This unimodular integral lattice is the Leech lattice.
Lattice Invariants
Dimension: | $24$ |
Determinant: | $1$ |
Level: | $1$ |
Density: | $0.00192957430940392304790334556369\dots$ |
Group order: | $8315553613086720000$ |
Hermite number: | $4.00000000000000000000000000000\dots$ |
Minimal vector length: | $4$ |
Kissing number: | $196560$ |
Normalized minimal vectors: |
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Theta Series
Gram Matrix
$\left(\begin{array}{rrrrrrrrrrrrrrrrrrrrrrrr} 8 & 4 & 4 & 4 & 4 & 4 & 4 & 2 & 4 & 4 & 4 & 2 & 4 & 2 & 2 & 2 & 4 & 2 & 2 & 2 & 0 & 0 & 0 & -3 \\ 4 & 4 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 1 & 1 & 2 & 1 & 1 & 2 & 1 & 0 & 0 & -1 \\ 4 & 2 & 4 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 1 & 2 & 1 & 2 & 2 & 1 & 1 & 1 & 0 & 0 & -1 \\ 4 & 2 & 2 & 4 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 1 & 1 & 2 & 2 & 1 & 2 & 1 & 1 & 0 & 0 & -1 \\ 4 & 2 & 2 & 2 & 4 & 2 & 2 & 2 & 2 & 2 & 2 & 1 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 1 & 0 & 0 & -1 \\ 4 & 2 & 2 & 2 & 2 & 4 & 2 & 2 & 2 & 2 & 2 & 1 & 2 & 2 & 1 & 1 & 2 & 1 & 2 & 1 & 0 & 0 & 0 & -1 \\ 4 & 2 & 2 & 2 & 2 & 2 & 4 & 2 & 2 & 2 & 2 & 1 & 2 & 1 & 2 & 1 & 2 & 1 & 1 & 2 & 0 & 0 & 0 & -1 \\ 2 & 2 & 2 & 2 & 2 & 2 & 2 & 4 & 1 & 1 & 1 & 2 & 1 & 2 & 2 & 2 & 1 & 2 & 2 & 2 & 2 & 0 & 0 & 1 \\ 4 & 2 & 2 & 2 & 2 & 2 & 2 & 1 & 4 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 1 & 1 & 1 & -1 \\ 4 & 2 & 2 & 2 & 2 & 2 & 2 & 1 & 2 & 4 & 2 & 2 & 2 & 2 & 1 & 1 & 2 & 2 & 1 & 1 & 0 & 1 & 0 & -1 \\ 4 & 2 & 2 & 2 & 2 & 2 & 2 & 1 & 2 & 2 & 4 & 2 & 2 & 1 & 2 & 1 & 2 & 1 & 2 & 1 & 0 & 0 & 1 & -1 \\ 2 & 2 & 2 & 2 & 1 & 1 & 1 & 2 & 2 & 2 & 2 & 4 & 1 & 2 & 2 & 2 & 1 & 2 & 2 & 2 & 2 & 1 & 1 & 1 \\ 4 & 2 & 2 & 2 & 2 & 2 & 2 & 1 & 2 & 2 & 2 & 1 & 4 & 2 & 2 & 2 & 2 & 1 & 1 & 1 & 1 & 1 & 1 & -1 \\ 2 & 2 & 1 & 1 & 2 & 2 & 1 & 2 & 2 & 2 & 1 & 2 & 2 & 4 & 2 & 2 & 1 & 2 & 2 & 2 & 2 & 2 & 1 & 1 \\ 2 & 1 & 2 & 1 & 2 & 1 & 2 & 2 & 2 & 1 & 2 & 2 & 2 & 2 & 4 & 2 & 1 & 2 & 2 & 2 & 2 & 1 & 2 & 1 \\ 2 & 1 & 1 & 2 & 2 & 1 & 1 & 2 & 2 & 1 & 1 & 2 & 2 & 2 & 2 & 4 & 1 & 2 & 2 & 2 & 2 & 1 & 1 & 1 \\ 4 & 2 & 2 & 2 & 2 & 2 & 2 & 1 & 2 & 2 & 2 & 1 & 2 & 1 & 1 & 1 & 4 & 2 & 2 & 2 & 1 & 1 & 1 & -1 \\ 2 & 1 & 2 & 1 & 2 & 1 & 1 & 2 & 2 & 2 & 1 & 2 & 1 & 2 & 2 & 2 & 2 & 4 & 2 & 2 & 2 & 2 & 1 & 1 \\ 2 & 1 & 1 & 2 & 2 & 2 & 1 & 2 & 2 & 1 & 2 & 2 & 1 & 2 & 2 & 2 & 2 & 2 & 4 & 2 & 2 & 1 & 2 & 1 \\ 2 & 2 & 1 & 1 & 2 & 1 & 2 & 2 & 2 & 1 & 1 & 2 & 1 & 2 & 2 & 2 & 2 & 2 & 2 & 4 & 2 & 1 & 1 & 1 \\ 0 & 1 & 1 & 1 & 1 & 0 & 0 & 2 & 1 & 0 & 0 & 2 & 1 & 2 & 2 & 2 & 1 & 2 & 2 & 2 & 4 & 2 & 2 & 2 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 0 & 1 & 1 & 2 & 1 & 1 & 1 & 2 & 1 & 1 & 2 & 4 & 2 & 2 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 1 & 1 & 1 & 1 & 2 & 1 & 1 & 1 & 2 & 1 & 2 & 2 & 4 & 2 \\ -3 & -1 & -1 & -1 & -1 & -1 & -1 & 1 & -1 & -1 & -1 & 1 & -1 & 1 & 1 & 1 & -1 & 1 & 1 & 1 & 2 & 2 & 2 & 4 \end{array}\right)$
Genus Structure
Class number: | $24$ |
$\left(\begin{array}{rrrrrrrrrrrrrrrrrrrrrrrr} 4 & 2 & 2 & 0 & 0 & 1 & 2 & 0 & 2 & 0 & 0 & 2 & 1 & 0 & -1 & 1 & -1 & -2 & 0 & -1 & -1 & 1 & -1 & 2 \\ 2 & 4 & 2 & -2 & 1 & 2 & 1 & 1 & 2 & 2 & 2 & 2 & 2 & 1 & 1 & 1 & -1 & -1 & 2 & 0 & 1 & -1 & -2 & 2 \\ 2 & 2 & 4 & 0 & 1 & 2 & 0 & 1 & 2 & 2 & 2 & 2 & 0 & 0 & 1 & 0 & -1 & -2 & 1 & 1 & 1 & 1 & -1 & 2 \\ 0 & -2 & 0 & 4 & 1 & -2 & 0 & 1 & -1 & -1 & -1 & 0 & -2 & -1 & 0 & 0 & -1 & 1 & -2 & -1 & 0 & 0 & 2 & -2 \\ 0 & 1 & 1 & 1 & 4 & 1 & 0 & 1 & 0 & 0 & 1 & 2 & 1 & 0 & 1 & -1 & -2 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 1 & 2 & 2 & -2 & 1 & 4 & -1 & -1 & 2 & 1 & 1 & 2 & 2 & 1 & 0 & -1 & -1 & -1 & 2 & 0 & 0 & 1 & -1 & 2 \\ 2 & 1 & 0 & 0 & 0 & -1 & 4 & 1 & 1 & -1 & 0 & 1 & 1 & 0 & 0 & 2 & 0 & -1 & -1 & -1 & -1 & 0 & -1 & 1 \\ 0 & 1 & 1 & 1 & 1 & -1 & 1 & 4 & -1 & 2 & 2 & 0 & 0 & 1 & 2 & 1 & 0 & 0 & -1 & 1 & 1 & -1 & -1 & 0 \\ 2 & 2 & 2 & -1 & 0 & 2 & 1 & -1 & 4 & 0 & 0 & 2 & 1 & -1 & 0 & 1 & -1 & -2 & 2 & -1 & 1 & 1 & -1 & 1 \\ 0 & 2 & 2 & -1 & 0 & 1 & -1 & 2 & 0 & 4 & 2 & 0 & 0 & 1 & 2 & 0 & 0 & 0 & 1 & 2 & 2 & -1 & -1 & 1 \\ 0 & 2 & 2 & -1 & 1 & 1 & 0 & 2 & 0 & 2 & 4 & 0 & 0 & 1 & 2 & 1 & -1 & -1 & 1 & 1 & 1 & -1 & -2 & 1 \\ 2 & 2 & 2 & 0 & 2 & 2 & 1 & 0 & 2 & 0 & 0 & 4 & 2 & 1 & 0 & -1 & -1 & -1 & 1 & -1 & 0 & 1 & 0 & 2 \\ 1 & 2 & 0 & -2 & 1 & 2 & 1 & 0 & 1 & 0 & 0 & 2 & 4 & 2 & 0 & -1 & 0 & 0 & 1 & 0 & -1 & 0 & -1 & 2 \\ 0 & 1 & 0 & -1 & 0 & 1 & 0 & 1 & -1 & 1 & 1 & 1 & 2 & 4 & 0 & -1 & 1 & 0 & 1 & 0 & 0 & -1 & 0 & 2 \\ -1 & 1 & 1 & 0 & 1 & 0 & 0 & 2 & 0 & 2 & 2 & 0 & 0 & 0 & 4 & 1 & -1 & 1 & 0 & 1 & 2 & -1 & -1 & -1 \\ 1 & 1 & 0 & 0 & -1 & -1 & 2 & 1 & 1 & 0 & 1 & -1 & -1 & -1 & 1 & 4 & -1 & -1 & 0 & -1 & 1 & -1 & -2 & -1 \\ -1 & -1 & -1 & -1 & -2 & -1 & 0 & 0 & -1 & 0 & -1 & -1 & 0 & 1 & -1 & -1 & 4 & 0 & 0 & 1 & 0 & 0 & 1 & 1 \\ -2 & -1 & -2 & 1 & 1 & -1 & -1 & 0 & -2 & 0 & -1 & -1 & 0 & 0 & 1 & -1 & 0 & 4 & -1 & 0 & 0 & -2 & 2 & -2 \\ 0 & 2 & 1 & -2 & 0 & 2 & -1 & -1 & 2 & 1 & 1 & 1 & 1 & 1 & 0 & 0 & 0 & -1 & 4 & 0 & 2 & -1 & -1 & 1 \\ -1 & 0 & 1 & -1 & 0 & 0 & -1 & 1 & -1 & 2 & 1 & -1 & 0 & 0 & 1 & -1 & 1 & 0 & 0 & 4 & 0 & 0 & -1 & 1 \\ -1 & 1 & 1 & 0 & 0 & 0 & -1 & 1 & 1 & 2 & 1 & 0 & -1 & 0 & 2 & 1 & 0 & 0 & 2 & 0 & 4 & -1 & 0 & -1 \\ 1 & -1 & 1 & 0 & 0 & 1 & 0 & -1 & 1 & -1 & -1 & 1 & 0 & -1 & -1 & -1 & 0 & -2 & -1 & 0 & -1 & 4 & 0 & 1 \\ -1 & -2 & -1 & 2 & 0 & -1 & -1 & -1 & -1 & -1 & -2 & 0 & -1 & 0 & -1 & -2 & 1 & 2 & -1 & -1 & 0 & 0 & 4 & -1 \\ 2 & 2 & 2 & -2 & 0 & 2 & 1 & 0 & 1 & 1 & 1 & 2 & 2 & 2 & -1 & -1 & 1 & -2 & 1 & 1 & -1 & 1 & -1 & 4 \end{array}\right)$, $\left(\begin{array}{rrrrrrrrrrrrrrrrrrrrrrrr} 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & -1 & 0 & -1 & -1 & 0 & 1 & -1 & -1 & 0 & -1 & -1 & 1 & -1 & -1 & -1 \\ 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 1 & 0 & 0 & 0 & -1 & 0 & 0 & -1 & 0 & -1 & 0 & 0 & -1 & 0 \\ 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & -1 & 1 & -1 & -1 & 1 & 0 & 1 & -1 & 1 & -1 & -1 & 1 & -1 & -1 \\ 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 1 & 1 & 1 & -1 & 1 & -1 & 0 & -1 & 0 & -1 & 0 & -1 & 0 & -1 & 0 & -1 \\ 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 1 & 1 & 1 & 0 & 1 & -1 & 0 & -1 & -1 & -1 & 1 & 1 & -1 & -1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 1 & 0 & 1 & -1 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ -1 & 1 & 1 & 1 & 0 & 0 & 0 & 0 & 4 & 0 & 1 & 1 & 1 & 0 & 0 & -1 & 0 & -1 & 0 & -1 & 0 & 0 & -1 & -1 \\ -1 & 0 & 0 & 1 & 1 & 1 & 0 & 1 & 0 & 4 & 1 & 1 & 0 & -1 & -2 & 0 & 1 & -2 & 1 & 1 & 0 & -1 & 0 & 1 \\ 0 & 1 & -1 & 1 & 1 & 0 & 0 & 0 & 1 & 1 & 4 & -1 & 2 & 1 & -2 & 0 & -2 & -2 & -1 & 0 & 2 & -2 & -1 & 1 \\ -1 & 0 & 1 & -1 & 1 & 0 & 0 & 1 & 1 & 1 & -1 & 4 & -1 & 0 & 0 & 1 & 1 & 0 & 0 & 1 & 0 & 1 & -1 & 0 \\ -1 & 0 & -1 & 1 & 0 & 0 & 1 & -1 & 1 & 0 & 2 & -1 & 4 & 1 & -1 & 1 & -1 & 0 & -1 & 0 & 1 & -1 & 0 & 1 \\ 0 & 0 & -1 & -1 & 1 & 0 & 0 & -1 & 0 & -1 & 1 & 0 & 1 & 4 & -1 & 0 & -2 & 1 & -2 & 2 & 2 & -1 & 0 & 1 \\ 1 & -1 & 1 & 0 & -1 & 0 & 0 & 0 & 0 & -2 & -2 & 0 & -1 & -1 & 4 & 0 & 0 & 1 & 0 & -1 & 0 & 1 & 0 & -2 \\ -1 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 1 & 1 & 0 & 0 & 4 & 0 & 0 & 0 & 1 & 0 & 1 & 0 & 1 \\ -1 & 0 & 1 & 0 & -1 & 0 & 0 & 0 & 0 & 1 & -2 & 1 & -1 & -2 & 0 & 0 & 4 & 0 & 3 & -1 & -2 & 2 & 1 & 0 \\ 0 & -1 & -1 & -1 & -1 & 0 & 0 & 0 & -1 & -2 & -2 & 0 & 0 & 1 & 1 & 0 & 0 & 4 & -1 & 1 & 0 & 1 & 2 & 0 \\ -1 & 0 & 1 & 0 & -1 & 0 & 0 & 0 & 0 & 1 & -1 & 0 & -1 & -2 & 0 & 0 & 3 & -1 & 4 & -1 & -2 & 2 & 1 & 1 \\ -1 & -1 & -1 & -1 & 1 & 1 & -1 & 0 & -1 & 1 & 0 & 1 & 0 & 2 & -1 & 1 & -1 & 1 & -1 & 4 & 1 & 0 & 1 & 1 \\ 1 & 0 & -1 & 0 & 1 & 1 & 0 & 0 & 0 & 0 & 2 & 0 & 1 & 2 & 0 & 0 & -2 & 0 & -2 & 1 & 4 & -2 & -1 & 0 \\ -1 & 0 & 1 & -1 & -1 & 0 & 0 & 0 & 0 & -1 & -2 & 1 & -1 & -1 & 1 & 1 & 2 & 1 & 2 & 0 & -2 & 4 & 1 & 0 \\ -1 & -1 & -1 & 0 & -1 & 0 & 0 & 0 & -1 & 0 & -1 & -1 & 0 & 0 & 0 & 0 & 1 & 2 & 1 & 1 & -1 & 1 & 4 & 1 \\ -1 & 0 & -1 & -1 & 0 & 0 & 1 & 0 & -1 & 1 & 1 & 0 & 1 & 1 & -2 & 1 & 0 & 0 & 1 & 1 & 0 & 0 & 1 & 4 \end{array}\right)$ ... | |
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Comments
This integral lattice is the Leech lattice.
Additional information
The Leech lattice is the unique solution of the sphere packing problem in dimension 24, by a theorem of Cohn-Kumar-Miller-Radchenko-Viazovska [arXiv:1603.06518], [10.4007/annals.2017.185.3.8]. See Wikipedia for more discussion.