↑

Properties

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

Learn more about

The $E_8$ lattice is the root lattice associated to the $E_8$ root system.

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: $(1, 1, 0, 0, 0, 0, 0, -1)$, $(1, 1, 0, 0, 0, 0, 0, 0)$, $(1, 1, 1, 0, 0, 0, 0, -1)$, $(1, 1, 1, 0, 0, 0, 0, 0)$, $(1, 1, 1, 1, 0, 0, 0, -1)$, $(1, 1, 1, 1, 0, 0, 0, 0)$, $(1, 1, 1, 1, 1, 0, 0, -1)$, $(1, 1, 1, 1, 1, 0, 0, 0)$, $(1, 1, 1, 1, 1, 1, 0, -1)$, $(1, 1, 1, 1, 1, 1, 0, 0)$, $(1, 1, 1, 1, 1, 1, 1, -1)$, $(1, 1, 1, 1, 1, 1, 1, 0)$, $(1, 2, 1, 0, 0, 0, 0, -1)$ ...
Download the complete list for gp, magma, sage

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: $\left(\begin{array}{rrrrrrrr} 2 & 1 & 1 & -1 & -1 & 0 & -1 & -1 \\ 1 & 2 & 1 & 0 & -1 & -1 & -1 & -1 \\ 1 & 1 & 2 & -1 & -1 & -1 & -1 & -1 \\ -1 & 0 & -1 & 2 & 1 & 0 & 0 & 0 \\ -1 & -1 & -1 & 1 & 2 & 0 & 0 & 0 \\ 0 & -1 & -1 & 0 & 0 & 2 & 1 & 1 \\ -1 & -1 & -1 & 0 & 0 & 1 & 2 & 1 \\ -1 & -1 & -1 & 0 & 0 & 1 & 1 & 2 \end{array}\right)$
Download this matrix for gp, magma, sage

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.