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Properties

Name A6
Label 6.7.7.1.1
Class number $1$
Dimension $6$
Determinant $7$
Level $7$

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Lattice Invariants

Dimension:$6$
Determinant:$7$
Level:$7$
Density:$0.244151479696829436971332601919\dots$
Group order:$10080$
Hermite number:$1.44604005279896755250968511855\dots$
Minimal vector length:$2$
Kissing number:$42$
Normalized minimal vectors: $(1, 0, 0, 0, 0, 0)$, $(1, 1, 0, 0, 0, 0)$, $(1, 1, 1, 0, 0, 0)$, $(1, 1, 1, 1, 0, 0)$, $(1, 1, 1, 1, 1, 0)$, $(1, 1, 1, 1, 1, 1)$, $(0, 1, 0, 0, 0, 0)$, $(0, 1, 1, 0, 0, 0)$, $(0, 1, 1, 1, 0, 0)$, $(0, 1, 1, 1, 1, 0)$, $(0, 1, 1, 1, 1, 1)$, $(0, 0, 1, 0, 0, 0)$, $(0, 0, 1, 1, 0, 0)$, $(0, 0, 1, 1, 1, 0)$, $(0, 0, 1, 1, 1, 1)$, $(0, 0, 0, 1, 0, 0)$, $(0, 0, 0, 1, 1, 0)$, $(0, 0, 0, 1, 1, 1)$ ...
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Theta Series

\(1 \) \(\mathstrut +\mathstrut 42 q^{2} \) \(\mathstrut +\mathstrut 210 q^{4} \) \(\mathstrut +\mathstrut 350 q^{6} \) \(\mathstrut +\mathstrut 882 q^{8} \) \(\mathstrut +\mathstrut 1050 q^{10} \) \(\mathstrut +\mathstrut 1750 q^{12} \) \(\mathstrut +\mathstrut 2100 q^{14} \) \(\mathstrut +\mathstrut 3570 q^{16} \) \(\mathstrut +\mathstrut 3066 q^{18} \) \(\mathstrut +\mathstrut 5250 q^{20} \) \(\mathstrut +\mathstrut O(q^{21}) \)

Gram Matrix

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

Genus Structure

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

This integral lattice is the A6 lattice.

This is a root lattice.