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Properties

Label 7.4.8.1.2
Class number $1$
Dimension $7$
Determinant $4$
Level $8$

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Lattice Invariants

Dimension:$7$
Determinant:$4$
Level:$8$
Density:$0.208807128571298248711975947784\dots$
Group order:$645120$
Hermite number:$1.64067071201527586234056936574\dots$
Minimal vector length:$2$
Kissing number:$84$
Normalized minimal vectors: $(1, -1, 0, 0, 0, 0, 0)$, $(1, -1, 0, 0, 1, 0, 0)$, $(1, -1, 0, 0, 1, 0, 1)$, $(1, -1, 0, 0, 1, 1, 0)$, $(1, -1, 0, 1, 1, 0, 0)$, $(1, -1, 1, 0, 1, 0, 0)$, $(1, 0, -1, 0, 0, 0, 0)$, $(1, 0, 0, -1, 0, 0, 0)$, $(1, 0, 0, 0, 0, -1, 0)$, $(1, 0, 0, 0, 0, 0, -1)$, $(1, 0, 0, 0, 0, 0, 0)$, $(1, 0, 0, 0, 1, 0, 0)$, $(0, 1, -1, -1, -1, 0, 0)$, $(0, 1, -1, 0, -1, -1, 0)$, $(0, 1, -1, 0, -1, 0, -1)$ ...
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Theta Series

\(1 \) \(\mathstrut +\mathstrut 84 q^{2} \) \(\mathstrut +\mathstrut 574 q^{4} \) \(\mathstrut +\mathstrut 1288 q^{6} \) \(\mathstrut +\mathstrut 3444 q^{8} \) \(\mathstrut +\mathstrut 4424 q^{10} \) \(\mathstrut +\mathstrut 9240 q^{12} \) \(\mathstrut +\mathstrut 11088 q^{14} \) \(\mathstrut +\mathstrut 18494 q^{16} \) \(\mathstrut +\mathstrut 19740 q^{18} \) \(\mathstrut +\mathstrut 34440 q^{20} \) \(\mathstrut +\mathstrut O(q^{21}) \)

Gram Matrix

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

Genus Structure

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