Properties

 Label 1.2.3.1.1a Name $$\mathrm{SU}(2)$$ Weight 1 Degree 2 Real dimension 3 Components 1 Contained in $$\mathrm{USp}(2)$$ Identity Component $$\mathrm{SU}(2)$$ Component group $$C_1$$

$\mathrm{SU}(2)$ is the Sato-Tate group of any non-CM elliptic curve $E$ over a number field $K$.

As conjectured by Mikio Sato and John Tate and proved by Richard Taylor et al., when $K=\Q$ the normalized Frobenius traces of $E$ are equidistributed with respect to the Haar measure on $\mathrm{SU}(2)$.

Invariants

 Weight: $1$ Degree: $2$ $\mathbb{R}$-dimension: $3$ Components: $1$ Contained in: $\mathrm{USp}(2)$ Rational: $\mathrm{True}$

Identity Component

 Name: $\mathrm{SU}(2)$ Index: $1$ $\mathbb{R}$-dimension: $3$ Description: $\left\{\begin{bmatrix}\alpha&\beta\\-\bar\beta&\bar\alpha\end{bmatrix}:\alpha\bar\alpha+\beta\bar\beta = 1,\ \alpha,\beta\in\mathbb{C}\right\}$

Moment Statistics

$x$ $\mathrm{E}[x^{0}]$ $\mathrm{E}[x^{1}]$ $\mathrm{E}[x^{2}]$ $\mathrm{E}[x^{3}]$ $\mathrm{E}[x^{4}]$ $\mathrm{E}[x^{5}]$ $\mathrm{E}[x^{6}]$ $\mathrm{E}[x^{7}]$ $\mathrm{E}[x^{8}]$ $\mathrm{E}[x^{9}]$ $\mathrm{E}[x^{10}]$ $\mathrm{E}[x^{11}]$ $\mathrm{E}[x^{12}]$
$a_1$ $1$ $0$ $1$ $0$ $2$ $0$ $5$ $0$ $14$ $0$ $42$ $0$ $132$

The Sato-Tate conjecture for elliptic curves over number fields has been proved for totally real fields (including $\Q$) [10.4007/annals.2010.171.779, 10.1007/s10240-008-0015-2], and CM fields [arXiv:1812.09999].