Galois orbit of Artin representations 4.2e4_3e2_5e2_11e2.8t22.10

• Label: 4.2e4_3e2_5e2_11e2.8t22.10
• Dimension: 4
• Group: $C_2^3 : D_4 $
• Conductor: $ 2^{4} \cdot 3^{2} \cdot 5^{2} \cdot 11^{2}$
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$4$
Group:	$C_2^3 : D_4 $
Conductor:	$435600= 2^{4} \cdot 3^{2} \cdot 5^{2} \cdot 11^{2} $
Artin number field:	Splitting field of $f= x^{8} + 5 x^{6} - 2 x^{5} + 18 x^{4} - 14 x^{3} + 30 x^{2} - 14 x + 13 $ over $\Q$
Size of Galois orbit:	1
Smallest containing permutation representation:	$C_2^3 : D_4 $
Parity:	Even

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 181 }$ to precision 5.

Roots:

$r_{ 1 }$	$=$	$ 3 + 20\cdot 181 + 59\cdot 181^{2} + 145\cdot 181^{3} + 94\cdot 181^{4} +O\left(181^{ 5 }\right)$
$r_{ 2 }$	$=$	$ 23 + 154\cdot 181 + 160\cdot 181^{2} + 108\cdot 181^{3} + 154\cdot 181^{4} +O\left(181^{ 5 }\right)$
$r_{ 3 }$	$=$	$ 57 + 160\cdot 181 + 39\cdot 181^{2} + 129\cdot 181^{3} + 40\cdot 181^{4} +O\left(181^{ 5 }\right)$
$r_{ 4 }$	$=$	$ 70 + 72\cdot 181 + 38\cdot 181^{2} + 165\cdot 181^{3} + 102\cdot 181^{4} +O\left(181^{ 5 }\right)$
$r_{ 5 }$	$=$	$ 76 + 11\cdot 181 + 93\cdot 181^{2} + 63\cdot 181^{3} + 83\cdot 181^{4} +O\left(181^{ 5 }\right)$
$r_{ 6 }$	$=$	$ 79 + 176\cdot 181 + 48\cdot 181^{2} + 44\cdot 181^{3} + 29\cdot 181^{4} +O\left(181^{ 5 }\right)$
$r_{ 7 }$	$=$	$ 117 + 46\cdot 181 + 40\cdot 181^{2} + 26\cdot 181^{3} + 42\cdot 181^{4} +O\left(181^{ 5 }\right)$
$r_{ 8 }$	$=$	$ 118 + 82\cdot 181 + 62\cdot 181^{2} + 41\cdot 181^{3} + 176\cdot 181^{4} +O\left(181^{ 5 }\right)$

Generators of the action on the roots $r_1, \ldots, r_{ 8 }$

Cycle notation

$(1,4)(2,8)(3,5)(6,7)$

$(1,5)(2,6)(3,8)(4,7)$

$(1,2)(4,8)$

$(1,8)(2,4)(3,5)(6,7)$

$(1,8,2,4)(3,5,7,6)$

$(1,5)(2,6)(3,4)(7,8)$

$(1,8,2,4)(3,6,7,5)$

Character values on conjugacy classes

Size	Order	Action on $r_1, \ldots, r_{ 8 }$	Character values
			$c1$
$1$	$1$	$()$	$4$
$1$	$2$	$(1,2)(3,7)(4,8)(5,6)$	$-4$
$2$	$2$	$(1,4)(2,8)(3,5)(6,7)$	$0$
$2$	$2$	$(1,5)(2,6)(3,8)(4,7)$	$0$
$2$	$2$	$(1,5)(2,6)(3,4)(7,8)$	$0$
$2$	$2$	$(3,7)(4,8)$	$0$
$2$	$2$	$(1,2)(4,8)$	$0$
$2$	$2$	$(1,8)(2,4)(3,5)(6,7)$	$0$
$2$	$2$	$(1,3)(2,7)(4,6)(5,8)$	$0$
$2$	$2$	$(1,2)(3,7)$	$0$
$2$	$2$	$(1,3)(2,7)(4,5)(6,8)$	$0$
$2$	$4$	$(1,8,2,4)(3,6,7,5)$	$0$
$2$	$4$	$(1,7,2,3)(4,5,8,6)$	$0$
$2$	$4$	$(1,6,2,5)(3,8,7,4)$	$0$
$2$	$4$	$(1,6,2,5)(3,4,7,8)$	$0$
$2$	$4$	$(1,4,2,8)(3,6,7,5)$	$0$
$2$	$4$	$(1,7,2,3)(4,6,8,5)$	$0$

The blue line marks the conjugacy class containing complex conjugation.