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| Main Authors: | , , , , , , , |
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| Format: | Preprint |
| Published: |
2023
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2311.07740 |
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| _version_ | 1866915111248068608 |
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| author | Bourdon, Abbey Hashimoto, Sachi Keller, Timo Klagsbrun, Zev Lowry-Duda, David Morrison, Travis Najman, Filip Shukla, Himanshu |
| author_facet | Bourdon, Abbey Hashimoto, Sachi Keller, Timo Klagsbrun, Zev Lowry-Duda, David Morrison, Travis Najman, Filip Shukla, Himanshu |
| contents | 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 $-162677523113838677$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2311_07740 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Towards a classification of isolated $j$-invariants Bourdon, Abbey Hashimoto, Sachi Keller, Timo Klagsbrun, Zev Lowry-Duda, David Morrison, Travis Najman, Filip Shukla, Himanshu Number Theory 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 $-162677523113838677$. |
| title | Towards a classification of isolated $j$-invariants |
| topic | Number Theory |
| url | https://arxiv.org/abs/2311.07740 |