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Properties

Label 5.4.8.1.3
Class number $1$
Dimension $5$
Determinant $4$
Level $8$

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The $D_5$ lattice is the root lattice associated to the $D_5$ root system..

Lattice Invariants

Dimension:$5$
Determinant:$4$
Level:$8$
Density:$0.465257613309258635610504062411\dots$
Group order:$3840$
Hermite number:$1.51571656651039808234725980131\dots$
Minimal vector length:$2$
Kissing number:$40$
Normalized minimal vectors: $(1, -1, 0, 0, 0)$, $(1, -1, 0, 0, 1)$, $(1, -1, 0, 1, 1)$, $(1, -1, 1, 0, 1)$, $(1, 0, -1, 0, 0)$, $(1, 0, 0, -1, 0)$, $(1, 0, 0, 0, 0)$, $(1, 0, 0, 0, 1)$, $(0, 1, -1, -1, -1)$, $(0, 1, -1, 0, -1)$, $(0, 1, -1, 0, 0)$, $(0, 1, 0, -1, -1)$, $(0, 1, 0, -1, 0)$, $(0, 1, 0, 0, 0)$, $(0, 0, 1, -1, 0)$, $(0, 0, 1, 0, 0)$, $(0, 0, 1, 0, 1)$, $(0, 0, 0, 1, 0)$, $(0, 0, 0, 1, 1)$, $(0, 0, 0, 0, 1)$ ...
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Theta Series

\(1 \) \(\mathstrut +\mathstrut 40 q^{2} \) \(\mathstrut +\mathstrut 90 q^{4} \) \(\mathstrut +\mathstrut 240 q^{6} \) \(\mathstrut +\mathstrut 200 q^{8} \) \(\mathstrut +\mathstrut 560 q^{10} \) \(\mathstrut +\mathstrut 400 q^{12} \) \(\mathstrut +\mathstrut 800 q^{14} \) \(\mathstrut +\mathstrut 730 q^{16} \) \(\mathstrut +\mathstrut 1240 q^{18} \) \(\mathstrut +\mathstrut 752 q^{20} \) \(\mathstrut +\mathstrut O(q^{21}) \)

Gram Matrix

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

Genus Structure

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

The $D_5$ lattice is the unique solution of the lattice packing problem in dimension 5.