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Properties

Label 7.24.48.1.1
Class number $1$
Dimension $7$
Determinant $24$
Level $48$

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Lattice Invariants

Dimension:$7$
Determinant:$24$
Level:$48$
Density:$0.0852451532759005576145091562695\dots$
Group order:$5760$
Hermite number:$1.27015569889384018743399056159\dots$
Minimal vector length:$2$
Kissing number:$32$
Normalized minimal vectors: $(1, 0, 0, 0, 0, 0, 0)$, $(1, 0, 0, 0, 1, 0, 0)$, $(1, 0, 0, 1, 0, 0, 0)$, $(1, 0, 1, 0, 0, 0, 0)$, $(1, 1, 0, 0, 0, 0, 0)$, $(0, 1, -1, 0, 0, 0, 0)$, $(0, 1, 0, -1, 0, 0, 0)$, $(0, 1, 0, 0, -1, 0, 0)$, $(0, 1, 0, 0, 0, 0, 0)$, $(0, 0, 1, -1, 0, 0, 0)$, $(0, 0, 1, 0, -1, 0, 0)$, $(0, 0, 1, 0, 0, 0, 0)$, $(0, 0, 0, 1, -1, 0, 0)$, $(0, 0, 0, 1, 0, 0, 0)$, $(0, 0, 0, 0, 1, 0, 0)$ ...
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Theta Series

\(1 \) \(\mathstrut +\mathstrut 32 q^{2} \) \(\mathstrut +\mathstrut 230 q^{4} \) \(\mathstrut +\mathstrut 560 q^{6} \) \(\mathstrut +\mathstrut 1354 q^{8} \) \(\mathstrut +\mathstrut 1792 q^{10} \) \(\mathstrut +\mathstrut 3930 q^{12} \) \(\mathstrut +\mathstrut 4208 q^{14} \) \(\mathstrut +\mathstrut 7774 q^{16} \) \(\mathstrut +\mathstrut 8672 q^{18} \) \(\mathstrut +\mathstrut 12700 q^{20} \) \(\mathstrut +\mathstrut O(q^{21}) \)

Gram Matrix

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

Genus Structure

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