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 $$X_0(n)$$ X 0 ( n ) , for any $$n \in \mathbb {Z}^+$$ n ∈ Z + , using key results from Menendez [23] and Zywina [31]. This work is inspired by the prior algorithm of Bourdon et al. [6] which tests whether a rational, non-CM j -invariant gives rise to an isolated point on any modular curve $$X_1(n)$$ X 1 ( n ) . From the implementation of our algorithm, we determine that the set of j -invariants corresponding to isolated points on $$X_1(n)$$ X 1 ( n ) is neither a subset nor a superset of those corresponding to isolated points on $$X_0(n)$$ X 0 ( n ) .