A classification of ECM-friendly families using modular curves

Razvan Barbulescu and Sudarshan Shinde

The main object in our work is the image of Galois : if $E/\mathbb{Q}$ is an elliptic curve and $m$ is an integer, and a basis $P,Q$ of $E[m]$ , we have the following embedding

\begin{array}{llll} ρ_{E, m} : & G a l (\bar{Q} / Q) & \to & {G L}_{2} (Z / m Z) \\ σ & \mapsto & (\begin{matrix} a & c \\ b & d \end{matrix}), \end{array}

$\begin{array}{llll} \rho_{E,m}: & \mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q}) & \rightarrow & \mathrm{GL}_2(\mathbb{Z}/m\mathbb{Z}) \\ & \sigma & \mapsto & \begin{pmatrix} a & c \\ b & d \end{pmatrix}, \end{array}$ where

a, b, c, d

$a,b,c,d$ are such that

P^{σ} = a P + b Q

$P^\sigma=aP+bQ$ and

Q^{σ} = c P + d Q

$Q^\sigma=cP+dQ$ . By Serre's open image theorem,

ρ_{E, m}

$\rho_{E,m}$ is surjective for all integers not divisible by a finite set of primes

L_{E}

$\mathcal{L}_E$ . Moreover, for all

ℓ \in L_{E}

$\ell\in \mathcal{L}_E$ , there exists an integer

k \geq 1

$k\geq 1$ such that

Im (ρ_{E, ℓ^{k}})

$\textrm{Im}(\rho_{E,\ell^k})$ has the same index as

ρ_{E, ℓ^{k}}

$\rho_{E,\ell^k}$ for all

k^{'} \geq k

$k'\geq k$ .

Mazur's program B consists in classifying the elliptic curves $E$ which correspond to a given congruence subgrop of $\text{GL}_2(\mathbb{Z})$ . In our paper (available here), we solve the case of subgroups $H$ whose levels are not prime-powers and which occur for infinitely many $j$ -invariants.

Data

A complete list of the 1525 ECM-friendly families is available in

COMPLETE LIST. Click on the label of any family to find a sage script allowing to parametrize the family.

We also provide this list in the form of two sage scripts:

GALDATA.sage contains the list of subgroups of prime-power level which contain the Galois image associated to elliptic curves for infinitely many j-invariants.
GALDATA_COMPOSITE.sage does the same for levels which are not prime-powers.

Alpha

We pre-computed $\alpha$ (resp. $\alpha_K$ for $K=\mathbb{Q}(\zeta_3)$ , $\mathbb{Q}(i)$ and $\mathbb{Q}(\zeta_5)$ ) for each family n Theorem 4.1 following the method in Sections 5.2 and 5.4 of our article. One can do so for any set corresponding to a subgroup of GL $_2(\mathbb{Z})$ without even knowing if the set is empty, finite or infinite. Given any elliptic curve $E$ with rational coefficients, one tests if its $j$ -invariant is coordinate on the modular curve of each subgroup which can occur as Galois image and hence obtains $\alpha(E)$ .

TABQQ.sage contains the $\alpha$ value of all the families occuring for infinitely many times.
TABZ3.sage TABZ4.sage TABZ5.sage contains the $\alpha$ relative to the primes congruent to $3$ , $4$ and $5$ respectively, for the same families.
alpha.sage prints the family of $E$ as well as $\alpha(E)$ .

Verifications

The following script allows to do the verifications in Appendix B, which proves the equality among various families in the literature and the families in Theorem 4.1.

verification-literature.sage

Several scripts allow to verify the results of Theorem 4.1:

miscel.sage is an auxiliary file.
genus_0_maps.sage certifies the maps given in the database of composite levels.
genus_0_twists.sage certifies the twists of cartesian products upto isomorphism of models over $\mathbb{Q}(t)$ .
genus_1_all_labels.sage contains 163 pairs of groups with genus 1.
genus_1_elimination.sage eliminates the rank 0 pairs out of 163 pairs.
genus_1_maps.sage obtains Weierstrass models of rank for the remaining curves and constructs j-map.
genus_1_twists.sage certifies the twists of genus 1 models.