Galois orbit of Artin representations 4.2e17_5e2.8t21.3

• Label: 4.2e17_5e2.8t21.3
• Dimension: 4
• Group: $C_2^3 : C_4 $
• Conductor: $ 2^{17} \cdot 5^{2}$
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$4$
Group:	$C_2^3 : C_4 $
Conductor:	$3276800= 2^{17} \cdot 5^{2} $
Artin number field:	Splitting field of $f= x^{8} - 4 x^{6} + 12 x^{4} + 40 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 $\Q_{ 137 }$ to precision 8.

Roots:

$r_{ 1 }$	$=$	$ 31 + 105\cdot 137 + 29\cdot 137^{2} + 29\cdot 137^{3} + 45\cdot 137^{4} + 72\cdot 137^{5} + 97\cdot 137^{6} + 51\cdot 137^{7} +O\left(137^{ 8 }\right)$
$r_{ 2 }$	$=$	$ 34 + 57\cdot 137 + 125\cdot 137^{2} + 18\cdot 137^{3} + 89\cdot 137^{4} + 111\cdot 137^{5} + 48\cdot 137^{6} + 53\cdot 137^{7} +O\left(137^{ 8 }\right)$
$r_{ 3 }$	$=$	$ 43 + 5\cdot 137 + 10\cdot 137^{2} + 49\cdot 137^{3} + 106\cdot 137^{4} + 66\cdot 137^{5} + 82\cdot 137^{6} + 91\cdot 137^{7} +O\left(137^{ 8 }\right)$
$r_{ 4 }$	$=$	$ 55 + 27\cdot 137 + 111\cdot 137^{2} + 131\cdot 137^{3} + 16\cdot 137^{4} + 72\cdot 137^{5} + 19\cdot 137^{6} + 108\cdot 137^{7} +O\left(137^{ 8 }\right)$
$r_{ 5 }$	$=$	$ 82 + 109\cdot 137 + 25\cdot 137^{2} + 5\cdot 137^{3} + 120\cdot 137^{4} + 64\cdot 137^{5} + 117\cdot 137^{6} + 28\cdot 137^{7} +O\left(137^{ 8 }\right)$
$r_{ 6 }$	$=$	$ 94 + 131\cdot 137 + 126\cdot 137^{2} + 87\cdot 137^{3} + 30\cdot 137^{4} + 70\cdot 137^{5} + 54\cdot 137^{6} + 45\cdot 137^{7} +O\left(137^{ 8 }\right)$
$r_{ 7 }$	$=$	$ 103 + 79\cdot 137 + 11\cdot 137^{2} + 118\cdot 137^{3} + 47\cdot 137^{4} + 25\cdot 137^{5} + 88\cdot 137^{6} + 83\cdot 137^{7} +O\left(137^{ 8 }\right)$
$r_{ 8 }$	$=$	$ 106 + 31\cdot 137 + 107\cdot 137^{2} + 107\cdot 137^{3} + 91\cdot 137^{4} + 64\cdot 137^{5} + 39\cdot 137^{6} + 85\cdot 137^{7} +O\left(137^{ 8 }\right)$

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

Cycle notation

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

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

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

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.