Sato-Tate group A(2,2)A(2,2) of weight 1 and degree 6

• Label: 1.6.N.4.2a
• Name: $A(2,2)$
• Weight: $1$
• Degree: $6$
• Real dimension: $1$
• Components: $4$
• Contained in: $\mathrm{USp}(6)$
• Identity component: $\mathrm{U}(1)_3$
• Component group: $C_2^2$

Invariants

Weight:	$1$
Degree:	$6$
$\mathbb{R}$-dimension:	$1$
Components:	$4$
Contained in:	$\mathrm{USp}(6)$
Rational:	yes

Identity component

Name:	$\mathrm{U}(1)_3$
$\mathbb{R}$-dimension:	$1$
Description:	$\left\{\begin{bmatrix}\alpha I_3&0,\\ 0&\bar \alpha I_3\end{bmatrix}: \alpha\bar\alpha=1,\ \alpha\in\mathbb{C}\right\}$	Symplectic form:	$\begin{bmatrix} 0 & I_3\\ -I_3 & 0\end{bmatrix}$
Hodge circle:	$u\mapsto \mathrm{diag}(u,u, u, \bar u, \bar u, \bar u)$

Component group

Name:	$C_2^2$
Order:	$4$
Abelian:	yes
Generators:	$\begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 \\0 & -1 & 0 & 0 & 0 & 0 \\0 & 0 & -1 & 0& 0 & 0 \\0 & 0 & 0 & 1 & 0 & 0 \\0 & 0 & 0 & 0 & -1 & 0 \\0 & 0 & 0 & 0 & 0 & -1 \\\end{bmatrix}, \begin{bmatrix}\zeta_{6}^{1} & 0 & 0 & 0 & 0 & 0 \\0 & \zeta_{6}^{1} & 0 & 0 & 0 & 0 \\0 & 0 & \zeta_{3}^{2} & 0 & 0 & 0 \\0 & 0 & 0 & \zeta_{6}^{5} & 0 & 0 \\0 & 0 & 0 & 0 & \zeta_{6}^{5} & 0 \\0 & 0 & 0 & 0 & 0 & \zeta_{3}^{1} \\\end{bmatrix}$

Subgroups and supergroups

Maximal subgroups:	$A(1,2)$${}^{\times 3}$
Minimal supergroups:	$A(2,4)$, $J(A(2,2))$, $J_s(A(2,2))$, $B(3,2)$, $A(6,2)$, $B(1,4)_2$${}^{\times 2}$, $C(2,2)$, $A(2,6)$

Moment sequences

$x$	$\mathrm{E}[x^{0}]$	$\mathrm{E}[x^{1}]$	$\mathrm{E}[x^{2}]$	$\mathrm{E}[x^{3}]$	$\mathrm{E}[x^{4}]$	$\mathrm{E}[x^{5}]$	$\mathrm{E}[x^{6}]$	$\mathrm{E}[x^{7}]$	$\mathrm{E}[x^{8}]$	$\mathrm{E}[x^{9}]$	$\mathrm{E}[x^{10}]$	$\mathrm{E}[x^{11}]$	$\mathrm{E}[x^{12}]$
$a_1$	$1$	$0$	$6$	$0$	$126$	$0$	$3660$	$0$	$114870$	$0$	$3720276$	$0$	$122763564$
$a_2$	$1$	$3$	$27$	$309$	$3963$	$53073$	$727101$	$10105875$	$141907059$	$2008185033$	$28592728257$	$409122034335$	$5877905052117$
$a_3$	$1$	$0$	$44$	$0$	$11820$	$0$	$3864140$	$0$	$1340497564$	$0$	$479997013104$	$0$	$175364875326036$

Moment simplex

$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=2\right)\colon$	$3$	$6$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=4\right)\colon$	$27$	$12$	$54$	$126$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=6\right)\colon$	$44$	$309$	$168$	$690$	$372$	$1578$	$3660$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=8\right)\colon$	$516$	$3963$	$2160$	$1194$	$9114$	$4986$	$21144$	$11520$	$49200$	$114870$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=10\right)\colon$	$6816$	$53073$	$3708$	$28920$	$15774$	$123570$	$67308$	$36732$	$288684$	$157116$	$675792$	$367332$	$1584450$	$3720276$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=12\right)\colon$	$11820$	$92112$	$727101$	$50208$	$395244$	$215022$	$1703682$	$116976$	$925470$	$502980$	$3998934$	$2170584$	$1178940$	$9398628$
$$	$5097960$	$22113882$	$11986632$	$52081092$	$122763564$

Moment matrix

$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&2&0&3&0&4&12&0&10&0&15&0&0&40\\0&6&0&6&0&36&0&0&66&0&30&0&90&162&0\\2&0&22&0&24&0&86&102&0&80&0&198&0&0&500\\0&6&0&26&0&76&0&0&126&0&50&0&240&332&0\\3&0&24&0&45&0&108&150&0&144&0&285&0&0&720\\0&36&0&76&0&320&0&0&576&0&256&0&936&1480&0\\4&0&86&0&108&0&382&432&0&364&0&900&0&0&2272\\12&0&102&0&150&0&432&582&0&516&0&1110&0&0&2844\\0&66&0&126&0&576&0&0&1062&0&480&0&1686&2736&0\\10&0&80&0&144&0&364&516&0&492&0&984&0&0&2528\\0&30&0&50&0&256&0&0&480&0&224&0&744&1232&0\\15&0&198&0&285&0&900&1110&0&984&0&2259&0&0&5760\\0&90&0&240&0&936&0&0&1686&0&744&0&2928&4464&0\\0&162&0&332&0&1480&0&0&2736&0&1232&0&4464&7154&0\\40&0&500&0&720&0&2272&2844&0&2528&0&5760&0&0&14784\end{bmatrix}$

$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&6&22&26&45&320&382&582&1062&492&224&2259&2928&7154&14784&7782&7911&19760&17520&5350\end{bmatrix}$

Event probabilities

$\mathrm{Pr}[a_i=n]=0$ for $i=1,2,3$ and $n\in\mathbb{Z}$.