ECQ

ECQ is a library based on FLINT to do arithmetic with elliptic curves over $\mathbb Q$. In particular, computing rational points on the curve.

$2$-descent

Complete $2$-descent is work in progress and is the first goal of this project.

Basic setup

• Basic data structure
• Minimal model (expect at $p = 2$ to keep $a_1 = a_3 = 0$)
• Computation of the possible value for $b_i$
• The $2$-descent itself for full $2$-torsion

Ternary quadratic forms

• Reduction of TQF
• Finding primitive solution with LLL
• Parameterization of the solution space

Quartics

• Equivalence of quartics
• Minimal model and reduction

Hyper-elliptic curve of genus $2, 3$

• Local solubility criterion
• Basic enumeration
• Sieving for rational points
• Higher order descent

General $2$-descent

• Finding quartics in full $2$-torsion case
• Finding quartics in $2$-isogeny case
• Finding quartics without $2$-torsion

Installation

For now, ECQ is dependent on FLINT, specifically on this PR, I might just add the full content of the PR to factor_addition is it ends up not being merged.

To make the current demo :

make

and run it with :

./ecq

Performance

For performance we consider the following curve :

$$y^2 = (x - 7265)(x - 649)(x + 7557)$$

which as of last build find the point (-6591312805886080952551/905772661979601025 : 154861252810623357814342127836368/862042768525758359255917375) in :

real    0m0,674s
user    0m0,662s
sys     0m0,009s