# Properties

 Label 1.2.B.1.1a Name $$\mathrm{U}(1)$$ Weight $1$ Degree $2$ Real dimension $1$ Components $1$ Contained in $$\mathrm{USp}(2)$$ Identity component $$\mathrm{U}(1)$$ Component group $$C_1$$

## Invariants

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

## Identity component

 Name: $\mathrm{U}(1)$ $\mathbb{R}$-dimension: $1$ Description: $\left\{\begin{bmatrix}\alpha&0\\0&\bar\alpha\end{bmatrix}:\alpha\bar\alpha = 1,\ \alpha\in\mathbb{C}\right\}$ Symplectic form: $\begin{bmatrix}0&1\\-1&0\end{bmatrix}$ Hodge circle: $u\mapsto\mathrm{diag}(u,\bar u)$

## Moment sequences

$x$ $\mathrm{E}[x^{0}]$ $\mathrm{E}[x^{1}]$ $\mathrm{E}[x^{2}]$ $\mathrm{E}[x^{3}]$ $\mathrm{E}[x^{4}]$ $\mathrm{E}[x^{5}]$ $\mathrm{E}[x^{6}]$ $\mathrm{E}[x^{7}]$ $\mathrm{E}[x^{8}]$ $\mathrm{E}[x^{9}]$ $\mathrm{E}[x^{10}]$ $\mathrm{E}[x^{11}]$ $\mathrm{E}[x^{12}]$
$a_1$ $1$ $0$ $2$ $0$ $6$ $0$ $20$ $0$ $70$ $0$ $252$ $0$ $924$

$\mathrm{U}(1)$ is the Sato-Tate group of an elliptic curve with complex multiplication over any number field that contains the CM field. In all of these cases the Sato-Tate conjecture follows from a classical result of Deuring; see Lemma 2.14 in [arXiv:1604.01256 ] for a proof.