An algorithm for determining torsion growth of elliptic curves
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arXiv
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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2019
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| _version_ | 1866916117881028608 |
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| author | González-Jiménez, Enrique Najman, Filip |
| author_facet | González-Jiménez, Enrique Najman, Filip |
| contents | We present a fast algorithm that takes as input an elliptic curve defined over $\mathbb Q$ and an integer $d$ and returns all the number fields $K$ of degree $d'$ dividing $d$ such that $E(K)_{tors}$ contains $E(F)_{tors}$ as a proper subgroup, for all $F \varsubsetneq K$. We ran this algorithm on all elliptic curves of conductor less than 400.000 (a total of 2.483.649 curves) and all $d \leq 23$ and collected various interesting data. In particular, we find a degree 6 sporadic point on $X_1(4,12)$, which is so far the lowest known degree a sporadic point on $X_1(m,n)$, for $m\geq 2$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1904_07071 |
| institution | arXiv |
| publishDate | 2019 |
| record_format | arxiv |
| spellingShingle | An algorithm for determining torsion growth of elliptic curves González-Jiménez, Enrique Najman, Filip Number Theory We present a fast algorithm that takes as input an elliptic curve defined over $\mathbb Q$ and an integer $d$ and returns all the number fields $K$ of degree $d'$ dividing $d$ such that $E(K)_{tors}$ contains $E(F)_{tors}$ as a proper subgroup, for all $F \varsubsetneq K$. We ran this algorithm on all elliptic curves of conductor less than 400.000 (a total of 2.483.649 curves) and all $d \leq 23$ and collected various interesting data. In particular, we find a degree 6 sporadic point on $X_1(4,12)$, which is so far the lowest known degree a sporadic point on $X_1(m,n)$, for $m\geq 2$. |
| title | An algorithm for determining torsion growth of elliptic curves |
| topic | Number Theory |
| url | https://arxiv.org/abs/1904.07071 |