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Properties

Label 7.8.16.1.3
Class number $1$
Dimension $7$
Determinant $8$
Level $16$

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Lattice Invariants

Dimension:$7$
Determinant:$8$
Level:$16$
Density:$0.147648936572856286549887214605\dots$
Group order:$80640$
Hermite number:$1.48599428913694842479985328671\dots$
Minimal vector length:$2$
Kissing number:$56$
Normalized minimal vectors: $(1, 0, -1, 0, 0, 0, 0)$, $(1, 0, 0, 0, 0, -1, 0)$, $(1, 0, 0, 0, 0, 0, -1)$, $(1, 0, 0, 0, 0, 0, 0)$, $(1, 0, 0, 0, 1, 0, 0)$, $(1, 0, 0, 1, 0, 0, 0)$, $(1, 1, 0, 0, 0, 0, 0)$, $(0, 1, 0, -1, 0, 0, 0)$, $(0, 1, 0, 0, -1, 0, 0)$, $(0, 1, 0, 0, 0, 0, 0)$, $(0, 1, 0, 0, 0, 0, 1)$, $(0, 1, 0, 0, 0, 1, 0)$, $(0, 1, 1, 0, 0, 0, 0)$, $(0, 0, 1, 0, 0, -1, 0)$, $(0, 0, 1, 0, 0, 0, -1)$ ...
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Theta Series

\(1 \) \(\mathstrut +\mathstrut 56 q^{2} \) \(\mathstrut +\mathstrut 420 q^{4} \) \(\mathstrut +\mathstrut 896 q^{6} \) \(\mathstrut +\mathstrut 2366 q^{8} \) \(\mathstrut +\mathstrut 3360 q^{10} \) \(\mathstrut +\mathstrut 6440 q^{12} \) \(\mathstrut +\mathstrut 7168 q^{14} \) \(\mathstrut +\mathstrut 14196 q^{16} \) \(\mathstrut +\mathstrut 14168 q^{18} \) \(\mathstrut +\mathstrut 22120 q^{20} \) \(\mathstrut +\mathstrut O(q^{21}) \)

Gram Matrix

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

Genus Structure

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