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Properties

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

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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: $(1, -1, 0)$, $(1, -1, 1)$, $(1, 0, 0)$, $(1, 0, 1)$, $(0, 1, 0)$, $(0, 0, 1)$
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Theta Series

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

Gram Matrix

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

Genus Structure

Genus representatives:
Class number:$1$
 
$\left(\begin{array}{rrr} 2 & 1 & -1 \\ 1 & 2 & 0 \\ -1 & 0 & 2 \end{array}\right)$
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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.