Properties

 Name A3*, D3* Label 3.128.8.1.1 Class number 1 Dimension 3 Determinant 128 Level 8

The $A_3^*$ lattice, also known as the body-centered cubic lattice, is the dual of the $A_3$ lattice.

Lattice Invariants

Dimension:$3$
Determinant:$128$
Level:$8$
Label:$3.128.8.1.1$
Density:$0.680174761587831693972779346617\dots$
Group order:$48$
Hermite number:$1.19055078897614960606377922945\dots$
Minimal vector length:$6$
Kissing number:$8$
Normalized minimal vectors: $(1, 0, 0)$, $(1, 1, 1)$, $(0, 1, 0)$, $(0, 0, 1)$

Theta Series

$$1$$ $$\mathstrut +\mathstrut 8q^{6}$$ $$\mathstrut +\mathstrut 6q^{8}$$ $$\mathstrut +\mathstrut 12q^{16}$$ $$\mathstrut +\mathstrut O(q^{21})$$

Gram Matrix

$\left(\begin{array}{rrr} 6 & -2 & -2 \\ -2 & 6 & -2 \\ -2 & -2 & 6 \end{array}\right)$

Genus Structure

 Class number: $1$ Genus representatives: $\left(\begin{array}{rrr} 6 & -2 & -2 \\ -2 & 6 & -2 \\ -2 & -2 & 6 \end{array}\right)$ Download this matrix for gp, magma, sage