An algorithm for determining torsion growth of elliptic curves

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 $d^{\prime }$ dividing $d$ such that $E(K{)}_{tors}$ contains $E(F{)}_{tors}$ as a proper subgroup, for all $FK$. We ran this algorithm on all elliptic curves of conductor less than 400.000 (a total of 2.483.649 curves) and all $d\le 23$ 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 $m\ge 2$.

15 pages, Added Supplementary material

References in corpus (6)

• Growth of torsion groups of elliptic curves upon base change
• Torsion points and Galois representations on CM elliptic curves
• Torsion subgroups of elliptic curves over quintic and sextic number fields
• Complete classification of the torsion structures of rational elliptic curves over quintic number fields
• Torsion of rational elliptic curves over quadratic fields
• On the torsion of rational elliptic curves over sextic fields

Cited by in corpus (1)

• Squares in arithmetic progression over certain non-primitive quartic number fields