Computing Isogenies at Singular Points of the Modular Polynomial
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866917581486555136 |
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| author | Mahaney, William E. Morrison, Travis |
| author_facet | Mahaney, William E. Morrison, Travis |
| contents | In this paper we present a method which, given a singular point $(j_1, j_2)$ on $Y_0(\ell)$ with $j_1, j_2 \neq 0, 1728$ and an elliptic curve $E$ with $j$-invariant ${j_1}$, returns an elliptic curve $\widetilde{E}$ with $j$-invariant ${j_2}$ that admits a normalized $\ell$-isogeny $ϕ\colon E\to \widetilde{E}$. The isogeny $ϕ$ can then be efficiently computed from $E$ and $\widetilde{E}$ using an algorithm of Bostan, Morain, Salvy, and Schost. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_02038 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Computing Isogenies at Singular Points of the Modular Polynomial Mahaney, William E. Morrison, Travis Number Theory In this paper we present a method which, given a singular point $(j_1, j_2)$ on $Y_0(\ell)$ with $j_1, j_2 \neq 0, 1728$ and an elliptic curve $E$ with $j$-invariant ${j_1}$, returns an elliptic curve $\widetilde{E}$ with $j$-invariant ${j_2}$ that admits a normalized $\ell$-isogeny $ϕ\colon E\to \widetilde{E}$. The isogeny $ϕ$ can then be efficiently computed from $E$ and $\widetilde{E}$ using an algorithm of Bostan, Morain, Salvy, and Schost. |
| title | Computing Isogenies at Singular Points of the Modular Polynomial |
| topic | Number Theory |
| url | https://arxiv.org/abs/2402.02038 |