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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.

Lattice Invariants

Dimension:$8$
Determinant:$1$
Level:$1$
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

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

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

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

This integral lattice is the E8 lattice.

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

Additional information

The $E_8$ lattice is the unique unimodular integral lattice of smallest positive dimension. It is the unique solution of the sphere packing problem in dimension 8, by a theorem of Viazovska [10.4007/annals.2017.185.3.7, MR:3664816], and of the general kissing number problem in dimension 8.