An algorithm for determining torsion growth of elliptic curves

Enrique González-Jiménez and Filip Najman

Algorithm

TorsionGrowth(E,d):

INPUT: An elliptic curve \(E/\mathbb Q\) and a positive integer \(d\).
OUTPUT: A sequence of all pairs \((K, T)\) of a number field \(K\) of degree \(d'\) such that \(d'|d\) and that \(E\) has primitive torsion growth over \(K\), together with the group \(T:=E(K)_{tors}\).

Magma script related to the algorithm

step1	step2	step3	program

Download the zipped file program.zip

Auxiliary files from other articles:

galrep folder from Sutherland's webpage included in the zipped file.
A.V. Sutherland. Computing images of Galois representations attached to elliptic curves. Forum Math. Sigma 4 (2016), e4.

Example: 256a1

Magma V2.24-5 Mon Apr 8 2019 14:57:05 on luna [Seed = 300288142]
Type ? for help. Type <Ctrl>-D to quit.
> load "programNF_optimized.m";
Loading "programNF_optimized.m"
-----------------------------------------------------
Printing level (1,2,3,Gal)=(false,false,false,true)
Using Optimized Representation? false
-----------------------------------------------------
Loading "step1NF.m"
Loading "galrep_Sutherland/galrep.m"
Loading "galrep_Sutherland/subgroups.m"
Loading "step2NF_optimized.m"
Loading "step3NF.m"
> E:=EllipticCurve("256a1");
> TorsionGrowth(E,2);
[
[
[ 2 ],
[ -1, 1 ]
],
[
[ 2, 2 ],
[ -1, 2, 1 ]
]
]
>

The output shows the torsion over (\\mathbb Q\) and the primitive torsion over number fields of degree divisind 2:

Torsion \(\mathbb Z/2\mathbb Z\) over \(\mathbb Q\) and
\(\mathbb Z/2\mathbb Z\times \mathbb Z/2\mathbb Z\) over the quadratic field \(\mathbb Q(\alpha)\) where \(\alpha^2+2\alpha-1=0\).

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