Images of the 2-adic Galois representation attached to elliptic curves

Jeremy Rouse and David Zureick-Brown

In our paper "Elliptic curves over $\mathbb{Q}$ and 2-adic images of Galois" (on arXiv here), we classify the possible images of the 2-adic Galois representations attached to non-CM elliptic curves $E/\mathbb{Q}$. This website serves as a companion to the paper, and includes links to websites documenting the relevant lattice of modular curves, as well as files with data, computation logs, and Magma scripts that are necessary for our results.

The lattice of modular curves can be accessed at the link for $X_{1}$.

Facts about the subgroups of ${\rm GL}_{2}(\mathbb{Z}_{2})$, the corresponding modular curves, and the rational points on them:

Below is a table with links to data files.

Filename File type Description
gl2data.gz Gzipped Magma list Data on the 727 arithmetically maximal subgroups containing $-I$. The index, level, genus, maximal subgroups, and chosen covering are listed.
gl2finedata.gz Gzipped Magma list Data on 1006 subgroups not containing $-I$. The subgroup containing it in gl2data, the index, level, and generating matrices are given.
models.tar.gz Gzipped tar archive of Magma scripts The 345 models computed (for subgroups $H$ with $-I \in H$).
maps.tar.gz Gzipped tar archive of Magma scripts Magma scripts giving the covering maps $X_{K} \to X_{H}$ when $K \subseteq H$.
finemodels.tar.gz Gzipped tar archive of Magma scripts Magma scripts giving the universal elliptic curve $E/U$ for the subgroups $H$ with $-I \not\in H$.
curvelist1.txt Magma list A list of numbers (corresponding to subgroups in gl2data.txt), an elliptic curve $E$ (given by the standard list of $5$ numbers) whose $2$-adic image is $H$ (when such a curve exists).
curvelist2.txt Magma list A list of names of curves $X_{H}$ with $-I \not\in H$, together with an elliptic curve $E$ whose $2$-adic image is $H$.

Below is a table with links to log files.

Filename File type Description
modellog.tar.gz Gzipped tar archive of text files Log files for the computations of the models for the 727 curves $X_{H}$ for those $H$ with $-I \in H$.
finemodelslog.tar.gz Gzipped tar archive of text files Log files from the computations of the universal elliptic curves.
unram.out Text file Output of the program unram.txt that searches for modular unramified double covers of the genus 5 and 7 curves.
desing3.gz Gzipped text file Output of the program desing3.txt that computes non-singular models and the rational points on them of the genus 5 and 7 curves (for which the unramified double cover method applies).

Below is a table with links to program files.

Filename File type Description
2adicimage.txt Magma script This script takes as input a non-CM elliptic curve and determines its $2$-adic image. It must be in the same directory as gl2data.txt, gl2finedata.txt, and the models and maps between them.
leven2n-gl.txt Magma script This script generates the lattice of arithmetically maximal subgroups $H$ with $-I \in H$. It outputs the file gl2data.txt
models10.txt Magma script This script computes the models of the curves $X_{H}$ for which $-I \in H$.
finesearch2.txt Magma script This script compiles a list of subgroups $H$ with $-I \not\in H$ that must be analyzed. It writes the file gl2finedata.txt
whichtwist2.txt Magma script This script computes the universal elliptic curve $E/U$ for the subgroups $H$ with $-I \not\in H$.
autocompute.txt Magma script This script attempts to guess the automorphism group of a canonically embedded curve defined over a number field by writing down the automorphism group scheme, using Magma's built in routines to construct points on it modulo a prime $p$, using Hensel's lemma to lift those points, and using lattice reduction to construct automorphisms over the number field.
magscript.txt Magma script This script proves that the $2$-adic representation is surjective for each of the non-CM elliptic curves corresponding to a rational point on one of the genus $2$ or $3$ curves.
genus-2.m Magma script This script determines all the rational points on the genus 2 curves (except $X_{441}$ and $X_{520}$).
X441.m Magma script This script determines all the rational points on $X_{441}$.
X520.m Magma script This script determines all the rational points on $X_{520}$.
genus-3-hyperelliptic.m Magma script This script determines all the rational points on the genus three hyperelliptic curves.
X618.m Magma script This script determines all the rational points on $X_{618}$.
X619.m Magma script This script uses elliptic curve Chabauty to determine the rational points on (a family of unramified double covers of) $X_{619}$.
x619.tar.gz Gzipped tar archive of Magma scripts This script provides the programs used to compute the maps to elliptic curves from the unramified double covers of $X_{619}$.
X641.m Magma script This script computes the rational points on $X_{641}$.
X650.m Magma script This script computes the rational points on $X_{650}$.
X672.m Magma script This script computes the rational points on the genus five curve $X_{672} \cong X_{689}$. (The model of $X_{689}$ is used.)
X686.m Magma script This script computes the rational points on the genus five curve $X_{686}$.
unram.txt Magma script This script determines which genus five and seven curves admit modular unramified double covers.
desing3.txt Magma script This script computes canonical models of genus five and seven curves that admit modular unramified double covers, and computes all the rational points on them.