# Properties

 Label 1.4.F.3.1a Name $$C_3$$ Weight $1$ Degree $4$ Real dimension $1$ Components $3$ Contained in $$\mathrm{USp}(4)$$ Identity component $$\mathrm{U}(1)_2$$ Component group $$C_3$$

## Invariants

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

## Identity component

 Name: $\mathrm{U}(1)_2$ $\mathbb{R}$-dimension: $1$ Description: $\left\{\begin{bmatrix}\alpha I_2&0\\0&\bar\alpha I_2\end{bmatrix}: \alpha\bar\alpha = 1,\ \alpha\in\mathbb{C}\right\}$ Symplectic form: $\begin{bmatrix}0&I_2\\-I_2&0\end{bmatrix}$ Hodge circle: $u\mapsto\mathrm{diag}(u, u,\bar u,\bar u)$

## Component group

 Name: $C_3$ Order: $3$ Abelian: yes Generators: $\begin{bmatrix}\zeta_6&0&0&0\\0&\zeta_6^5&0&0\\0&0&\zeta_6^5&0\\0&0&0&\zeta_6\end{bmatrix}$

## Moment sequences

$x$ $\mathrm{E}[x^{0}]$ $\mathrm{E}[x^{1}]$ $\mathrm{E}[x^{2}]$ $\mathrm{E}[x^{3}]$ $\mathrm{E}[x^{4}]$ $\mathrm{E}[x^{5}]$ $\mathrm{E}[x^{6}]$ $\mathrm{E}[x^{7}]$ $\mathrm{E}[x^{8}]$ $\mathrm{E}[x^{9}]$ $\mathrm{E}[x^{10}]$ $\mathrm{E}[x^{11}]$ $\mathrm{E}[x^{12}]$
$a_1$ $1$ $0$ $4$ $0$ $36$ $0$ $440$ $0$ $6020$ $0$ $86184$ $0$ $1262184$
$a_2$ $1$ $2$ $8$ $34$ $164$ $842$ $4506$ $24726$ $137892$ $777418$ $4417178$ $25244606$ $144936754$

## Moment simplex

 $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=2\right)\colon$ $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=4\right)\colon$ $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=6\right)\colon$ $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=8\right)\colon$ $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=10\right)\colon$ $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=12\right)\colon$ $2$ $4$ $8$ $16$ $36$ $34$ $76$ $180$ $440$ $164$ $388$ $952$ $2380$ $6020$ $842$ $2068$ $5184$ $13144$ $33572$ $86184$ $4506$ $11312$ $28740$ $73540$ $189084$ $487872$ $1262184$

## Moment matrix

$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&1&2&0&2&0&3&0&4\\0&4&0&0&8&0&8&0&16&0\\1&0&5&6&0&8&0&17&0&14\\2&0&6&12&0&14&0&32&0&26\\0&8&0&0&28&0&28&0&56&0\\2&0&8&14&0&20&0&42&0&34\\0&8&0&0&28&0&36&0&64&0\\3&0&17&32&0&42&0&101&0&78\\0&16&0&0&56&0&64&0&124&0\\4&0&14&26&0&34&0&78&0&66\end{bmatrix}$

$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&4&5&12&28&20&36&101&124&66\end{bmatrix}$

## Event probabilities

$\mathrm{Pr}[a_i=n]=0$ for $i=1,2$ and $n\in\mathbb{Z}$.