An algorithm for determining torsion growth of elliptic curves

Enrique González-Jiménez and Filip Najman


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 \(E(K)_{tors} \supsetneq E(F)_{tors}\), for all \( F\subsetneq K\). We ran this algorithm on all elliptic curves of conductor less than 400.000 (a total of 2.483.649 curves) and all \(d \leq 23\).

Algorithm (Magma)

Computational Results

Magma script related to prove some results at section 2 and 5 README

Lemma 2.6

Lemma 2.8

Lemma 2.9

Lemma 2.10 and 2.11

Theorem 5.1

Auxiliary files from other articles:


Download the PDF and then click in the text to open a file with magma source proving the corresponding result.

Here you can download all the necessary files zipped

Last modified: 14/4/2019