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An algorithm for determining torsion growth of elliptic curves

Enrique González-Jiménez, Filip Najman

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 $m\geq 2$.

An algorithm for determining torsion growth of elliptic curves

Abstract

We present a fast algorithm that takes as input an elliptic curve defined over and an integer and returns all the number fields of degree dividing such that contains as a proper subgroup, for all . We ran this algorithm on all elliptic curves of conductor less than 400.000 (a total of 2.483.649 curves) and all and collected various interesting data. In particular, we find a degree 6 sporadic point on , which is so far the lowest known degree a sporadic point on , for .

Paper Structure

This paper contains 15 sections, 11 theorems, 10 equations, 4 tables.

Table of Contents

  1. Introduction
  2. Auxiliary results
  3. Points of order $125$
  4. Points of order $49$
  5. Points of order $\ell^2$ for $\ell > 7$
  6. Points of order $37$
  7. Points of order $17$
  8. Some special degrees
  9. The algorithm
  10. Computational results
  11. Primitive torsion growth
  12. Heuristical complexity
  13. Timing
  14. On sporadic torsion
  15. A degree 6 sporadic point on $X_1(4,12)$

Key Result

Proposition 2.2

Let $E$ be an elliptic curve defined over a number field $K$ such that its $\ell$-adic representation is defined modulo $\ell^n$. Then for any point $P\in E({\overline{K}})$ of order $\ell^{n+1}$, we have $[K(P):K(\ell P)]=\ell^2$.

Theorems & Definitions (36)

  • Conjecture 2.1
  • Proposition 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • proof
  • Proposition 2.6
  • ...and 26 more