Properties

Label 3.412.824.12.12
Class number $12$
Dimension $3$
Determinant $412$
Level $824$

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Lattice Invariants

Dimension:$3$
Determinant:$412$
Level:$824$
Density:$0.583693685123204920936501247995\dots$
Group order:$2$
Hermite number:$1.07512158613286143022520345691\dots$
Minimal vector length:$8$
Kissing number:$6$
Normalized minimal vectors: $(1, 0, 0)$, $(0, 1, 0)$, $(0, 0, 1)$
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Theta Series

\(1 \) \(\mathstrut +\mathstrut 6 q^{8} \) \(\mathstrut +\mathstrut 2 q^{10} \) \(\mathstrut +\mathstrut 2 q^{12} \) \(\mathstrut +\mathstrut 2 q^{14} \) \(\mathstrut +\mathstrut 2 q^{16} \) \(\mathstrut +\mathstrut 2 q^{18} \) \(\mathstrut +\mathstrut 4 q^{20} \) \(\mathstrut +\mathstrut O(q^{21}) \)

Gram Matrix

$\left(\begin{array}{rrr} 8 & 3 & 2 \\ 3 & 8 & 1 \\ 2 & 1 & 8 \end{array}\right)$

Genus Structure

Genus representatives:
Class number:$12$
 
$\left(\begin{array}{rrr} 8 & 3 & 2 \\ 3 & 8 & 1 \\ 2 & 1 & 8 \end{array}\right)$, $\left(\begin{array}{rrr} 2 & -1 & -1 \\ -1 & 4 & 2 \\ -1 & 2 & 60 \end{array}\right)$, $\left(\begin{array}{rrr} 4 & 0 & 1 \\ 0 & 4 & 0 \\ 1 & 0 & 26 \end{array}\right)$, $\left(\begin{array}{rrr} 4 & 0 & -2 \\ 0 & 8 & 1 \\ -2 & 1 & 14 \end{array}\right)$, $\left(\begin{array}{rrr} 2 & 1 & 0 \\ 1 & 8 & 2 \\ 0 & 2 & 28 \end{array}\right)$, $\left(\begin{array}{rrr} 2 & -1 & 0 \\ -1 & 12 & 1 \\ 0 & 1 & 18 \end{array}\right)$, $\left(\begin{array}{rrr} 4 & -1 & 2 \\ -1 & 6 & 2 \\ 2 & 2 & 20 \end{array}\right)$, $\left(\begin{array}{rrr} 4 & 0 & 0 \\ 0 & 8 & -3 \\ 0 & -3 & 14 \end{array}\right)$, $\left(\begin{array}{rrr} 2 & 1 & 1 \\ 1 & 8 & 1 \\ 1 & 1 & 28 \end{array}\right)$, $\left(\begin{array}{rrr} 2 & 0 & 1 \\ 0 & 2 & -1 \\ 1 & -1 & 104 \end{array}\right)$, $\left(\begin{array}{rrr} 4 & -2 & 0 \\ -2 & 8 & 3 \\ 0 & 3 & 16 \end{array}\right)$, $\left(\begin{array}{rrr} 2 & 0 & 1 \\ 0 & 4 & 0 \\ 1 & 0 & 52 \end{array}\right)$
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Comments

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