Galois orbit of Artin representations 4.2e6_3e2_5e2_7e2.8t22.9

• Label: 4.2e6_3e2_5e2_7e2.8t22.9
• Dimension: 4
• Group: $C_2^3 : D_4 $
• Conductor: $ 2^{6} \cdot 3^{2} \cdot 5^{2} \cdot 7^{2}$
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$4$
Group:	$C_2^3 : D_4 $
Conductor:	$705600= 2^{6} \cdot 3^{2} \cdot 5^{2} \cdot 7^{2} $
Artin number field:	Splitting field of $f= x^{8} - 2 x^{7} + 7 x^{6} - 20 x^{5} + 42 x^{4} - 58 x^{3} + 52 x^{2} - 28 x + 7 $ 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 $\Q_{ 151 }$ to precision 5.

Roots:

$r_{ 1 }$	$=$	$ 23 + 118\cdot 151 + 28\cdot 151^{2} + 95\cdot 151^{3} + 151^{4} +O\left(151^{ 5 }\right)$
$r_{ 2 }$	$=$	$ 24 + 2\cdot 151 + 113\cdot 151^{2} + 87\cdot 151^{3} + 70\cdot 151^{4} +O\left(151^{ 5 }\right)$
$r_{ 3 }$	$=$	$ 47 + 127\cdot 151 + 5\cdot 151^{2} + 106\cdot 151^{3} + 62\cdot 151^{4} +O\left(151^{ 5 }\right)$
$r_{ 4 }$	$=$	$ 66 + 106\cdot 151 + 39\cdot 151^{2} + 48\cdot 151^{3} + 121\cdot 151^{4} +O\left(151^{ 5 }\right)$
$r_{ 5 }$	$=$	$ 80 + 17\cdot 151 + 54\cdot 151^{2} + 40\cdot 151^{3} + 69\cdot 151^{4} +O\left(151^{ 5 }\right)$
$r_{ 6 }$	$=$	$ 104 + 44\cdot 151 + 130\cdot 151^{2} + 89\cdot 151^{3} + 108\cdot 151^{4} +O\left(151^{ 5 }\right)$
$r_{ 7 }$	$=$	$ 114 + 27\cdot 151 + 61\cdot 151^{2} + 84\cdot 151^{3} + 26\cdot 151^{4} +O\left(151^{ 5 }\right)$
$r_{ 8 }$	$=$	$ 148 + 8\cdot 151 + 20\cdot 151^{2} + 52\cdot 151^{3} + 143\cdot 151^{4} +O\left(151^{ 5 }\right)$

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

Cycle notation

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

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

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

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

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

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.