Du Sautoy nilpotent group for an elliptic curve

Definition

Consider an elliptic curve $E$ given by an equation of the form $y^2+x^3-Dx=0$ where $D\in \mathbb{Z}$. The du Sautoy nilpotent group corresponding to $E$ is the group $G(E)$ with the following presentation:

$G(E)=⟨x_1,x_2,x_3,x_4,x_5,x_6,y_1,y_2,y_3\mid [x_1,x_4]=y_3^D,[x_1,x_5]=y_1,[x_1,x_6]=y_2,[x_2,x_4]=y_1,[x_2,x_5]=y_3,[x_3,x_4]=y_2,[x_3,x_6]=y_1,\text{other commutators between generating set elements all trivial}⟩$

The group $G(E)$ is canonically attached to the elliptic curve $E$.

Arithmetic functions

Function	Value	Similar groups	Explanation
Hirsch length	9
nilpotency class	2
derived length	2

Related notions

• Du Sautoy nilpotent Lie ring for an elliptic curve

References

• Counting subgroups in nilpotent groups and points on elliptic curves by Marcus du Sautoy, Journal fur die reine und angewandte Mathematik, Volume 549, Page 1 - 21(Year 2002): ^{Gated copy (PDF)More info}