Integral lattice 8.64.4.1.1

• Label: 8.64.4.1.1
• Class number: $1$
• Dimension: $8$
• Determinant: $64$
• Level: $4$

Lattice Invariants

Dimension:

$8$

Determinant:

$64$

Level:

$4$

Density:

$0.0317086884876310017045704207970\dots$

Group order:

$442368$

Hermite number:

$1.18920711500272106671749997056\dots$

Minimal vector length:

$2$

Kissing number:

$32$

Normalized minimal vectors:

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

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Theta Series

Gram Matrix

$\left(\begin{array}{rrrrrrrr} 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 2 & 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 2 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 2 & 1 & 0 & 0 & 0 \\ 0 & 1 & 1 & 1 & 2 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 \end{array}\right)$

Genus Structure

Genus representatives:

Class number:	$1$
$\left(\begin{array}{rrrrrrrr} 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 2 & -1 & -1 & 1 & 0 & 0 & 0 \\ 0 & -1 & 2 & 0 & -1 & 0 & 0 & 0 \\ 0 & -1 & 0 & 2 & -1 & 0 & 0 & 0 \\ 0 & 1 & -1 & -1 & 2 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 \end{array}\right)$
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