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Properties

Label 4.4.2.1.1
Class number $1$
Dimension $4$
Determinant $4$
Level $2$

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The $D_4$ lattice is the root lattice associated to the $D_4$ and $F_4$ root systems.

Lattice Invariants

Dimension:$4$
Determinant:$4$
Level:$2$
Density:$0.616850275068084913677155687491\dots$
Group order:$1152$
Hermite number:$1.41421356237309504880168872421\dots$
Minimal vector length:$2$
Kissing number:$24$
Normalized minimal vectors: $(1, 0, 0, -1)$, $(1, 0, 0, 0)$, $(1, 0, 1, -1)$, $(1, 1, 0, -1)$, $(1, 1, 1, -2)$, $(1, 1, 1, -1)$, $(0, 1, 0, -1)$, $(0, 1, 0, 0)$, $(0, 1, 1, -1)$, $(0, 0, 1, -1)$, $(0, 0, 1, 0)$, $(0, 0, 0, 1)$
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Theta Series

\(1 \) \(\mathstrut +\mathstrut 24 q^{2} \) \(\mathstrut +\mathstrut 24 q^{4} \) \(\mathstrut +\mathstrut 96 q^{6} \) \(\mathstrut +\mathstrut 24 q^{8} \) \(\mathstrut +\mathstrut 144 q^{10} \) \(\mathstrut +\mathstrut 96 q^{12} \) \(\mathstrut +\mathstrut 192 q^{14} \) \(\mathstrut +\mathstrut 24 q^{16} \) \(\mathstrut +\mathstrut 312 q^{18} \) \(\mathstrut +\mathstrut 144 q^{20} \) \(\mathstrut +\mathstrut O(q^{21}) \)

Gram Matrix

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

Genus Structure

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

The $D_4$ lattice is the unique solution of the lattice packing problem in dimension 4, and a solution (conjectured to be unique) of the general kissing number problem in dimension 4, by a theorem of Musin [MR:2415397].