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Properties

Label 6.27.6.1.1
Class number $1$
Dimension $6$
Determinant $27$
Level $6$

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Lattice Invariants

Dimension:$6$
Determinant:$27$
Level:$6$
Density:$0.0155394810659193724818115649450\dots$
Group order:$768$
Hermite number:$0.577350269189625764509148780502\dots$
Minimal vector length:$1$
Kissing number:$4$
Normalized minimal vectors: $(1, 0, 0, 0, 0, 0)$, $(0, 1, 0, 0, 0, 0)$
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Theta Series

\(1 \) \(\mathstrut +\mathstrut 4 q \) \(\mathstrut +\mathstrut 10 q^{2} \) \(\mathstrut +\mathstrut 28 q^{3} \) \(\mathstrut +\mathstrut 44 q^{4} \) \(\mathstrut +\mathstrut 48 q^{5} \) \(\mathstrut +\mathstrut 130 q^{6} \) \(\mathstrut +\mathstrut 200 q^{7} \) \(\mathstrut +\mathstrut 106 q^{8} \) \(\mathstrut +\mathstrut 244 q^{9} \) \(\mathstrut +\mathstrut 480 q^{10} \) \(\mathstrut +\mathstrut 240 q^{11} \) \(\mathstrut +\mathstrut 308 q^{12} \) \(\mathstrut +\mathstrut 680 q^{13} \) \(\mathstrut +\mathstrut 500 q^{14} \) \(\mathstrut +\mathstrut 624 q^{15} \) \(\mathstrut +\mathstrut 812 q^{16} \) \(\mathstrut +\mathstrut 576 q^{17} \) \(\mathstrut +\mathstrut 1210 q^{18} \) \(\mathstrut +\mathstrut 1448 q^{19} \) \(\mathstrut +\mathstrut 528 q^{20} \) \(\mathstrut +\mathstrut O(q^{21}) \)

Gram Matrix

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

Genus Structure

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