Galois orbit of Artin representations 4.2e14_5e3.8t21.4

• Label: 4.2e14_5e3.8t21.4
• Dimension: 4
• Group: $C_2^3 : C_4 $
• Conductor: $ 2^{14} \cdot 5^{3}$
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$4$
Group:	$C_2^3 : C_4 $
Conductor:	$2048000= 2^{14} \cdot 5^{3} $
Artin number field:	Splitting field of $f= x^{8} - 4 x^{6} + 10 x^{4} + 20 x^{2} + 25 $ 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 8.

Roots:

$r_{ 1 }$	$=$	$ 8 + 36\cdot 149 + 40\cdot 149^{2} + 22\cdot 149^{3} + 120\cdot 149^{4} + 120\cdot 149^{5} + 61\cdot 149^{6} + 3\cdot 149^{7} +O\left(149^{ 8 }\right)$
$r_{ 2 }$	$=$	$ 17 + 47\cdot 149 + 148\cdot 149^{2} + 18\cdot 149^{3} + 74\cdot 149^{4} + 102\cdot 149^{5} + 92\cdot 149^{6} + 145\cdot 149^{7} +O\left(149^{ 8 }\right)$
$r_{ 3 }$	$=$	$ 27 + 76\cdot 149 + 62\cdot 149^{2} + 25\cdot 149^{3} + 126\cdot 149^{4} + 93\cdot 149^{5} + 56\cdot 149^{6} + 26\cdot 149^{7} +O\left(149^{ 8 }\right)$
$r_{ 4 }$	$=$	$ 73 + 21\cdot 149 + 60\cdot 149^{2} + 81\cdot 149^{3} + 20\cdot 149^{4} + 141\cdot 149^{5} + 137\cdot 149^{6} + 100\cdot 149^{7} +O\left(149^{ 8 }\right)$
$r_{ 5 }$	$=$	$ 76 + 127\cdot 149 + 88\cdot 149^{2} + 67\cdot 149^{3} + 128\cdot 149^{4} + 7\cdot 149^{5} + 11\cdot 149^{6} + 48\cdot 149^{7} +O\left(149^{ 8 }\right)$
$r_{ 6 }$	$=$	$ 122 + 72\cdot 149 + 86\cdot 149^{2} + 123\cdot 149^{3} + 22\cdot 149^{4} + 55\cdot 149^{5} + 92\cdot 149^{6} + 122\cdot 149^{7} +O\left(149^{ 8 }\right)$
$r_{ 7 }$	$=$	$ 132 + 101\cdot 149 + 130\cdot 149^{3} + 74\cdot 149^{4} + 46\cdot 149^{5} + 56\cdot 149^{6} + 3\cdot 149^{7} +O\left(149^{ 8 }\right)$
$r_{ 8 }$	$=$	$ 141 + 112\cdot 149 + 108\cdot 149^{2} + 126\cdot 149^{3} + 28\cdot 149^{4} + 28\cdot 149^{5} + 87\cdot 149^{6} + 145\cdot 149^{7} +O\left(149^{ 8 }\right)$

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

Cycle notation

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

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

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

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

$(2,7)(3,6)$

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.