Galois orbit of Artin representations 4.2e8_3e2_5e3.8t21.4

• Label: 4.2e8_3e2_5e3.8t21.4
• Dimension: 4
• Group: $C_2^3 : C_4 $
• Conductor: $ 2^{8} \cdot 3^{2} \cdot 5^{3}$
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$4$
Group:	$C_2^3 : C_4 $
Conductor:	$288000= 2^{8} \cdot 3^{2} \cdot 5^{3} $
Artin number field:	Splitting field of $f= x^{8} - 2 x^{7} + 10 x^{6} - 20 x^{5} + 30 x^{4} - 52 x^{3} + 64 x^{2} - 40 x + 10 $ 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:

$r_{ 1 }$	$=$	$ 3 + 58\cdot 149 + 102\cdot 149^{2} + 99\cdot 149^{3} + 81\cdot 149^{4} + 81\cdot 149^{5} +O\left(149^{ 6 }\right)$
$r_{ 2 }$	$=$	$ 11 + 34\cdot 149 + 21\cdot 149^{2} + 122\cdot 149^{3} + 7\cdot 149^{4} + 90\cdot 149^{5} +O\left(149^{ 6 }\right)$
$r_{ 3 }$	$=$	$ 16 + 83\cdot 149 + 43\cdot 149^{2} + 105\cdot 149^{3} + 90\cdot 149^{4} + 131\cdot 149^{5} +O\left(149^{ 6 }\right)$
$r_{ 4 }$	$=$	$ 19 + 28\cdot 149 + 55\cdot 149^{2} + 48\cdot 149^{3} + 43\cdot 149^{4} + 42\cdot 149^{5} +O\left(149^{ 6 }\right)$
$r_{ 5 }$	$=$	$ 59 + 17\cdot 149 + 135\cdot 149^{2} + 144\cdot 149^{3} + 45\cdot 149^{4} + 74\cdot 149^{5} +O\left(149^{ 6 }\right)$
$r_{ 6 }$	$=$	$ 76 + 134\cdot 149 + 96\cdot 149^{2} + 27\cdot 149^{3} + 131\cdot 149^{4} + 128\cdot 149^{5} +O\left(149^{ 6 }\right)$
$r_{ 7 }$	$=$	$ 120 + 14\cdot 149 + 70\cdot 149^{2} + 31\cdot 149^{3} + 12\cdot 149^{4} + 79\cdot 149^{5} +O\left(149^{ 6 }\right)$
$r_{ 8 }$	$=$	$ 145 + 76\cdot 149 + 71\cdot 149^{2} + 16\cdot 149^{3} + 34\cdot 149^{4} + 117\cdot 149^{5} +O\left(149^{ 6 }\right)$

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

Cycle notation

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

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

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

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

$(4,5)(7,8)$

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.