Sato-Tate Group \(\mathrm{SU}(2)[C_{1215000000}]\) of Weight 3 and Degree 2

• Label: 3.2.3.c1215000000
• Name: \(\mathrm{SU}(2)[C_{1215000000}]\)
• Weight: 3
• Degree: 2
• Real dimension: 3
• Components: 1215000000
• Contained in: \(\mathrm{U}(2)\)
• Identity Component: \(\mathrm{SU}(2)\)
• Component group: \(C_{1215000000}\)

Invariants

Weight:	$3$
Degree:	$2$
$\mathbb{R}$-dimension:	$3$
Components:	$1215000000$
Contained in:	$\mathrm{U}(2)$
Rational:	$\mathrm{False}$

Identity Component

Name:	$\mathrm{SU}(2)$
Index:	$1215000000$
$\mathbb{R}$-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:	$C_{1215000000}$
Order:	$1215000000$
Abelian:	$\mathrm{True}$
Generators:	$\begin{bmatrix} 1 & 0 \\ 0 & \zeta_{1215000000}\end{bmatrix}$

Subgroups and Supergroups

Maximal Subgroups:	$\mathrm{SU}(2)[C_{607500000}]$, $\mathrm{SU}(2)[C_{405000000}]$, $\mathrm{SU}(2)[C_{243000000}]$
Minimal Supergroups:	$\mathrm{SU}(2)[C_{2430000000}]$, $\mathrm{SU}(2)[C_{3645000000}]$, $\mathrm{SU}(2)[C_{6075000000}]$, $\ldots$

Moment Statistics

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