Sato-Tate group J(A(1,4)1)J(A(1,4)_1) of weight 1 and degree 6

• Label: 1.6.N.8.3d
• Name: $J(A(1,4)_1)$
• 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:	$\begin{bmatrix}\zeta_{6}^{1} & 0 & 0 & 0 & 0 & 0 \\0 & \zeta_{12}^{5} & 0 & 0 & 0 & 0 \\0 & 0 & \zeta_{12}^{5} & 0 & 0 & 0 \\0 & 0 & 0 & \zeta_{6}^{5} & 0 & 0\\0 & 0 & 0 & 0 & \zeta_{12}^{7} & 0 \\0 & 0 & 0 & 0 & 0 & \zeta_{12}^{7} \\\end{bmatrix}, \begin{bmatrix}0 & 0 & 0 & 1 & 0 & 0 \\0 & 0 & 0 & 0 & 1 & 0 \\0 & 0 & 0 & 0 & 0 & 1 \\-1 & 0 & 0 & 0 & 0 & 0 \\0 & -1 & 0 & 0 & 0 & 0 \\0 & 0 & -1 & 0 & 0 & 0 \\\end{bmatrix}$

Subgroups and supergroups

Maximal subgroups:	$A(1,4)_1$, $J(A(1,2))$${}^{\times 2}$
Minimal supergroups:	$J(A(2,4))$${}^{\times 2}$, $J_s(A(2,4))$, $J(A(3,4))$, $J_s(A(3,4))$, $J(A(1,8)_2)$${}^{\times 2}$, $J_s(A(1,8)_2)$

Moment sequences

$x$	$\mathrm{E}[x^{0}]$	$\mathrm{E}[x^{1}]$	$\mathrm{E}[x^{2}]$	$\mathrm{E}[x^{3}]$	$\mathrm{E}[x^{4}]$	$\mathrm{E}[x^{5}]$	$\mathrm{E}[x^{6}]$	$\mathrm{E}[x^{7}]$	$\mathrm{E}[x^{8}]$	$\mathrm{E}[x^{9}]$	$\mathrm{E}[x^{10}]$	$\mathrm{E}[x^{11}]$	$\mathrm{E}[x^{12}]$
$a_1$	$1$	$0$	$5$	$0$	$99$	$0$	$2450$	$0$	$68355$	$0$	$2056950$	$0$	$64990926$
$a_2$	$1$	$4$	$26$	$235$	$2586$	$31489$	$405911$	$5423464$	$74237002$	$1033441417$	$14560357821$	$206947415734$	$2960611792863$
$a_3$	$1$	$0$	$34$	$0$	$7222$	$0$	$2086690$	$0$	$689426766$	$0$	$242452083624$	$0$	$88002695271370$

Moment simplex

$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=2\right)\colon$	$4$	$5$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=4\right)\colon$	$26$	$12$	$45$	$99$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=6\right)\colon$	$34$	$235$	$126$	$487$	$276$	$1087$	$2450$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=8\right)\colon$	$362$	$2586$	$1428$	$807$	$5739$	$3215$	$13042$	$7280$	$29790$	$68355$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=10\right)\colon$	$4272$	$31489$	$2388$	$17404$	$9683$	$71871$	$39790$	$22090$	$165432$	$91386$	$382148$	$210644$	$885423$	$2056950$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=12\right)\colon$	$7222$	$53476$	$405911$	$29602$	$223068$	$122933$	$939463$	$67872$	$515807$	$283682$	$2182723$	$1196356$	$656802$	$5083740$
$$	$2782094$	$11865693$	$6484296$	$27746838$	$64990926$

Moment matrix

$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&3&0&1&0&4&10&0&3&0&10&0&0&24\\0&5&0&7&0&28&0&0&41&0&21&0&61&97&0\\3&0&19&0&18&0&55&73&0&46&0&123&0&0&284\\0&7&0&15&0&52&0&0&81&0&37&0&132&196&0\\1&0&18&0&35&0&78&83&0&80&0&182&0&0&400\\0&28&0&52&0&208&0&0&336&0&160&0&536&832&0\\4&0&55&0&78&0&225&256&0&218&0&518&0&0&1232\\10&0&73&0&83&0&256&345&0&260&0&619&0&0&1524\\0&41&0&81&0&336&0&0&585&0&264&0&929&1464&0\\3&0&46&0&80&0&218&260&0&252&0&532&0&0&1328\\0&21&0&37&0&160&0&0&264&0&132&0&426&672&0\\10&0&123&0&182&0&518&619&0&532&0&1262&0&0&3040\\0&61&0&132&0&536&0&0&929&0&426&0&1540&2384&0\\0&97&0&196&0&832&0&0&1464&0&672&0&2384&3763&0\\24&0&284&0&400&0&1232&1524&0&1328&0&3040&0&0&7664\end{bmatrix}$

$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&5&19&15&35&208&225&345&585&252&132&1262&1540&3763&7664&3949&4080&9988&8860&2613\end{bmatrix}$

Event probabilities

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