The modular curve $X_1(n)$ is an algebraic curve over $\mathbb{Q}$ whose non-cuspidal points parametrize isomorphism classes of pairs $(E,P)$ where $E$ is an elliptic curve and $P \in E$ is a point of exact order $n$. We wish to understand isolated points on this curve, which are roughly those not belonging to an infinite family of points of the same degree which are parametrized by a geometric object. For fixed $n$, there are only finitely many isolated points on $X_1(n)$ of any degree, but as $n$ ranges over all positive integers, there exist infinitely many isolated points (associated to infinitely many non-isomorphic elliptic curves).

In this talk, I will discuss a new algorithm which can be used to determine whether an elliptic curve with rational $j$-invariant gives rise to an isolated point on some modular curve $X_1(n)$. This builds on prior work of Zywina, which gives a method for computing the image of the adelic Galois representation of a non-CM elliptic curve over $\mathbb{Q}$. Running the algorithm on all elliptic curves presently in the LMFDB and the Stein-Watkins Database gives evidence for the conjecture that $j\in \mathbb{Q}$ is the image of an isolated point if and only if $j$ corresponds to an elliptic curve with complex multiplication or $j=-140625/8, -9317,$ $351/4$, or $-162677523113838677$.

This is joint work with Sachi Hashimoto, Timo Keller, Zev Klagsbrun, David Lowry-Duda, Travis Morrison, Filip Najman, and Himanshu Shukla.