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

• Label: 1.6.N.8.3a
• Name: $J_s(A(2,2))$
• Weight: $1$
• Degree: $6$
• Real dimension: $1$
• Components: $8$
• Contained in: $\mathrm{USp}(6)$
• Identity component: $\mathrm{U}(1)_3$
• Component group: $D_4$

Invariants

Weight:	$1$
Degree:	$6$
$\mathbb{R}$-dimension:	$1$
Components:	$8$
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:	$D_4$
Order:	$8$
Abelian:	no
Generators:	$\begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 \\0 & -1 & 0 & 0 & 0 & 0 \\0 & 0 & -1 & 0& 0 & 0 \\0 & 0 & 0 & 1 & 0 & 0 \\0 & 0 & 0 & 0 & -1 & 0 \\0 & 0 & 0 & 0 & 0 & -1 \\\end{bmatrix}, \begin{bmatrix}\zeta_{6}^{1} & 0 & 0 & 0 & 0 & 0 \\0 & \zeta_{6}^{1} & 0 & 0 & 0 & 0 \\0 & 0 & \zeta_{3}^{2} & 0 & 0 & 0 \\0 & 0 & 0 & \zeta_{6}^{5} & 0 & 0 \\0 & 0 & 0 & 0 & \zeta_{6}^{5} & 0 \\0 & 0 & 0 & 0 & 0 & \zeta_{3}^{1} \\\end{bmatrix}, \begin{bmatrix}0 & 0 & 0 & -1 & 0 & 0 \\0 & 0 & 0 & 0 & 0 & -1 \\0 & 0 & 0 & 0 & -1 & 0 \\1 & 0 & 0 & 0 & 0 & 0 \\0 & 0 & 1 & 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\\end{bmatrix}$

Subgroups and supergroups

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

Moment sequences

$x$	$\mathrm{E}[x^{0}]$	$\mathrm{E}[x^{1}]$	$\mathrm{E}[x^{2}]$	$\mathrm{E}[x^{3}]$	$\mathrm{E}[x^{4}]$	$\mathrm{E}[x^{5}]$	$\mathrm{E}[x^{6}]$	$\mathrm{E}[x^{7}]$	$\mathrm{E}[x^{8}]$	$\mathrm{E}[x^{9}]$	$\mathrm{E}[x^{10}]$	$\mathrm{E}[x^{11}]$	$\mathrm{E}[x^{12}]$
$a_1$	$1$	$0$	$3$	$0$	$63$	$0$	$1830$	$0$	$57435$	$0$	$1860138$	$0$	$61381782$
$a_2$	$1$	$2$	$16$	$161$	$2002$	$26597$	$363733$	$5053484$	$70955170$	$1004097437$	$14296378891$	$204561061454$	$2938952658919$
$a_3$	$1$	$0$	$22$	$0$	$5910$	$0$	$1932070$	$0$	$670248782$	$0$	$239998506552$	$0$	$87682437663018$

Moment simplex

$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=2\right)\colon$	$2$	$3$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=4\right)\colon$	$16$	$6$	$27$	$63$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=6\right)\colon$	$22$	$161$	$84$	$345$	$186$	$789$	$1830$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=8\right)\colon$	$258$	$2002$	$1080$	$597$	$4557$	$2493$	$10572$	$5760$	$24600$	$57435$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=10\right)\colon$	$3408$	$26597$	$1854$	$14460$	$7887$	$61785$	$33654$	$18366$	$144342$	$78558$	$337896$	$183666$	$792225$	$1860138$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=12\right)\colon$	$5910$	$46056$	$363733$	$25104$	$197622$	$107511$	$851841$	$58488$	$462735$	$251490$	$1999467$	$1085292$	$589470$	$4699314$
$$	$2548980$	$11056941$	$5993316$	$26040546$	$61381782$

Moment matrix

$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&1&0&1&0&2&8&0&3&0&6&0&0&20\\0&3&0&3&0&18&0&0&33&0&15&0&45&81&0\\1&0&13&0&10&0&41&53&0&40&0&99&0&0&250\\0&3&0&13&0&38&0&0&63&0&25&0&120&166&0\\1&0&10&0&25&0&56&71&0&74&0&144&0&0&360\\0&18&0&38&0&160&0&0&288&0&128&0&468&740&0\\2&0&41&0&56&0&193&214&0&182&0&450&0&0&1136\\8&0&53&0&71&0&214&301&0&250&0&549&0&0&1422\\0&33&0&63&0&288&0&0&531&0&240&0&843&1368&0\\3&0&40&0&74&0&182&250&0&254&0&498&0&0&1264\\0&15&0&25&0&128&0&0&240&0&112&0&372&616&0\\6&0&99&0&144&0&450&549&0&498&0&1134&0&0&2880\\0&45&0&120&0&468&0&0&843&0&372&0&1464&2232&0\\0&81&0&166&0&740&0&0&1368&0&616&0&2232&3577&0\\20&0&250&0&360&0&1136&1422&0&1264&0&2880&0&0&7392\end{bmatrix}$

$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&3&13&13&25&160&193&301&531&254&112&1134&1464&3577&7392&3891&3978&9880&8778&2675\end{bmatrix}$

Event probabilities

	$-$	$a_2\in\mathbb{Z}$	$a_2=-1$	$a_2=0$	$a_2=1$	$a_2=2$	$a_2=3$
$-$	$1$	$1/2$	$1/4$	$0$	$0$	$0$	$1/4$
$a_1=0$	$1/2$	$1/2$	$1/4$	$0$	$0$	$0$	$1/4$
$a_3=0$	$1/2$	$1/2$	$1/4$	$0$	$0$	$0$	$1/4$
$a_1=a_3=0$	$1/2$	$1/2$	$1/4$	$0$	$0$	$0$	$1/4$