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arXiv · 2311.07740

Towards a classification of isolated $j$-invariants

Abbey Bourdon·Sachi Hashimoto·Timo Keller·Zev Klagsbrun·David Lowry-Duda·Travis Morrison·Filip Najman·Himanshu Shukla
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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 $-162677523113838677$.

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Abbey Bourdon, Sachi Hashimoto, Timo Keller, Zev Klagsbrun, David Lowry-Duda, Travis Morrison, Filip Najman, Himanshu Shukla. 2025-01-20. Towards a classification of isolated $j$-invariants. https://doi.org/10.1090/mcom%2F3956

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