# An algorithm for determining torsion growth of elliptic curves

## TL;DR

This paper introduces a fast algorithm to determine torsion growth of elliptic curves over number fields, applied to a large dataset of curves, revealing new sporadic points and expanding understanding of torsion structures.

## Contribution

The paper presents a novel, efficient algorithm for analyzing torsion subgroup growth of elliptic curves over number fields, with extensive computational results.

## Key findings

- Identified a degree 6 sporadic point on X_1(4,12)
- Analyzed 2,483,649 elliptic curves of conductor < 400,000
- Collected data on torsion subgroup behavior across various degrees

## 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 $m\geq 2$.