Towards a classification of isolated 𝑗-invariants

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Abstract
We develop an algorithm to test whether a non-complex multiplication elliptic curve E / Q E/\mathbf {Q} gives rise to an isolated point of any degree on any modular curve of the form X 1 ( N ) 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 E . Running this algorithm on all elliptic curves presently in the L L -functions and Modular Forms Database and the Stein–Watkins Database gives strong evidence for the conjecture that E E gives rise to an isolated point on X 1 ( N ) X_1(N) if and only if j ( E ) = − 140625 / 8 , − 9317 j(E)=-140625/8, -9317 , 351 / 4 351/4 , or − 162677523113838677 -162677523113838677 .
Funding Information
  • National Science Foundation (DMS-2145270, DMS-1928930, 01064790 -- ExplicitRatPoints, grant 546235, HRZZ IP-2022-10-5008, STO 299/17-1, 2007959)

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