Properties

Label 1.2.B.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 \(C_2\)

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Invariants

Weight:$1$
Degree:$2$
$\mathbb{R}$-dimension:$1$
Components:$2$
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:$u\mapsto\mathrm{diag}(u,\bar u)$

Component group

Name:$C_2$
Order:$2$
Abelian:yes
Generators:$\begin{bmatrix}0&1\\1&0\end{bmatrix}$

Subgroups and supergroups

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

Moment sequences

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

Additional information

$N(\mathrm{U}(1)$ 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].