# Properties

 Label 1.4.E.1.1a Name $$E_1$$ Weight $1$ Degree $4$ Real dimension $3$ Components $1$ Contained in $$\mathrm{USp}(4)$$ Identity component $$\mathrm{SU}(2)_2$$ Component group $$C_1$$

## Invariants

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

## Identity component

 Name: $\mathrm{SU}(2)_2$ Index: $1$ $\mathbb{R}$-dimension: $3$ Description: $\left\{\begin{bmatrix}A&0\\0&\bar{A}\end{bmatrix}: A\in \mathrm{SU}(2)\right\}$ Symplectic form: $\begin{bmatrix}0&I_2\\-I_2&0\end{bmatrix}$ Hodge circle: $u\mapsto\mathrm{diag}(u,\bar u,\bar u,u)$

## Moment sequences

$x$ $\mathrm{E}[x^{0}]$ $\mathrm{E}[x^{1}]$ $\mathrm{E}[x^{2}]$ $\mathrm{E}[x^{3}]$ $\mathrm{E}[x^{4}]$ $\mathrm{E}[x^{5}]$ $\mathrm{E}[x^{6}]$ $\mathrm{E}[x^{7}]$ $\mathrm{E}[x^{8}]$ $\mathrm{E}[x^{9}]$ $\mathrm{E}[x^{10}]$ $\mathrm{E}[x^{11}]$ $\mathrm{E}[x^{12}]$
$a_1$ $1$ $0$ $4$ $0$ $32$ $0$ $320$ $0$ $3584$ $0$ $43008$ $0$ $540672$
$a_2$ $1$ $3$ $10$ $37$ $150$ $654$ $3012$ $14445$ $71398$ $361114$ $1859628$ $9716194$ $51373180$

## Moment simplex

 $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=2\right)\colon$ $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=4\right)\colon$ $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=6\right)\colon$ $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=8\right)\colon$ $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=10\right)\colon$ $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=12\right)\colon$ $3$ $4$ $10$ $16$ $32$ $37$ $68$ $144$ $320$ $150$ $304$ $672$ $1536$ $3584$ $654$ $1416$ $3232$ $7552$ $17920$ $43008$ $3012$ $6816$ $15936$ $37888$ $91136$ $221184$ $540672$

## Moment matrix

$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&2&1&0&3&0&2&0&4\\0&4&0&0&8&0&4&0&12&0\\2&0&5&5&0&8&0&10&0&11\\1&0&5&10&0&9&0&20&0&13\\0&8&0&0&20&0&16&0&32&0\\3&0&8&9&0&14&0&21&0&20\\0&4&0&0&16&0&20&0&28&0\\2&0&10&20&0&21&0&49&0&32\\0&12&0&0&32&0&28&0&56&0\\4&0&11&13&0&20&0&32&0&30\end{bmatrix},\qquad\mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&4&5&10&20&14&20&49&56&30\end{bmatrix}$