Galois orbit of Artin representations 4.2e8_17e3.8t21.3

• Label: 4.2e8_17e3.8t21.3
• Dimension: 4
• Group: $C_2^3 : C_4 $
• Conductor: $ 2^{8} \cdot 17^{3}$
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$4$
Group:	$C_2^3 : C_4 $
Conductor:	$1257728= 2^{8} \cdot 17^{3} $
Artin number field:	Splitting field of $f= x^{8} - 2 x^{7} + 12 x^{5} - 2 x^{4} - 16 x^{3} + 24 x^{2} + 48 x + 20 $ over $\Q$
Size of Galois orbit:	1
Smallest containing permutation representation:	$C_2^3: C_4$
Parity:	Even

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 149 }$ to precision 6.

Roots:

$r_{ 1 }$	$=$	$ 44 + 74\cdot 149 + 88\cdot 149^{2} + 88\cdot 149^{3} + 11\cdot 149^{4} + 31\cdot 149^{5} +O\left(149^{ 6 }\right)$
$r_{ 2 }$	$=$	$ 51 + 123\cdot 149 + 25\cdot 149^{2} + 119\cdot 149^{3} + 9\cdot 149^{4} + 124\cdot 149^{5} +O\left(149^{ 6 }\right)$
$r_{ 3 }$	$=$	$ 73 + 7\cdot 149 + 48\cdot 149^{2} + 139\cdot 149^{3} + 126\cdot 149^{4} + 46\cdot 149^{5} +O\left(149^{ 6 }\right)$
$r_{ 4 }$	$=$	$ 87 + 104\cdot 149 + 101\cdot 149^{2} + 7\cdot 149^{3} + 14\cdot 149^{4} + 81\cdot 149^{5} +O\left(149^{ 6 }\right)$
$r_{ 5 }$	$=$	$ 91 + 103\cdot 149 + 10\cdot 149^{2} + 95\cdot 149^{3} + 72\cdot 149^{4} + 90\cdot 149^{5} +O\left(149^{ 6 }\right)$
$r_{ 6 }$	$=$	$ 129 + 139\cdot 149 + 124\cdot 149^{2} + 73\cdot 149^{3} + 137\cdot 149^{4} + 121\cdot 149^{5} +O\left(149^{ 6 }\right)$
$r_{ 7 }$	$=$	$ 130 + 71\cdot 149 + 57\cdot 149^{2} + 136\cdot 149^{3} + 99\cdot 149^{4} + 132\cdot 149^{5} +O\left(149^{ 6 }\right)$
$r_{ 8 }$	$=$	$ 142 + 119\cdot 149 + 138\cdot 149^{2} + 84\cdot 149^{3} + 123\cdot 149^{4} + 116\cdot 149^{5} +O\left(149^{ 6 }\right)$

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

Cycle notation

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

$(5,8)(6,7)$

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

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.