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Properties

Label 7.64.2.1.1
Class number $1$
Dimension $7$
Determinant $64$
Level $2$

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The $E_7^*$ lattice is the dual of the $E_7$ lattice.

Lattice Invariants

Dimension:$7$
Determinant:$64$
Level:$2$
Density:$0.215776794229623286461726301808\dots$
Group order:$2903040$
Hermite number:$1.65613427051071850647425808144\dots$
Minimal vector length:$3$
Kissing number:$56$
Normalized minimal vectors: $(1, -1, -1, -1, 0, 0, 0)$, $(1, -1, -1, 0, 0, 0, 0)$, $(1, 0, 0, -1, -1, -1, 0)$, $(1, 0, 0, -1, -1, -1, 1)$, $(1, 0, 0, -1, -1, 0, 0)$, $(1, 0, 0, -1, 0, 0, 0)$, $(1, 0, 0, 0, -1, -1, 0)$, $(1, 0, 0, 0, -1, -1, 1)$, $(1, 0, 0, 0, -1, 0, 0)$, $(1, 0, 0, 0, 0, 0, 0)$, $(2, 0, -1, -1, -2, -2, 1)$, $(2, 0, -1, -1, -2, -1, 0)$, $(2, 0, -1, -1, -2, -1, 1)$, $(2, 0, -1, -1, -1, -1, 0)$, $(2, 0, -1, -1, -1, -1, 1)$ ...
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Theta Series

\(1 \) \(\mathstrut +\mathstrut 56 q^{3} \) \(\mathstrut +\mathstrut 126 q^{4} \) \(\mathstrut +\mathstrut 576 q^{7} \) \(\mathstrut +\mathstrut 756 q^{8} \) \(\mathstrut +\mathstrut 1512 q^{11} \) \(\mathstrut +\mathstrut 2072 q^{12} \) \(\mathstrut +\mathstrut 4032 q^{15} \) \(\mathstrut +\mathstrut 4158 q^{16} \) \(\mathstrut +\mathstrut 5544 q^{19} \) \(\mathstrut +\mathstrut 7560 q^{20} \) \(\mathstrut +\mathstrut O(q^{21}) \)

Gram Matrix

$\left(\begin{array}{rrrrrrr} 3 & 1 & 1 & 2 & 2 & 0 & 0 \\ 1 & 3 & -1 & 0 & 2 & 0 & 0 \\ 1 & -1 & 3 & 0 & 0 & 0 & 0 \\ 2 & 0 & 0 & 4 & 0 & 0 & 0 \\ 2 & 2 & 0 & 0 & 4 & -2 & 0 \\ 0 & 0 & 0 & 0 & -2 & 4 & 2 \\ 0 & 0 & 0 & 0 & 0 & 2 & 4 \end{array}\right)$

Genus Structure

Genus representatives:
Class number:$1$
 
$\left(\begin{array}{rrrrrrr} 3 & -1 & 1 & -1 & 1 & 1 & 1 \\ -1 & 3 & 1 & 1 & -1 & 1 & 1 \\ 1 & 1 & 3 & -1 & -1 & 1 & 1 \\ -1 & 1 & -1 & 3 & -1 & -1 & -1 \\ 1 & -1 & -1 & -1 & 3 & 1 & 1 \\ 1 & 1 & 1 & -1 & 1 & 3 & 1 \\ 1 & 1 & 1 & -1 & 1 & 1 & 3 \end{array}\right)$
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