Properties

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

Learn more about

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: $(1, -1, 0)$, $(1, -1, 1)$, $(1, 0, 0)$, $(1, 0, 1)$, $(0, 1, 0)$, $(0, 0, 1)$
Download this list for gp, magma, sage

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

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.