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arXiv · 1904.07071

An algorithm for determining torsion growth of elliptic curves

Enrique González-Jiménez·Filip Najman
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Abstract

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 $m\geq 2$.

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Enrique González-Jiménez, Filip Najman. 2020-06-21. An algorithm for determining torsion growth of elliptic curves. https://doi.org/10.1080/10586458.2020.1771638

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