Towards a classification of isolated 𝑗-invariants

Bourdon, Abbey; Hashimoto, Sachi; Keller, Timo; Klagsbrun, Zev; Lowry-Duda, David; Morrison, Travis; Najman, Filip; Shukla, Himanshu

doi:10.1090/mcom/3956

Towards a classification of isolated $j$-invariants
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by Abbey Bourdon, Sachi Hashimoto, Timo Keller, Zev Klagsbrun, David Lowry-Duda, Travis Morrison, Filip Najman and Himanshu Shukla; with an appendix by Maarten Derickx and Mark van Hoeij

Math. Comp. 94 (2025), 447-473

DOI: https://doi.org/10.1090/mcom/3956

Published electronically: April 25, 2024

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Abstract:

We develop an algorithm to test whether a non-complex multiplication elliptic curve $E/\mathbf {Q}$ 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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Towards a classification of isolated $j$-invariants
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Abstract: