Properties

Name A14
Label 14.15.15.9.1
Class number 9
Dimension 14
Determinant 15
Level 15

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Lattice Invariants

Dimension:$14$
Determinant:$15$
Level:$15$
Label:$14.15.15.9.1$
Density:$0.00120882371980846561842974135302\dots$
Group order:$2615348736000$
Hermite number:$1.64825149055779494991198141680\dots$
Minimal vector length:$2$
Kissing Number:$210$
Normalized minimal vectors: $(1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0)$, $(1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0)$, $(1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0)$, $(1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0)$, $(1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0)$, $(1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0)$, $(1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0)$, $(1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0)$ ...
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Theta Series

\(1 \) \(\mathstrut +\mathstrut 210q \) \(\mathstrut +\mathstrut 8190q^{2} \) \(\mathstrut +\mathstrut 102830q^{3} \) \(\mathstrut +\mathstrut 570780q^{4} \) \(\mathstrut +\mathstrut 2140866q^{5} \) \(\mathstrut +\mathstrut 6527430q^{6} \) \(\mathstrut +\mathstrut 16213080q^{7} \) \(\mathstrut +\mathstrut 36584730q^{8} \) \(\mathstrut +\mathstrut 73035900q^{9} \) \(\mathstrut +\mathstrut 139912500q^{10} \) \(\mathstrut +\mathstrut 243455940q^{11} \) \(\mathstrut +\mathstrut 417269580q^{12} \) \(\mathstrut +\mathstrut 664812330q^{13} \) \(\mathstrut +\mathstrut 1050179130q^{14} \) \(\mathstrut +\mathstrut 1567036926q^{15} \) \(\mathstrut +\mathstrut 2347625490q^{16} \) \(\mathstrut +\mathstrut 3315827970q^{17} \) \(\mathstrut +\mathstrut 4752128290q^{18} \) \(\mathstrut +\mathstrut 6479581290q^{19} \) \(\mathstrut +\mathstrut 8930870676q^{20} \) \(\mathstrut +\mathstrut O(q^{21}) \)

Gram Matrix

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

Genus Structure

Class number:$9$
 
Genus representatives: $\left(\begin{array}{rrrrrrrrrrrrrr} 2 & 0 & -1 & -1 & -1 & -1 & 0 & -1 & -1 & -1 & 1 & 1 & -1 & -1 \\ 0 & 2 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & -1 & 0 & 0 \\ -1 & 0 & 2 & 1 & 1 & 1 & 0 & 1 & 1 & 1 & -1 & -1 & 1 & 1 \\ -1 & 0 & 1 & 2 & 1 & 1 & 0 & 1 & 1 & 1 & -1 & -1 & 1 & 1 \\ -1 & 0 & 1 & 1 & 2 & 1 & 0 & 1 & 1 & 1 & -1 & -1 & 1 & 1 \\ -1 & 0 & 1 & 1 & 1 & 2 & 0 & 1 & 1 & 1 & -1 & -1 & 1 & 1 \\ 0 & 1 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & -1 & 0 & 0 \\ -1 & 0 & 1 & 1 & 1 & 1 & 0 & 2 & 1 & 1 & -1 & -1 & 1 & 1 \\ -1 & 0 & 1 & 1 & 1 & 1 & 0 & 1 & 2 & 1 & -1 & -1 & 1 & 1 \\ -1 & 0 & 1 & 1 & 1 & 1 & 0 & 1 & 1 & 2 & -1 & -1 & 1 & 1 \\ 1 & 0 & -1 & -1 & -1 & -1 & 0 & -1 & -1 & -1 & 2 & 1 & -1 & -1 \\ 1 & -1 & -1 & -1 & -1 & -1 & -1 & -1 & -1 & -1 & 1 & 2 & -1 & -1 \\ -1 & 0 & 1 & 1 & 1 & 1 & 0 & 1 & 1 & 1 & -1 & -1 & 2 & 1 \\ -1 & 0 & 1 & 1 & 1 & 1 & 0 & 1 & 1 & 1 & -1 & -1 & 1 & 2 \end{array}\right)$, $\left(\begin{array}{rrrrrrrrrrrrrr} 2 & 1 & 1 & 0 & 0 & 1 & -1 & 0 & 0 & 0 & 0 & 0 & -1 & 1 \\ 1 & 2 & 0 & 0 & 0 & 1 & -1 & 0 & 0 & 0 & 0 & 0 & -1 & 1 \\ 1 & 0 & 2 & -1 & 0 & 1 & -1 & 0 & 0 & 0 & 0 & 0 & -1 & 1 \\ 0 & 0 & -1 & 2 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 1 & -1 \\ 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & -1 & -1 & 1 & 0 & -1 \\ 1 & 1 & 1 & 0 & 0 & 2 & -1 & 0 & 0 & 0 & 0 & 0 & -1 & 1 \\ -1 & -1 & -1 & 1 & 0 & -1 & 2 & 0 & 0 & 0 & 0 & 0 & 1 & -1 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 1 & -1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 2 & -1 & 0 & 0 & 0 & -1 \\ 0 & 0 & 0 & 0 & -1 & 0 & 0 & -1 & -1 & 2 & 1 & -1 & 0 & 1 \\ 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 1 & 2 & -1 & 0 & 1 \\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & -1 & -1 & 2 & 0 & -1 \\ -1 & -1 & -1 & 1 & 0 & -1 & 1 & 0 & 0 & 0 & 0 & 0 & 2 & -1 \\ 1 & 1 & 1 & -1 & -1 & 1 & -1 & 0 & -1 & 1 & 1 & -1 & -1 & 4 \end{array}\right)$ ...
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Comments

This integral lattice is the A14 lattice.

This is a root lattice.