Sato-Tate Group \(\mathrm{SU}(2)[C_{4194304000}]\) of Weight 5 and Degree 2

• Label: 5.2.3.c4194304000
• Name: \(\mathrm{SU}(2)[C_{4194304000}]\)
• Weight: 5
• Degree: 2
• Real dimension: 3
• Components: 4194304000
• Contained in: \(\mathrm{U}(2)\)
• Identity Component: \(\mathrm{SU}(2)\)
• Component group: \(C_{4194304000}\)

Invariants

Weight:	$5$
Degree:	$2$
$\mathbb{R}$-dimension:	$3$
Components:	$4194304000$
Contained in:	$\mathrm{U}(2)$
Rational:	$\mathrm{False}$

Identity Component

Name:	$\mathrm{SU}(2)$
Index:	$4194304000$
$\mathbb{R}$-dimension:	$3$
Description:	$\left\{\begin{bmatrix}\alpha&\beta\\-\bar\beta&\bar\alpha\end{bmatrix}:\alpha\bar\alpha+\beta\bar\beta = 1,\ \alpha,\beta\in\mathbb{C}\right\}$

Component Group

Name:	$C_{4194304000}$
Order:	$4194304000$
Abelian:	$\mathrm{True}$
Generators:	$\begin{bmatrix} 1 & 0 \\ 0 & \zeta_{4194304000}\end{bmatrix}$

Subgroups and Supergroups

Maximal Subgroups:	$\mathrm{SU}(2)[C_{2097152000}]$, $\mathrm{SU}(2)[C_{838860800}]$
Minimal Supergroups:	$\mathrm{SU}(2)[C_{8388608000}]$, $\mathrm{SU}(2)[C_{12582912000}]$, $\mathrm{SU}(2)[C_{20971520000}]$, $\ldots$

Moment Statistics

$x$	$\mathrm{E}[x^{0}]$	$\mathrm{E}[x^{1}]$	$\mathrm{E}[x^{2}]$	$\mathrm{E}[x^{3}]$	$\mathrm{E}[x^{4}]$	$\mathrm{E}[x^{5}]$	$\mathrm{E}[x^{6}]$	$\mathrm{E}[x^{7}]$	$\mathrm{E}[x^{8}]$	$\mathrm{E}[x^{9}]$	$\mathrm{E}[x^{10}]$	$\mathrm{E}[x^{11}]$	$\mathrm{E}[x^{12}]$
$a_1$	$1$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$