# Properties

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

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: $(1, 1, 0, 0, 0, 0, 0, -1)$, $(1, 1, 0, 0, 0, 0, 0, 0)$, $(1, 1, 1, 0, 0, 0, 0, -1)$, $(1, 1, 1, 0, 0, 0, 0, 0)$, $(1, 1, 1, 1, 0, 0, 0, -1)$, $(1, 1, 1, 1, 0, 0, 0, 0)$, $(1, 1, 1, 1, 1, 0, 0, -1)$, $(1, 1, 1, 1, 1, 0, 0, 0)$, $(1, 1, 1, 1, 1, 1, 0, -1)$, $(1, 1, 1, 1, 1, 1, 0, 0)$, $(1, 1, 1, 1, 1, 1, 1, -1)$, $(1, 1, 1, 1, 1, 1, 1, 0)$, $(1, 2, 1, 0, 0, 0, 0, -1)$ ...

## 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

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.