# Properties

 Label 1.4.F.2.1a Name $$J(C_1)$$ Weight $1$ Degree $4$ Real dimension $1$ Components $2$ Contained in $$\mathrm{USp}(4)$$ Identity component $$\mathrm{U}(1)_2$$ Component group $$C_2$$

## Invariants

 Weight: $1$ Degree: $4$ $\mathbb{R}$-dimension: $1$ Components: $2$ 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: $C_2$ Order: $2$ Abelian: yes Generators: $\begin{bmatrix}0&0&0&1\\0&0&-1&0\\0&-1&0&0\\1&0&0&0\end{bmatrix}$

## Moment sequences

$x$ $\mathrm{E}[x^{0}]$ $\mathrm{E}[x^{1}]$ $\mathrm{E}[x^{2}]$ $\mathrm{E}[x^{3}]$ $\mathrm{E}[x^{4}]$ $\mathrm{E}[x^{5}]$ $\mathrm{E}[x^{6}]$ $\mathrm{E}[x^{7}]$ $\mathrm{E}[x^{8}]$ $\mathrm{E}[x^{9}]$ $\mathrm{E}[x^{10}]$ $\mathrm{E}[x^{11}]$ $\mathrm{E}[x^{12}]$
$a_1$ $1$ $0$ $4$ $0$ $48$ $0$ $640$ $0$ $8960$ $0$ $129024$ $0$ $1892352$
$a_2$ $1$ $1$ $11$ $40$ $235$ $1196$ $6650$ $36632$ $205859$ $1162732$ $6617326$ $37840232$ $217333390$

## Moment simplex

 $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=2\right)\colon$ $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=4\right)\colon$ $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=6\right)\colon$ $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=8\right)\colon$ $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=10\right)\colon$ $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=12\right)\colon$ $1$ $4$ $11$ $20$ $48$ $40$ $104$ $256$ $640$ $235$ $556$ $1392$ $3520$ $8960$ $1196$ $3032$ $7680$ $19584$ $50176$ $129024$ $6650$ $16776$ $42848$ $109952$ $283136$ $731136$ $1892352$

## Moment matrix

$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&0&3&0&6&0&3&0&0\\0&4&0&0&12&0&12&0&24&0\\0&0&10&6&0&3&0&33&0&36\\3&0&6&19&0&27&0&44&0&28\\0&12&0&0&40&0&44&0&84&0\\6&0&3&27&0&46&0&48&0&21\\0&12&0&0&44&0&52&0&96&0\\3&0&33&44&0&48&0&163&0&143\\0&24&0&0&84&0&96&0&184&0\\0&0&36&28&0&21&0&143&0&146\end{bmatrix}$

$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&4&10&19&40&46&52&163&184&146\end{bmatrix}$

## Event probabilities

$-$$a_2\in\mathbb{Z}$$a_2=-2$$a_2=-1$$a_2=0$$a_2=1$$a_2=2$
$-$$1$$1/2$$1/2$$0$$0$$0$$0 a_1=0$$1/2$$1/2$$1/2$$0$$0$$0$$0$