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Properties

Label 3.498.996.12.1
Class number $12$
Dimension $3$
Determinant $498$
Level $996$

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Lattice Invariants

Dimension:$3$
Determinant:$498$
Level:$996$
Density:$0.0663634483283132495870930953503\dots$
Group order:$24$
Hermite number:$0.252321087666175453759585413518\dots$
Minimal vector length:$2$
Kissing number:$6$
Normalized minimal vectors: $(1, -1, 0)$, $(1, 0, 0)$, $(0, 1, 0)$
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Theta Series

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

Gram Matrix

$\left(\begin{array}{rrr} 2 & 1 & 0 \\ 1 & 2 & 0 \\ 0 & 0 & 166 \end{array}\right)$

Genus Structure

Genus representatives:
Class number:$12$
 
$\left(\begin{array}{rrr} 2 & 1 & 0 \\ 1 & 2 & 0 \\ 0 & 0 & 166 \end{array}\right)$, $\left(\begin{array}{rrr} 2 & 1 & 0 \\ 1 & 12 & -2 \\ 0 & -2 & 22 \end{array}\right)$, $\left(\begin{array}{rrr} 6 & 1 & -3 \\ 1 & 10 & 4 \\ -3 & 4 & 12 \end{array}\right)$, $\left(\begin{array}{rrr} 6 & -2 & 2 \\ -2 & 8 & -1 \\ 2 & -1 & 12 \end{array}\right)$, $\left(\begin{array}{rrr} 2 & 1 & 0 \\ 1 & 6 & -2 \\ 0 & -2 & 46 \end{array}\right)$, $\left(\begin{array}{rrr} 4 & -1 & -1 \\ -1 & 6 & 1 \\ -1 & 1 & 22 \end{array}\right)$, $\left(\begin{array}{rrr} 6 & 0 & 2 \\ 0 & 6 & -3 \\ 2 & -3 & 16 \end{array}\right)$, $\left(\begin{array}{rrr} 6 & 0 & 0 \\ 0 & 6 & -1 \\ 0 & -1 & 14 \end{array}\right)$, $\left(\begin{array}{rrr} 6 & 3 & -2 \\ 3 & 6 & 1 \\ -2 & 1 & 20 \end{array}\right)$, $\left(\begin{array}{rrr} 6 & 0 & -3 \\ 0 & 8 & -1 \\ -3 & -1 & 12 \end{array}\right)$, $\left(\begin{array}{rrr} 2 & 0 & 1 \\ 0 & 6 & 0 \\ 1 & 0 & 42 \end{array}\right)$, $\left(\begin{array}{rrr} 6 & -3 & -3 \\ -3 & 10 & -1 \\ -3 & -1 & 12 \end{array}\right)$
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Comments

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