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

• Label: 1.6.N.4.2b
• Name: $J(A(1,2))$
• 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:	$\begin{bmatrix}\zeta_{3}^{1} & 0 & 0 & 0 & 0 & 0 \\0 & \zeta_{6}^{5} & 0 & 0 &0 & 0 \\0 & 0 & \zeta_{6}^{5} & 0 & 0 & 0 \\0 & 0 & 0 & \zeta_{3}^{2} & 0 & 0 \\0 & 0 & 0 & 0 & \zeta_{6}^{1} & 0 \\0 & 0 & 0 & 0 & 0 & \zeta_{6}^{1} \\\end{bmatrix}, \begin{bmatrix}0 & 0 & 0 & 1 & 0 & 0 \\0 & 0 & 0 & 0 & 1 & 0 \\0 & 0 & 0 & 0 & 0 & 1 \\-1 & 0 & 0 & 0 & 0 & 0 \\0 & -1 & 0 & 0 & 0 & 0 \\0 & 0 & -1 & 0 & 0 & 0 \\\end{bmatrix}$

Subgroups and supergroups

Maximal subgroups:	$J(A(1,1))$${}^{\times 2}$, $A(1,2)$
Minimal supergroups:	$J(A(1,4)_1)$${}^{\times 2}$, $J_s(A(1,4)_2)$, $J(A(1,6)_1)$, $J(A(3,2))$, $J(A(1,4)_2)$${}^{\times 2}$, $J(A(1,6)_2)$, $J(B(3,1))$, $J_s(A(2,2))$, $J(A(2,2))$${}^{\times 6}$, $J_s(A(3,2))$

Moment sequences

$x$	$\mathrm{E}[x^{0}]$	$\mathrm{E}[x^{1}]$	$\mathrm{E}[x^{2}]$	$\mathrm{E}[x^{3}]$	$\mathrm{E}[x^{4}]$	$\mathrm{E}[x^{5}]$	$\mathrm{E}[x^{6}]$	$\mathrm{E}[x^{7}]$	$\mathrm{E}[x^{8}]$	$\mathrm{E}[x^{9}]$	$\mathrm{E}[x^{10}]$	$\mathrm{E}[x^{11}]$	$\mathrm{E}[x^{12}]$
$a_1$	$1$	$0$	$5$	$0$	$123$	$0$	$3650$	$0$	$114835$	$0$	$3720150$	$0$	$122763102$
$a_2$	$1$	$4$	$30$	$319$	$3994$	$53169$	$727395$	$10106772$	$141909786$	$2008193305$	$28592753305$	$409122110082$	$5877905280943$
$a_3$	$1$	$0$	$42$	$0$	$11798$	$0$	$3863850$	$0$	$1340493518$	$0$	$479996954952$	$0$	$175364874474218$

Moment simplex

$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=2\right)\colon$	$4$	$5$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=4\right)\colon$	$30$	$13$	$54$	$123$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=6\right)\colon$	$42$	$319$	$167$	$689$	$376$	$1579$	$3650$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=8\right)\colon$	$517$	$3994$	$2160$	$1187$	$9113$	$4983$	$21142$	$11535$	$49205$	$114835$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=10\right)\colon$	$6814$	$53169$	$3720$	$28918$	$15779$	$123567$	$67310$	$36706$	$288684$	$157105$	$675786$	$367388$	$1584471$	$3720150$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=12\right)\colon$	$11798$	$92111$	$727395$	$50198$	$395241$	$215017$	$1703676$	$117022$	$925467$	$503000$	$3998931$	$2170593$	$1178842$	$9398631$
$$	$5097918$	$22113861$	$11986842$	$52081176$	$122763102$

Moment matrix

$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&3&0&1&0&5&13&0&6&0&14&0&0&40\\0&5&0&8&0&36&0&0&64&0&28&0&96&159&0\\3&0&23&0&23&0&82&109&0&81&0&195&0&0&500\\0&8&0&21&0&76&0&0&132&0&56&0&224&339&0\\1&0&23&0&45&0&117&137&0&138&0&290&0&0&720\\0&36&0&76&0&320&0&0&576&0&256&0&936&1480&0\\5&0&82&0&117&0&371&438&0&388&0&900&0&0&2272\\13&0&109&0&137&0&438&587&0&487&0&1105&0&0&2844\\0&64&0&132&0&576&0&0&1053&0&472&0&1708&2727&0\\6&0&81&0&138&0&388&487&0&462&0&994&0&0&2528\\0&28&0&56&0&256&0&0&472&0&216&0&764&1224&0\\14&0&195&0&290&0&900&1105&0&994&0&2262&0&0&5760\\0&96&0&224&0&936&0&0&1708&0&764&0&2872&4488&0\\0&159&0&339&0&1480&0&0&2727&0&1224&0&4488&7143&0\\40&0&500&0&720&0&2272&2844&0&2528&0&5760&0&0&14784\end{bmatrix}$

$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&5&23&21&45&320&371&587&1053&462&216&2262&2872&7143&14784&7703&7930&19640&17474&5175\end{bmatrix}$

Event probabilities

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