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: |
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Theta Series
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: | $1$ |
$\left(\begin{array}{rrrrrrrr} 2 & 1 & 1 & -1 & -1 & 0 & -1 & -1 \\ 1 & 2 & 1 & 0 & -1 & -1 & -1 & -1 \\ 1 & 1 & 2 & -1 & -1 & -1 & -1 & -1 \\ -1 & 0 & -1 & 2 & 1 & 0 & 0 & 0 \\ -1 & -1 & -1 & 1 & 2 & 0 & 0 & 0 \\ 0 & -1 & -1 & 0 & 0 & 2 & 1 & 1 \\ -1 & -1 & -1 & 0 & 0 & 1 & 2 & 1 \\ -1 & -1 & -1 & 0 & 0 & 1 & 1 & 2 \end{array}\right)$ | |
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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.