Galois orbit of Artin representations 4.2e4_3e4_5e2_11e2.8t22.8

• Label: 4.2e4_3e4_5e2_11e2.8t22.8
• Dimension: 4
• Group: $C_2^3 : D_4 $
• Conductor: $ 2^{4} \cdot 3^{4} \cdot 5^{2} \cdot 11^{2}$
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$4$
Group:	$C_2^3 : D_4 $
Conductor:	$3920400= 2^{4} \cdot 3^{4} \cdot 5^{2} \cdot 11^{2} $
Artin number field:	Splitting field of $f= x^{8} - 3 x^{7} + 4 x^{6} - 9 x^{5} + 30 x^{4} - 27 x^{3} + 25 x^{2} - 126 x + 124 $ 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:

$r_{ 1 }$	$=$	$ 28 + 3\cdot 181 + 29\cdot 181^{2} + 71\cdot 181^{3} + 118\cdot 181^{4} +O\left(181^{ 5 }\right)$
$r_{ 2 }$	$=$	$ 44 + 90\cdot 181 + 180\cdot 181^{2} + 125\cdot 181^{3} + 123\cdot 181^{4} +O\left(181^{ 5 }\right)$
$r_{ 3 }$	$=$	$ 57 + 155\cdot 181 + 116\cdot 181^{2} + 146\cdot 181^{3} + 110\cdot 181^{4} +O\left(181^{ 5 }\right)$
$r_{ 4 }$	$=$	$ 64 + 135\cdot 181 + 52\cdot 181^{2} + 84\cdot 181^{3} + 37\cdot 181^{4} +O\left(181^{ 5 }\right)$
$r_{ 5 }$	$=$	$ 66 + 160\cdot 181 + 108\cdot 181^{2} + 181^{3} + 19\cdot 181^{4} +O\left(181^{ 5 }\right)$
$r_{ 6 }$	$=$	$ 67 + 180\cdot 181 + 103\cdot 181^{2} + 181^{3} + 41\cdot 181^{4} +O\left(181^{ 5 }\right)$
$r_{ 7 }$	$=$	$ 94 + 145\cdot 181 + 135\cdot 181^{2} + 115\cdot 181^{3} + 60\cdot 181^{4} +O\left(181^{ 5 }\right)$
$r_{ 8 }$	$=$	$ 126 + 34\cdot 181 + 177\cdot 181^{2} + 176\cdot 181^{3} + 31\cdot 181^{4} +O\left(181^{ 5 }\right)$

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

Cycle notation

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

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

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

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

$(1,6)(7,8)$

$(1,6)(4,5)$

$(1,6)(2,3)$

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.