# 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:

• We call a subgroup of $H \subset {\rm GL}_{2}(\mathbb{Z}_{2})$ arithmetically maximal if $\det \colon H \to \mathbb{Z}_{2}^{\times}$ is surjective, there is an $M \in H$ with determinant $-1$ and trace zero, and there is no subgroup $K$ with $H \subseteq K$ so that the modular curve $X_{K}$ has genus $\geq 2$.
• There are $727$ arithmetically maximal subgroups of ${\rm GL}_{2}(\mathbb{Z}_{2})$ that contain $-I$ (up to conjugacy in ${\rm GL}_{2}(\mathbb{Z}_{2})$), and each of them has a webpage (which can be found from the link above).
• The curves $X_{H}$ have genus $0$, $1$, $2$, $3$, $5$ or $7$. We find all the rational points on each $X_{H}$.
• Each curve $X_{H}$ has good reduction away from $2$.
• We compute a model for each $X_{H}$ unless there is a subgroup $K$ with $H \subset K$ so that $X_{K}$ has finitely many rational points. In total, we compute models of $345$ curves $X_{H}$ for subgroups $H$ containing $-I$. Of these, $185$ have genus zero, $58$ have genus one, $57$ have genus two, $22$ have genus three, $20$ have genus five, and $4$ have genus seven.
• If $H$ is a subgroup that contains $-I$ and $X_{H}$ has any non-cuspidal, non-CM rational points, and there is a subgroup $K \subseteq H$ with $-I \not\in K$ then (i) $X_{H} \cong \mathbb{P}^{1}$, and (ii) we compute a model for the universal elliptic curve $E/U$, where $U$ is the locus of $X_{H}$ with $j \ne 0$, $j \ne 1728$, and $j \ne \infty$. (Each of the $1006$ subgroups $K$ of this type also has a webpage.)
• If $E/\mathbb{Q}$ is a non-CM elliptic curve, the image of $\rho_{E} : {\rm Gal}(\overline{\mathbb{Q}}/\mathbb{Q}) \to {\rm GL}_{2}(\mathbb{Z}_{2})$ has index equal to $1$, $2$, $3$, $4$, $6$, $8$, $12$, $16$, $24$, $32$, $48$, $64$ or $96$.
• For a non-CM curve $E$, the image of $\rho_{E}$ in ${\rm GL}_{2}(\mathbb{Z}_{2})$ always contains the kernel of the map ${\rm GL}_{2}(\mathbb{Z}_{2}) \to {\rm GL}_{2}(\mathbb{Z}/32\mathbb{Z})$.
• Of the $1733$ subgroups $H$ we list, there are $1215$ that have non-cuspidal, non-CM rational points.
• There are $1208$ possibilities for the image of $\rho_{E} \colon {\rm Gal}(\overline{\mathbb{Q}}/\mathbb{Q}) \to {\rm GL}_{2}(\mathbb{Z}_{2})$.
• The seven curves $X_{150}$, $X_{153}$, $X_{155}$, $X_{156}$, $X_{165}$, $X_{166}$, and $X_{167}$ are all modular curves $X_{H}$ that are elliptic with positive rank, and every rational point on $X_{H}$ lifts to a rational point on $X_{K}$ for some index $2$ subgroup $K$ of $H$.
• Every rational point on a curve $X_{H}$ of genus one that has rank zero is a cusp or a CM point.
• The only genus $2$ curve with non-cuspidal, non-CM rational points is $X_{441}$, also known as $X_{ns}^{+}(16)$. This curve has two non-cuspidal, non-CM rational points.
• The genus $3$ curves $X_{556}$, $X_{558}$, $X_{563}$, and $X_{566}$ are hyperelliptic.
• The only genus $3$ curves with non-cuspidal, non-CM rational points are $X_{556}$, $X_{558}$, $X_{563}$, $X_{566}$, $X_{619}$, and $X_{649}$. Each of these gives rise to a single $j$-invariant.
• The genus $5$ and $7$ curves are represented (on this website) by a singular model that is a double cover of an elliptic curve. Non-singular canonical models are given below in the log file "desing3.out".
• All the rational points on the genus $5$ and $7$ curves are either cusps or CM points.
• There are many instances of non-conjugate subgroups $H$ and $K$ with $X_{H} \cong X_{K}$. Within the $22$ genus three curves, there are at most $7$ isomorphism classes. Within the $20$ genus five curves, there are at most $10$ isomorphism classes. The $4$ genus seven curves, fall into two isomorphism classes.
• The curve $X_{504}$ is a genus two curve with six rational points. Its Jacobian has rank zero, and torsion subgroup $\mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/10\mathbb{Z}$.
• If $E/\mathbb{Q}$ is a non-CM elliptic curve whose mod $2$ image has index $1$, the $2$-adic image can have index as large as $64$.
• If $E/\mathbb{Q}$ is a non-CM elliptic curve whose mod $2$ image has index $2$, the $2$-adic image has index $2$ or $4$.
• If $E/\mathbb{Q}$ is a non-CM elliptic curve whose mod $2$ image has index $3$, the $2$-adic image can have index as large as $96$.
• If $E/\mathbb{Q}$ is a non-CM elliptic curve whose mod $2$ image has index $6$, the $2$-adic image can have index as large as $96$, although some quadratic twist of $E$ must have $2$-adic image with index less than $96$.

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.