# Properties

 Label 1.6.H.4.1b Name $$H_{at}$$ Weight $1$ Degree $6$ Real dimension $3$ Components $4$ Contained in $$\mathrm{USp}(6)$$ Identity component $$\mathrm{U}(1)^3$$ Component group $$C_4$$

## Invariants

 Weight: $1$ Degree: $6$ $\mathbb{R}$-dimension: $3$ Components: $4$ Contained in: $\mathrm{USp}(6)$ Rational: yes

## Identity component

 Name: $\mathrm{U}(1)^3$ $\mathbb{R}$-dimension: $3$ Description: $\left\{\begin{bmatrix}A&0&0\\0&B&0\\0&0&C\end{bmatrix}: A,B,C\in\mathrm{U}(1)\subseteq\mathrm{SU}(2)\right\}$ Symplectic form: $\begin{bmatrix}J_2&0&0\\0&J_2&0\\0&0&J_2\end{bmatrix},\ J_2:=\begin{bmatrix}0&1\\-1&0\end{bmatrix}$ Hodge circle: $u\mapsto\mathrm{diag}(u,\bar u, u, \bar u, u, \bar u)$

## Component group

 Name: $C_4$ Order: $4$ Abelian: yes Generators: $\begin{bmatrix}0 & 0 & 0 & 1 & 0 & 0 \\0 & 0 & -1 & 0 & 0 & 0 \\1 & 0 & 0 & 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\0 & 0 & 0 & 0 & 1 & 0 \\0 & 0 & 0 & 0 & 0 & 1 \\\end{bmatrix}$

## Moment sequences

$x$ $\mathrm{E}[x^{0}]$ $\mathrm{E}[x^{1}]$ $\mathrm{E}[x^{2}]$ $\mathrm{E}[x^{3}]$ $\mathrm{E}[x^{4}]$ $\mathrm{E}[x^{5}]$ $\mathrm{E}[x^{6}]$ $\mathrm{E}[x^{7}]$ $\mathrm{E}[x^{8}]$ $\mathrm{E}[x^{9}]$ $\mathrm{E}[x^{10}]$ $\mathrm{E}[x^{11}]$ $\mathrm{E}[x^{12}]$
$a_1$ $1$ $0$ $3$ $0$ $27$ $0$ $480$ $0$ $11235$ $0$ $293328$ $0$ $8124732$
$a_2$ $1$ $2$ $8$ $53$ $482$ $5117$ $58715$ $704678$ $8716118$ $110244185$ $1419037673$ $18526707608$ $244745665961$
$a_3$ $1$ $0$ $10$ $0$ $1254$ $0$ $277600$ $0$ $73331510$ $0$ $21210900480$ $0$ $6499814269872$