# Properties

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

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