Properties

Label 1.2.1.1.1a
  
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 \(C_1\)

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Invariants

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

Identity Component

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

Subgroups and Supergroups

Maximal Subgroups:
Minimal Supergroups:$N(\mathrm{U}(1))$

Moment Statistics

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

Additional information

$\mathrm{U}(1)$ 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.