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.
Lattice Invariants
Dimension: | $3$ |
Determinant: | $4$ |
Level: | $8$ |
Density: | $0.740480489693061041169313498343\dots$ |
Group order: | $48$ |
Hermite number: | $1.25992104989487316476721060728\dots$ |
Minimal vector length: | $2$ |
Kissing number: | $12$ |
Normalized minimal vectors: |
|
Download this list for gp, magma, sage |
Theta Series
Gram Matrix
$\left(\begin{array}{rrr} 2 & 1 & -1 \\ 1 & 2 & 0 \\ -1 & 0 & 2 \end{array}\right)$
Genus Structure
Class number: | $1$ |
$\left(\begin{array}{rrr} 2 & 1 & -1 \\ 1 & 2 & 0 \\ -1 & 0 & 2 \end{array}\right)$ | |
Download this matrix for gp, magma, sage |
Comments
This integral lattice is the A3, D3 lattice.
This is the face centered cubic lattice. This is a root lattice. This is the cubic F Bravais lattice of classical holotype and even holotype.
Additional information
This lattice 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 in dimension 3, by a theorem of Ferguson-Hales [MR:3075372], and of the general kissing number problem in dimension 3.