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Properties

Name A10
Label 10.11.11.3.1
Class number $3$
Dimension $10$
Determinant $11$
Level $11$

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Lattice Invariants

Dimension:$10$
Determinant:$11$
Level:$11$
Density:$0.0240282308924150010610161580966\dots$
Group order:$79833600$
Hermite number:$1.57358688439354447091057738970\dots$
Minimal vector length:$2$
Kissing number:$110$
Normalized minimal vectors: $(1, 0, 0, 0, 0, 0, 0, 0, 0, 0)$, $(1, 1, 0, 0, 0, 0, 0, 0, 0, 0)$, $(1, 1, 1, 0, 0, 0, 0, 0, 0, 0)$, $(1, 1, 1, 1, 0, 0, 0, 0, 0, 0)$, $(1, 1, 1, 1, 1, 0, 0, 0, 0, 0)$, $(1, 1, 1, 1, 1, 1, 0, 0, 0, 0)$, $(1, 1, 1, 1, 1, 1, 1, 0, 0, 0)$, $(1, 1, 1, 1, 1, 1, 1, 1, 0, 0)$, $(1, 1, 1, 1, 1, 1, 1, 1, 1, 0)$, $(1, 1, 1, 1, 1, 1, 1, 1, 1, 1)$, $(0, 1, 0, 0, 0, 0, 0, 0, 0, 0)$ ...
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Theta Series

\(1 \) \(\mathstrut +\mathstrut 110 q^{2} \) \(\mathstrut +\mathstrut 1980 q^{4} \) \(\mathstrut +\mathstrut 10230 q^{6} \) \(\mathstrut +\mathstrut 30140 q^{8} \) \(\mathstrut +\mathstrut 79992 q^{10} \) \(\mathstrut +\mathstrut 155100 q^{12} \) \(\mathstrut +\mathstrut 302280 q^{14} \) \(\mathstrut +\mathstrut 487080 q^{16} \) \(\mathstrut +\mathstrut 839630 q^{18} \) \(\mathstrut +\mathstrut 1188660 q^{20} \) \(\mathstrut +\mathstrut O(q^{21}) \)

Gram Matrix

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

Genus Structure

Genus representatives:
Class number:$3$
 
$\left(\begin{array}{rrrrrrrrrr} 2 & 1 & 1 & 1 & 1 & 1 & -1 & -1 & 1 & 1 \\ 1 & 2 & 1 & 1 & 1 & 1 & -1 & -1 & 1 & 1 \\ 1 & 1 & 2 & 1 & 1 & 1 & -1 & -1 & 1 & 1 \\ 1 & 1 & 1 & 2 & 1 & 1 & -1 & -1 & 1 & 1 \\ 1 & 1 & 1 & 1 & 2 & 1 & -1 & -1 & 1 & 1 \\ 1 & 1 & 1 & 1 & 1 & 2 & -1 & -1 & 1 & 1 \\ -1 & -1 & -1 & -1 & -1 & -1 & 2 & 1 & -1 & -1 \\ -1 & -1 & -1 & -1 & -1 & -1 & 1 & 2 & -1 & -1 \\ 1 & 1 & 1 & 1 & 1 & 1 & -1 & -1 & 2 & 1 \\ 1 & 1 & 1 & 1 & 1 & 1 & -1 & -1 & 1 & 2 \end{array}\right)$, $\left(\begin{array}{rrrrrrrrrr} 2 & 0 & 1 & 0 & 0 & 0 & 1 & 1 & 1 & 1 \\ 0 & 2 & 1 & 0 & 0 & 1 & 0 & 0 & 1 & 0 \\ 1 & 1 & 2 & 0 & 0 & 1 & 1 & 1 & 1 & 1 \\ 0 & 0 & 0 & 2 & -1 & 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & -1 & 2 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 1 & 0 & 0 & 2 & 1 & 1 & 1 & 1 \\ 1 & 0 & 1 & 0 & 0 & 1 & 2 & 1 & 1 & 1 \\ 1 & 0 & 1 & 0 & 0 & 1 & 1 & 2 & 1 & 1 \\ 1 & 1 & 1 & 0 & 0 & 1 & 1 & 1 & 2 & 1 \\ 1 & 0 & 1 & 1 & 0 & 1 & 1 & 1 & 1 & 4 \end{array}\right)$, $\left(\begin{array}{rrrrrrrrrr} 2 & -1 & 0 & 1 & -1 & 1 & -1 & -1 & 0 & 0 \\ -1 & 2 & -1 & -1 & 1 & -1 & 1 & 1 & 0 & 0 \\ 0 & -1 & 2 & 1 & 0 & 0 & -1 & 0 & 0 & 0 \\ 1 & -1 & 1 & 2 & 0 & 0 & -1 & -1 & 0 & 0 \\ -1 & 1 & 0 & 0 & 2 & -1 & 1 & 1 & 0 & 0 \\ 1 & -1 & 0 & 0 & -1 & 2 & -1 & -1 & 0 & 0 \\ -1 & 1 & -1 & -1 & 1 & -1 & 2 & 1 & 0 & 0 \\ -1 & 1 & 0 & -1 & 1 & -1 & 1 & 2 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 1 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 6 \end{array}\right)$
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Comments

This integral lattice is the A10 lattice.

This is a root lattice.