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