# Properties

 Name A3, D3 Label 3.4.8.1.2 Class number 1 Dimension 3 Determinant 4 Level 8

The $A_3$ lattice, also known as the face-centered cubic lattice, is the root lattice associated to the $A_3$ and $D_3$ root systems. It is the cubic F Bravais lattice of classical holotype and even holotype. It is a solution (unique among lattices but not among all packings) of the sphere packing problem (a theorem of Ferguson and Hales) and the general kissing number problem in dimension 3.

## Lattice Invariants

Dimension:$3$
Determinant:$4$
Level:$8$
Label:$3.4.8.1.2$
Density:$0.740480489693061041169313498343\dots$
Group order:$48$
Hermite number:$1.25992104989487316476721060728\dots$
Minimal vector length:$2$
Kissing number:$12$
Normalized minimal vectors: $(1, -1, 0)$, $(1, -1, 1)$, $(1, 0, 0)$, $(1, 0, 1)$, $(0, 1, 0)$, $(0, 0, 1)$

## Theta Series

$1$ $\mathstrut +\mathstrut 12q^{2}$ $\mathstrut +\mathstrut 6q^{4}$ $\mathstrut +\mathstrut 24q^{6}$ $\mathstrut +\mathstrut 12q^{8}$ $\mathstrut +\mathstrut 24q^{10}$ $\mathstrut +\mathstrut 8q^{12}$ $\mathstrut +\mathstrut 48q^{14}$ $\mathstrut +\mathstrut 6q^{16}$ $\mathstrut +\mathstrut 36q^{18}$ $\mathstrut +\mathstrut 24q^{20}$ $\mathstrut +\mathstrut O(q^{21})$

## Gram Matrix

$\left(\begin{array}{rrr} 2 & 1 & -1 \\ 1 & 2 & 0 \\ -1 & 0 & 2 \end{array}\right)$

## Genus Structure

 Class number: $1$ Genus representatives: $\left(\begin{array}{rrr} 2 & 1 & -1 \\ 1 & 2 & 0 \\ -1 & 0 & 2 \end{array}\right)$ Download this matrix for gp, magma, sage