Properties

Name E8
Label 8.1.1.1.1
Class number 1
Dimension 8
Determinant 1
Level 1

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The $E_8$ lattice is the root lattice associated to the $E_8$ root system. It is the unique unimodular integral lattice of smallest positive dimension. It is the unique solution of the sphere packing problem (a theorem of Viazovska) and the general kissing number problem in dimension 8. See Wikipedia for additional information.

Lattice Invariants

Dimension:$8$
Determinant:$1$
Level:$1$
Label:$8.1.1.1.1$
Density:$0.253669507901048013636563366376\dots$
Group order:$696729600$
Hermite number:$2.00000000000000000000000000000\dots$
Minimal vector length:$2$
Kissing number:$240$
Normalized minimal vectors: $(1, 1, 0, 0, 0, 0, 0, -1)$, $(1, 1, 0, 0, 0, 0, 0, 0)$, $(1, 1, 1, 0, 0, 0, 0, -1)$, $(1, 1, 1, 0, 0, 0, 0, 0)$, $(1, 1, 1, 1, 0, 0, 0, -1)$, $(1, 1, 1, 1, 0, 0, 0, 0)$, $(1, 1, 1, 1, 1, 0, 0, -1)$, $(1, 1, 1, 1, 1, 0, 0, 0)$, $(1, 1, 1, 1, 1, 1, 0, -1)$, $(1, 1, 1, 1, 1, 1, 0, 0)$, $(1, 1, 1, 1, 1, 1, 1, -1)$, $(1, 1, 1, 1, 1, 1, 1, 0)$, $(1, 2, 1, 0, 0, 0, 0, -1)$ ...
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Theta Series

\(1 \) \(\mathstrut +\mathstrut 240q^{2} \) \(\mathstrut +\mathstrut 2160q^{4} \) \(\mathstrut +\mathstrut 6720q^{6} \) \(\mathstrut +\mathstrut 17520q^{8} \) \(\mathstrut +\mathstrut 30240q^{10} \) \(\mathstrut +\mathstrut 60480q^{12} \) \(\mathstrut +\mathstrut 82560q^{14} \) \(\mathstrut +\mathstrut 140400q^{16} \) \(\mathstrut +\mathstrut 181680q^{18} \) \(\mathstrut +\mathstrut 272160q^{20} \) \(\mathstrut +\mathstrut O(q^{21}) \)

Gram Matrix

$\left(\begin{array}{rrrrrrrr} 4 & -2 & 0 & 0 & 0 & 0 & 0 & 1 \\ -2 & 2 & -1 & 0 & 0 & 0 & 0 & 0 \\ 0 & -1 & 2 & -1 & 0 & 0 & 0 & 0 \\ 0 & 0 & -1 & 2 & -1 & 0 & 0 & 0 \\ 0 & 0 & 0 & -1 & 2 & -1 & 0 & 0 \\ 0 & 0 & 0 & 0 & -1 & 2 & -1 & 0 \\ 0 & 0 & 0 & 0 & 0 & -1 & 2 & 0 \\ 1 & 0 & 0 & 0 & 0 & 0 & 0 & 2 \end{array}\right)$

Genus Structure

Class number:$1$
 
Genus representatives: $\left(\begin{array}{rrrrrrrr} 2 & 1 & 1 & -1 & -1 & 0 & -1 & -1 \\ 1 & 2 & 1 & 0 & -1 & -1 & -1 & -1 \\ 1 & 1 & 2 & -1 & -1 & -1 & -1 & -1 \\ -1 & 0 & -1 & 2 & 1 & 0 & 0 & 0 \\ -1 & -1 & -1 & 1 & 2 & 0 & 0 & 0 \\ 0 & -1 & -1 & 0 & 0 & 2 & 1 & 1 \\ -1 & -1 & -1 & 0 & 0 & 1 & 2 & 1 \\ -1 & -1 & -1 & 0 & 0 & 1 & 1 & 2 \end{array}\right)$
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Comments

This integral lattice is the E8 lattice.

This is the unique positive definite, even, unimodular lattice of rank 8.