# Properties

 Label 1.2.1.2.1a Name $$N(\mathrm{U}(1))$$ Weight 1 Degree 2 Real dimension 1 Components 2 Contained in $$\mathrm{USp}(2)$$ Identity Component $$\mathrm{U}(1)$$ Component group $$C_2$$

## Invariants

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

## Identity Component

 Name: $\mathrm{U}(1)$ Index: $2$ $\mathbb{R}$-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: $C_2$ Order: $2$ Abelian: $\mathrm{True}$ Generators: $\begin{bmatrix}0&1\\1&0\end{bmatrix}$

## Moment Statistics

$x$ $\mathrm{E}[x^{0}]$ $\mathrm{E}[x^{1}]$ $\mathrm{E}[x^{2}]$ $\mathrm{E}[x^{3}]$ $\mathrm{E}[x^{4}]$ $\mathrm{E}[x^{5}]$ $\mathrm{E}[x^{6}]$ $\mathrm{E}[x^{7}]$ $\mathrm{E}[x^{8}]$ $\mathrm{E}[x^{9}]$ $\mathrm{E}[x^{10}]$ $\mathrm{E}[x^{11}]$ $\mathrm{E}[x^{12}]$
$a_1$ $1$ $0$ $1$ $0$ $3$ $0$ $10$ $0$ $35$ $0$ $126$ $0$ $462$

## Event Probabilities

 $\mathrm{P}[a_1=0]=\frac{1}{2}$

$N(\mathrm{U}(1)$ is the Sato-Tate group of any elliptic curve with complex multiplication defined over a number field that does not contain the CM field; the Sato-Tate conjecture in this case follows from a classical results of Deuring; see Lemma 2.15 of [arXiv:1604.01256].