# Properties

 Label 1.4.D.4.1a Name $$F_{ac}$$ Weight $1$ Degree $4$ Real dimension $2$ Components $4$ Contained in $$\mathrm{USp}(4)$$ Identity component $$\mathrm{U}(1)\times\mathrm{U}(1)$$ Component group $$C_4$$

## Invariants

 Weight: $1$ Degree: $4$ $\mathbb{R}$-dimension: $2$ Components: $4$ Contained in: $\mathrm{USp}(4)$ Rational: yes

## Identity component

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

## Component group

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

## Moment sequences

$x$ $\mathrm{E}[x^{0}]$ $\mathrm{E}[x^{1}]$ $\mathrm{E}[x^{2}]$ $\mathrm{E}[x^{3}]$ $\mathrm{E}[x^{4}]$ $\mathrm{E}[x^{5}]$ $\mathrm{E}[x^{6}]$ $\mathrm{E}[x^{7}]$ $\mathrm{E}[x^{8}]$ $\mathrm{E}[x^{9}]$ $\mathrm{E}[x^{10}]$ $\mathrm{E}[x^{11}]$ $\mathrm{E}[x^{12}]$
$a_1$ $1$ $0$ $1$ $0$ $9$ $0$ $100$ $0$ $1225$ $0$ $15876$ $0$ $213444$
$a_2$ $1$ $1$ $3$ $10$ $41$ $186$ $912$ $4656$ $24425$ $130402$ $705368$ $3854016$ $21230420$

## Moment simplex

 $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=2\right)\colon$ $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=4\right)\colon$ $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=6\right)\colon$ $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=8\right)\colon$ $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=10\right)\colon$ $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=12\right)\colon$ $1$ $1$ $3$ $4$ $9$ $10$ $18$ $42$ $100$ $41$ $86$ $206$ $500$ $1225$ $186$ $428$ $1044$ $2570$ $6370$ $15876$ $912$ $2192$ $5418$ $13480$ $33712$ $84672$ $213444$

## Moment matrix

$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&0&0&0&1&0&1&0&1\\0&1&0&0&2&0&2&0&3&0\\0&0&2&1&0&2&0&3&0&3\\0&0&1&4&0&2&0&7&0&3\\0&2&0&0&6&0&6&0&10&0\\1&0&2&2&0&5&0&7&0&6\\0&2&0&0&6&0&7&0&11&0\\1&0&3&7&0&7&0&19&0&12\\0&3&0&0&10&0&11&0&20&0\\1&0&3&3&0&6&0&12&0&11\end{bmatrix}$

$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&1&2&4&6&5&7&19&20&11\end{bmatrix}$

## Event probabilities

$-$$a_2\in\mathbb{Z}$$a_2=-2$$a_2=-1$$a_2=0$$a_2=1$$a_2=2$
$-$$1$$3/4$$0$$0$$1/2$$0$$1/4 a_1=0$$3/4$$3/4$$0$$0$$1/2$$0$$1/4$