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Properties

Name Leech
Label 24.1.1.24.1
Class number 24
Dimension 24
Determinant 1
Level 1

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This unimodular integral lattice is the Leech lattice.

Lattice Invariants

Dimension:$24$
Determinant:$1$
Level:$1$
Label:$24.1.1.24.1$
Density:$0.00192957430940392304790334556369\dots$
Group order:$8315553613086720000$
Hermite number:$4.00000000000000000000000000000\dots$
Minimal vector length:$4$
Kissing number:$196560$
Normalized minimal vectors: $[1, -2, -2, -2, 2, -1, -1, 3, 3, 0, 0, 2, 2, -1, -1, -2, 2, -2, -1, -1, 0, 0, -1, 2]$, $[1, -2, -2, -2, 2, -1, 0, 2, 3, 0, 0, 2, 2, -1, -1, -2, 2, -1, -1, -2, 1, -1, -1, 3]$, $[1, -2, -2, -1, 1, -1, -1, 2, 2, 0, 0, 2, 2, 0, 0, -2, 2, -1, -1, -1, 0, -1, -1, 2]$ ...
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Theta Series

\(1 \) \(\mathstrut +\mathstrut 196560q^{4} \) \(\mathstrut +\mathstrut 16773120q^{6} \) \(\mathstrut +\mathstrut 398034000q^{8} \) \(\mathstrut +\mathstrut 4629381120q^{10} \) \(\mathstrut +\mathstrut 34417656000q^{12} \) \(\mathstrut +\mathstrut 187489935360q^{14} \) \(\mathstrut +\mathstrut 814879774800q^{16} \) \(\mathstrut +\mathstrut 2975551488000q^{18} \) \(\mathstrut +\mathstrut 9486551299680q^{20} \) \(\mathstrut +\mathstrut O(q^{21}) \)

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:$24$
 
Genus representatives: $\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.