Properties

Label 7.64.8.1.1
Class number $1$
Dimension $7$
Determinant $64$
Level $8$

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Lattice Invariants

Dimension:$7$
Determinant:$64$
Level:$8$
Density:$0.0522017821428245621779939869460\dots$
Group order:$92160$
Hermite number:$1.10408951367381233764950538762\dots$
Minimal vector length:$2$
Kissing number:$12$
Normalized minimal vectors: $(1, 0, 0, 0, 0, 0, 0)$, $(0, 1, 0, 0, 0, 0, 0)$, $(0, 0, 1, 0, 0, 0, 0)$, $(0, 0, 0, 0, 1, 0, 0)$, $(0, 0, 0, 0, 0, 1, 0)$, $(0, 0, 0, 0, 0, 0, 1)$
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Theta Series

\(1 \) \(\mathstrut +\mathstrut 12 q^{2} \) \(\mathstrut +\mathstrut 190 q^{4} \) \(\mathstrut +\mathstrut 184 q^{6} \) \(\mathstrut +\mathstrut 1140 q^{8} \) \(\mathstrut +\mathstrut 632 q^{10} \) \(\mathstrut +\mathstrut 3096 q^{12} \) \(\mathstrut +\mathstrut 1584 q^{14} \) \(\mathstrut +\mathstrut 6206 q^{16} \) \(\mathstrut +\mathstrut 2820 q^{18} \) \(\mathstrut +\mathstrut 11400 q^{20} \) \(\mathstrut +\mathstrut O(q^{21}) \)

Gram Matrix

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

Genus Structure

Genus representatives:
Class number:$1$
 
$\left(\begin{array}{rrrrrrr} 2 & 0 & 0 & 0 & 0 & -1 & 1 \\ 0 & 2 & 0 & 0 & 0 & 1 & -1 \\ 0 & 0 & 2 & 0 & 0 & -1 & 1 \\ 0 & 0 & 0 & 2 & 0 & -1 & 1 \\ 0 & 0 & 0 & 0 & 2 & -1 & 1 \\ -1 & 1 & -1 & -1 & -1 & 4 & -3 \\ 1 & -1 & 1 & 1 & 1 & -3 & 4 \end{array}\right)$
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