# Properties

 Name A5 Label 5.6.12.1.1 Class number 1 Dimension 5 Determinant 6 Level 12

## Lattice Invariants

Dimension:$5$
Determinant:$6$
Level:$12$
Label:$5.6.12.1.1$
Density:$0.379881250517603758105399396698\dots$
Group order:$1440$
Hermite number:$1.39765423754315849046595420554\dots$
Minimal vector length:$2$
Kissing number:$30$
Normalized minimal vectors: $(1, 0, 0, 0, 0)$, $(1, 1, 0, 0, 0)$, $(1, 1, 1, 0, 0)$, $(1, 1, 1, 1, 0)$, $(1, 1, 1, 1, 1)$, $(0, 1, 0, 0, 0)$, $(0, 1, 1, 0, 0)$, $(0, 1, 1, 1, 0)$, $(0, 1, 1, 1, 1)$, $(0, 0, 1, 0, 0)$, $(0, 0, 1, 1, 0)$, $(0, 0, 1, 1, 1)$, $(0, 0, 0, 1, 0)$, $(0, 0, 0, 1, 1)$, $(0, 0, 0, 0, 1)$ ...

## Theta Series

$$1$$ $$\mathstrut +\mathstrut 30q^{2}$$ $$\mathstrut +\mathstrut 90q^{4}$$ $$\mathstrut +\mathstrut 140q^{6}$$ $$\mathstrut +\mathstrut 270q^{8}$$ $$\mathstrut +\mathstrut 360q^{10}$$ $$\mathstrut +\mathstrut 330q^{12}$$ $$\mathstrut +\mathstrut 660q^{14}$$ $$\mathstrut +\mathstrut 810q^{16}$$ $$\mathstrut +\mathstrut 570q^{18}$$ $$\mathstrut +\mathstrut 1020q^{20}$$ $$\mathstrut +\mathstrut O(q^{21})$$

## Gram Matrix

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

## Genus Structure

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

This integral lattice is the A5 lattice.

This is a root lattice.