# 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}$$E/\mathbb{Q}$ is an elliptic curve and $m$$m$ is an integer, and a basis $P,Q$$P,Q$ of $E\left[m\right]$$E[m]$, we have the following embedding

$\begin{array}{llll}{\rho }_{E,m}:& \mathrm{G}\mathrm{a}\mathrm{l}\left(\overline{\mathbb{Q}}/\mathbb{Q}\right)& \to & {\mathrm{G}\mathrm{L}}_{2}\left(\mathbb{Z}/m\mathbb{Z}\right)\\ & \sigma & ↦& \left(\begin{array}{cc}a& c\\ b& d\end{array}\right),\end{array}$
where $a,b,c,d$$a,b,c,d$ are such that ${P}^{\sigma }=aP+bQ$$P^\sigma=aP+bQ$ and ${Q}^{\sigma }=cP+dQ$$Q^\sigma=cP+dQ$. By Serre's open image theorem, ${\rho }_{E,m}$$\rho_{E,m}$ is surjective for all integers not divisible by a finite set of primes ${\mathcal{L}}_{E}$$\mathcal{L}_E$. Moreover, for all $\ell \in {\mathcal{L}}_{E}$$\ell\in \mathcal{L}_E$, there exists an integer $k\ge 1$$k\geq 1$ such that $\text{Im}\left({\rho }_{E,{\ell }^{k}}\right)$$\textrm{Im}(\rho_{E,\ell^k})$ has the same index as ${\rho }_{E,{\ell }^{k}}$$\rho_{E,\ell^k}$ for all ${k}^{\prime }\ge k$$k'\geq k$.

Mazur's program B consists in classifying the elliptic curves $E$$E$ which correspond to a given congruence subgrop of ${\text{GL}}_{2}\left(\mathbb{Z}\right)$$\text{GL}_2(\mathbb{Z})$. In our paper (available here), we solve the case of subgroups $H$$H$ whose levels are not prime-powers and which occur for infinitely many $j$$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$$\alpha$ (resp. ${\alpha }_{K}$$\alpha_K$ for $K=\mathbb{Q}\left({\zeta }_{3}\right)$$K=\mathbb{Q}(\zeta_3)$, $\mathbb{Q}\left(i\right)$$\mathbb{Q}(i)$ and $\mathbb{Q}\left({\zeta }_{5}\right)$$\mathbb{Q}(\zeta_5)$) for each family n Theorem 4.1 following the method in Sections 5.2 and 5.4 of our article. One can do so for any set corresponding to a subgroup of GL${}_{2}\left(\mathbb{Z}\right)$$_2(\mathbb{Z})$ without even knowing if the set is empty, finite or infinite. Given any elliptic curve $E$$E$ with rational coefficients, one tests if its $j$$j$-invariant is coordinate on the modular curve of each subgroup which can occur as Galois image and hence obtains $\alpha \left(E\right)$$\alpha(E)$.

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

Several scripts allow to verify the results of Theorem 4.1: