Sato-Tate Group \(\mathrm{U}(1)[D_{1582031250}]\) of Weight 1 and Degree 2

• Label: 1.2.1.d1582031250
• Name: \(\mathrm{U}(1)[D_{1582031250}]\)
• Weight: 1
• Degree: 2
• Real dimension: 1
• Components: 3164062500
• Contained in: \(\mathrm{U}(2)\)
• Identity Component: \(\mathrm{U}(1)\)
• Component group: \(D_{1582031250}\)

Invariants

Weight:	$1$
Degree:	$2$
$\mathbb{R}$-dimension:	$1$
Components:	$3164062500$
Contained in:	$\mathrm{U}(2)$
Rational:	$\mathrm{False}$

Identity Component

Name:	$\mathrm{U}(1)$
Index:	$3164062500$
$\mathbb{R}$-dimension:	$1$
Description:	$\left\{\begin{bmatrix}\alpha&0\\0&\bar\alpha\end{bmatrix}:\alpha\bar\alpha = 1,\ \alpha\in\mathbb{C}\right\}$

Component Group

Name:	$D_{1582031250}$
Order:	$3164062500$
Abelian:	$\mathrm{False}$
Generators:	$\left\{\begin{bmatrix} 0 & 1\\ -1 & 0\end{bmatrix}, \begin{bmatrix} 1 & 0 \\ 0 & \zeta_{1582031250}\end{bmatrix}\right\}$

Subgroups and Supergroups

Maximal Subgroups:	$\mathrm{U}(1)[D_{791015625}]$, $\mathrm{U}(1)[D_{527343750}]$, $\mathrm{U}(1)[D_{316406250}]$
Minimal Supergroups:	$\mathrm{U}(1)[D_{3164062500}]$, $\mathrm{U}(1)[D_{4746093750}]$, $\mathrm{U}(1)[D_{7910156250}]$, $\ldots$

Moment Statistics

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