1. Towards a classification of isolated $j$-invariants
- Author
-
Bourdon, Abbey, Hashimoto, Sachi, Keller, Timo, Klagsbrun, Zev, Lowry-Duda, David, Morrison, Travis, Najman, Filip, and Shukla, Himanshu
- Subjects
Mathematics - Number Theory - Abstract
We develop an algorithm to test whether a non-CM elliptic curve $E/\mathbb{Q}$ gives rise to an isolated point of any degree on any modular curve of the form $X_1(N)$. This builds on prior work of Zywina which gives a method for computing the image of the adelic Galois representation associated to $E$. Running this algorithm on all elliptic curves presently in the $L$-functions and Modular Forms Database and the Stein-Watkins Database gives strong evidence for the conjecture that $E$ gives rise to an isolated point on $X_1(N)$ if and only if $j(E)=-140625/8, -9317,$ $351/4$, or $-162677523113838677$., Comment: With an appendix by Maarten Derickx and Mark van Hoeij, to appear in Math. Comp
- Published
- 2023