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Properties

Label 10.19683.3.1.1
Class number $1$
Dimension $10$
Determinant $19683$
Level $3$

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Lattice Invariants

Dimension:$10$
Determinant:$19683$
Level:$3$
Density:$0.000568031595254972578974448122037\dots$
Group order:$8360755200$
Hermite number:$0.744082116022602956295076092252\dots$
Minimal vector length:$2$
Kissing number:$6$
Normalized minimal vectors: $(0, 0, 0, 0, 0, 0, 0, 0, 1, 0)$, $(0, 0, 0, 0, 0, 0, 0, 0, 1, 1)$, $(0, 0, 0, 0, 0, 0, 0, 0, 0, 1)$
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Theta Series

\(1 \) \(\mathstrut +\mathstrut 6 q^{2} \) \(\mathstrut +\mathstrut 246 q^{6} \) \(\mathstrut +\mathstrut 1446 q^{8} \) \(\mathstrut +\mathstrut 3600 q^{12} \) \(\mathstrut +\mathstrut 14412 q^{14} \) \(\mathstrut +\mathstrut 19686 q^{18} \) \(\mathstrut +\mathstrut 56160 q^{20} \) \(\mathstrut +\mathstrut O(q^{21}) \)

Gram Matrix

$\left(\begin{array}{rrrrrrrrrr} 6 & 3 & 3 & 3 & 3 & 3 & 3 & 3 & 0 & 0 \\ 3 & 6 & 3 & 0 & 3 & 3 & 0 & 3 & 0 & 0 \\ 3 & 3 & 6 & 0 & 0 & 0 & 3 & 0 & 0 & 0 \\ 3 & 0 & 0 & 6 & 0 & 0 & 0 & 3 & 0 & 0 \\ 3 & 3 & 0 & 0 & 6 & 3 & 0 & 3 & 0 & 0 \\ 3 & 3 & 0 & 0 & 3 & 6 & 0 & 3 & 0 & 0 \\ 3 & 0 & 3 & 0 & 0 & 0 & 6 & 0 & 0 & 0 \\ 3 & 3 & 0 & 3 & 3 & 3 & 0 & 6 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & -1 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 2 \end{array}\right)$

Genus Structure

Genus representatives:
Class number:$1$
 
$\left(\begin{array}{rrrrrrrrrr} 2 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ -1 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 6 & 3 & 0 & -3 & -3 & 3 & 0 & 0 \\ 0 & 0 & 3 & 6 & 3 & -3 & -3 & 3 & -3 & 3 \\ 0 & 0 & 0 & 3 & 6 & -3 & -3 & 0 & -3 & 3 \\ 0 & 0 & -3 & -3 & -3 & 6 & 3 & -3 & 3 & -3 \\ 0 & 0 & -3 & -3 & -3 & 3 & 6 & 0 & 0 & 0 \\ 0 & 0 & 3 & 3 & 0 & -3 & 0 & 6 & -3 & 3 \\ 0 & 0 & 0 & -3 & -3 & 3 & 0 & -3 & 6 & -3 \\ 0 & 0 & 0 & 3 & 3 & -3 & 0 & 3 & -3 & 6 \end{array}\right)$
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