Galois orbit of Artin representations 4.3e4_5e4_31e2.8t16.4

• Label: 4.3e4_5e4_31e2.8t16.4
• Dimension: 4
• Group: $(C_8:C_2):C_2$
• Conductor: $ 3^{4} \cdot 5^{4} \cdot 31^{2}$
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$4$
Group:	$(C_8:C_2):C_2$
Conductor:	$48650625= 3^{4} \cdot 5^{4} \cdot 31^{2} $
Artin number field:	Splitting field of $f= x^{8} - 2 x^{7} + 23 x^{6} - 29 x^{5} + 220 x^{4} - 124 x^{3} + 1013 x^{2} - 142 x + 1801 $ over $\Q$
Size of Galois orbit:	1
Smallest containing permutation representation:	$(C_8:C_2):C_2$
Parity:	Even

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 191 }$ to precision 7.

Roots:

$r_{ 1 }$	$=$	$ 7 + 60\cdot 191 + 16\cdot 191^{2} + 146\cdot 191^{3} + 152\cdot 191^{4} + 190\cdot 191^{5} + 11\cdot 191^{6} +O\left(191^{ 7 }\right)$
$r_{ 2 }$	$=$	$ 29 + 182\cdot 191 + 182\cdot 191^{2} + 86\cdot 191^{3} + 123\cdot 191^{4} + 4\cdot 191^{5} + 3\cdot 191^{6} +O\left(191^{ 7 }\right)$
$r_{ 3 }$	$=$	$ 74 + 46\cdot 191 + 112\cdot 191^{2} + 187\cdot 191^{3} + 47\cdot 191^{4} + 36\cdot 191^{5} + 122\cdot 191^{6} +O\left(191^{ 7 }\right)$
$r_{ 4 }$	$=$	$ 82 + 93\cdot 191 + 70\cdot 191^{2} + 152\cdot 191^{3} + 57\cdot 191^{4} + 150\cdot 191^{5} + 53\cdot 191^{6} +O\left(191^{ 7 }\right)$
$r_{ 5 }$	$=$	$ 105 + 83\cdot 191 + 153\cdot 191^{2} + 163\cdot 191^{3} + 7\cdot 191^{4} + 59\cdot 191^{5} + 53\cdot 191^{6} +O\left(191^{ 7 }\right)$
$r_{ 6 }$	$=$	$ 129 + 2\cdot 191 + 60\cdot 191^{2} + 25\cdot 191^{3} + 152\cdot 191^{4} + 5\cdot 191^{5} + 88\cdot 191^{6} +O\left(191^{ 7 }\right)$
$r_{ 7 }$	$=$	$ 151 + 150\cdot 191 + 26\cdot 191^{2} + 82\cdot 191^{3} + 58\cdot 191^{4} + 144\cdot 191^{5} + 168\cdot 191^{6} +O\left(191^{ 7 }\right)$
$r_{ 8 }$	$=$	$ 189 + 144\cdot 191 + 141\cdot 191^{2} + 110\cdot 191^{3} + 163\cdot 191^{4} + 172\cdot 191^{5} + 71\cdot 191^{6} +O\left(191^{ 7 }\right)$

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

Cycle notation

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

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

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.