# Properties

 Name A2 Label 2.3.3.1.1 Class number 1 Dimension 2 Determinant 3 Level 3

The $A_2$ lattice, also known as the hexagonal lattice, is the root lattice associated to the $A_2$ and $G_2$ root systems. It is the unique solution of the sphere packing problem and the general kissing number problem in dimension 2.

## Lattice Invariants

Dimension:$2$
Determinant:$3$
Level:$3$
Label:$2.3.3.1.1$
Density:$0.906899682117108925297039128820\dots$
Group order:$12$
Hermite number:$1.15470053837925152901829756100\dots$
Minimal vector length:$2$
Kissing number:$6$
Normalized minimal vectors: $(1, 0)$, $(1, 1)$, $(0, 1)$

## Theta Series

$1$ $\mathstrut +\mathstrut 2q^{2}$ $\mathstrut +\mathstrut 2q^{8}$ $\mathstrut +\mathstrut 2q^{18}$ $\mathstrut +\mathstrut O(q^{21})$

## Gram Matrix

$\left(\begin{array}{rr} 2 & -1 \\ -1 & 2 \end{array}\right)$

## Genus Structure

 Class number: $1$ Genus representatives: $\left(\begin{array}{rr} 2 & 1 \\ 1 & 2 \end{array}\right)$ Download this matrix for gp, magma, sage