Sato-Tate group \(\mathrm{U}(1)\) of weight 1 and degree 2

• 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:	$u\mapsto\mathrm{diag}(u,\bar u)$

Subgroups and supergroups

Maximal subgroups:
Minimal supergroups:	$N(\mathrm{U}(1))$

Moment sequences

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

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.