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Properties

Label 8.16.4.1.3
Class number $1$
Dimension $8$
Determinant $16$
Level $4$

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Lattice Invariants

Dimension:$8$
Determinant:$16$
Level:$4$
Density:$0.0634173769752620034091408415940\dots$
Group order:$622080$
Hermite number:$1.41421356237309504880168872421\dots$
Minimal vector length:$2$
Kissing number:$72$
Normalized minimal vectors: $(1, -2, 1, 1, 1, 1, 0, 0)$, $(1, -1, 0, 0, 0, 0, 0, 0)$, $(1, -1, 0, 1, 0, 0, 0, 0)$, $(1, -1, 0, 1, 0, 1, 0, 0)$, $(1, -1, 0, 1, 1, 0, 0, 0)$, $(1, -1, 1, 0, 0, 0, 0, 0)$, $(1, -1, 1, 0, 0, 1, 0, 0)$, $(1, -1, 1, 0, 1, 0, 0, 0)$, $(1, -1, 1, 1, 0, 1, 0, 0)$, $(1, -1, 1, 1, 1, 0, 0, 0)$, $(1, -1, 1, 1, 1, 1, 0, 0)$, $(1, 0, 0, 0, -1, 0, 0, 0)$, $(1, 0, 0, 0, 0, -1, 0, 0)$ ...
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Theta Series

\(1 \) \(\mathstrut +\mathstrut 72 q^{2} \) \(\mathstrut +\mathstrut 432 q^{4} \) \(\mathstrut +\mathstrut 2016 q^{6} \) \(\mathstrut +\mathstrut 3696 q^{8} \) \(\mathstrut +\mathstrut 9072 q^{10} \) \(\mathstrut +\mathstrut 12096 q^{12} \) \(\mathstrut +\mathstrut 24768 q^{14} \) \(\mathstrut +\mathstrut 29808 q^{16} \) \(\mathstrut +\mathstrut 54504 q^{18} \) \(\mathstrut +\mathstrut 54432 q^{20} \) \(\mathstrut +\mathstrut O(q^{21}) \)

Gram Matrix

$\left(\begin{array}{rrrrrrrr} 2 & 1 & -1 & -1 & 1 & 1 & 1 & 1 \\ 1 & 2 & 0 & 0 & 1 & 1 & 0 & 0 \\ -1 & 0 & 2 & 1 & -1 & -1 & 0 & 0 \\ -1 & 0 & 1 & 2 & -1 & -1 & -1 & -1 \\ 1 & 1 & -1 & -1 & 2 & 1 & 0 & 0 \\ 1 & 1 & -1 & -1 & 1 & 2 & 0 & 0 \\ 1 & 0 & 0 & -1 & 0 & 0 & 4 & 0 \\ 1 & 0 & 0 & -1 & 0 & 0 & 0 & 4 \end{array}\right)$

Genus Structure

Genus representatives:
Class number:$1$
 
$\left(\begin{array}{rrrrrrrr} 2 & 1 & -1 & -1 & 1 & -1 & 0 & 0 \\ 1 & 2 & -1 & 0 & 1 & -1 & 1 & 1 \\ -1 & -1 & 2 & 1 & -1 & 1 & 0 & 0 \\ -1 & 0 & 1 & 2 & 0 & 1 & 0 & 0 \\ 1 & 1 & -1 & 0 & 2 & -1 & 0 & 0 \\ -1 & -1 & 1 & 1 & -1 & 2 & -1 & -1 \\ 0 & 1 & 0 & 0 & 0 & -1 & 4 & 0 \\ 0 & 1 & 0 & 0 & 0 & -1 & 0 & 4 \end{array}\right)$
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