Integral lattice 7.2.4.1.2

• Label: 7.2.4.1.2
• Class number: $1$
• Dimension: $7$
• Determinant: $2$
• Level: $4$

The $E_7$ lattice is the root lattice associated to the $E_7$ root system.

Lattice Invariants

Dimension:

$7$

Determinant:

$2$

Level:

$4$

Density:

$0.295297873145712573099774429210\dots$

Group order:

$2903040$

Hermite number:

$1.81144732852781334318834574643\dots$

Minimal vector length:

$2$

Kissing number:

$126$

Normalized minimal vectors:

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

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Theta Series

Gram Matrix

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

Genus Structure

Genus representatives:

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

The $E_7$ lattice is the unique solution of the lattice packing problem in dimension 7.