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Properties

Label 6.12.6.1.3
Class number $1$
Dimension $6$
Determinant $12$
Level $6$

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Lattice Invariants

Dimension:$6$
Determinant:$12$
Level:$6$
Density:$0.0233092215988790587227173474175\dots$
Group order:$768$
Hermite number:$0.660901076083364714113932338225\dots$
Minimal vector length:$1$
Kissing number:$4$
Normalized minimal vectors: $(1, 0, 0, 0, 0, 0)$, $(0, 0, 0, 0, 0, 1)$
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Theta Series

\(1 \) \(\mathstrut +\mathstrut 4 q \) \(\mathstrut +\mathstrut 10 q^{2} \) \(\mathstrut +\mathstrut 40 q^{3} \) \(\mathstrut +\mathstrut 104 q^{4} \) \(\mathstrut +\mathstrut 120 q^{5} \) \(\mathstrut +\mathstrut 82 q^{6} \) \(\mathstrut +\mathstrut 200 q^{7} \) \(\mathstrut +\mathstrut 430 q^{8} \) \(\mathstrut +\mathstrut 364 q^{9} \) \(\mathstrut +\mathstrut 192 q^{10} \) \(\mathstrut +\mathstrut 600 q^{11} \) \(\mathstrut +\mathstrut 1040 q^{12} \) \(\mathstrut +\mathstrut 680 q^{13} \) \(\mathstrut +\mathstrut 500 q^{14} \) \(\mathstrut +\mathstrut 984 q^{15} \) \(\mathstrut +\mathstrut 1448 q^{16} \) \(\mathstrut +\mathstrut 1440 q^{17} \) \(\mathstrut +\mathstrut 730 q^{18} \) \(\mathstrut +\mathstrut 1448 q^{19} \) \(\mathstrut +\mathstrut 3120 q^{20} \) \(\mathstrut +\mathstrut O(q^{21}) \)

Gram Matrix

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

Genus Structure

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