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Properties

Label 3.172.344.5.4
Class number $5$
Dimension $3$
Determinant $172$
Level $344$

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Lattice Invariants

Dimension:$3$
Determinant:$172$
Level:$344$
Density:$0.319392258826571474509662595280\dots$
Group order:$4$
Hermite number:$0.719256577858467052414592468450\dots$
Minimal vector length:$4$
Kissing number:$4$
Normalized minimal vectors: $(1, 0, 0)$, $(0, 1, 0)$
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Theta Series

\(1 \) \(\mathstrut +\mathstrut 4 q^{4} \) \(\mathstrut +\mathstrut 4 q^{8} \) \(\mathstrut +\mathstrut 4 q^{12} \) \(\mathstrut +\mathstrut 4 q^{14} \) \(\mathstrut +\mathstrut 4 q^{16} \) \(\mathstrut +\mathstrut 4 q^{18} \) \(\mathstrut +\mathstrut 12 q^{20} \) \(\mathstrut +\mathstrut O(q^{21}) \)

Gram Matrix

$\left(\begin{array}{rrr} 4 & 0 & 2 \\ 0 & 4 & 1 \\ 2 & 1 & 12 \end{array}\right)$

Genus Structure

Genus representatives:
Class number:$5$
 
$\left(\begin{array}{rrr} 4 & 0 & 2 \\ 0 & 4 & 1 \\ 2 & 1 & 12 \end{array}\right)$, $\left(\begin{array}{rrr} 4 & -1 & -1 \\ -1 & 6 & -1 \\ -1 & -1 & 8 \end{array}\right)$, $\left(\begin{array}{rrr} 2 & 1 & 0 \\ 1 & 8 & -2 \\ 0 & -2 & 12 \end{array}\right)$, $\left(\begin{array}{rrr} 2 & -1 & -1 \\ -1 & 8 & 1 \\ -1 & 1 & 12 \end{array}\right)$, $\left(\begin{array}{rrr} 2 & 0 & -1 \\ 0 & 2 & -1 \\ -1 & -1 & 44 \end{array}\right)$
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Comments

This lattice appears in the Brandt-Intrau-Schiemann Table of Even Ternary Quadratic Forms.