X_0(n)$

Abstract

An isolated point of degree $d$ is a closed point on an algebraic curve which does not belong to an infinite family of degree $d$ points that can be parameterized by some geometric object. We provide an algorithm to test whether a rational, non-CM $j$-invariant gives rise to an isolated point on the modular curve $X_0(n)$, for any $n \in \mathbb{Z}^+$, using key results from Menendez and Zywina. This work is inspired by the prior algorithm of Bourdon et al. which tests whether a rational, non-CM $j$-invariant gives rise to an isolated point on any modular curve $X_1(n)$. From the implementation of our algorithm, we determine that the set of $j$-invariants corresponding to isolated points on $X_1(n)$ is neither a subset nor a superset of those corresponding to isolated points on $X_0(n)$.