Galois orbit of Artin representations 1.11_17.8t1.1

• Label: 1.11_17.8t1.1
• Dimension: 1
• Group: $C_8$
• Conductor: $ 11 \cdot 17 $
• Frobenius-Schur indicator: 0

Basic invariants

Dimension:	$1$
Group:	$C_8$
Conductor:	$187= 11 \cdot 17 $
Artin number field:	Splitting field of $f= x^{8} - x^{7} + 44 x^{6} - 45 x^{5} + 423 x^{4} - 877 x^{3} + 1826 x^{2} - 3515 x + 4591 $ over $\Q$
Size of Galois orbit:	4
Smallest containing permutation representation:	$C_8$
Parity:	Odd

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 137 }$ to precision 5.

Roots:

$r_{ 1 }$	$=$	$ 12 + 41\cdot 137 + 101\cdot 137^{2} + 69\cdot 137^{3} + 38\cdot 137^{4} +O\left(137^{ 5 }\right)$
$r_{ 2 }$	$=$	$ 19 + 79\cdot 137 + 34\cdot 137^{2} + 124\cdot 137^{3} + 54\cdot 137^{4} +O\left(137^{ 5 }\right)$
$r_{ 3 }$	$=$	$ 66 + 32\cdot 137 + 113\cdot 137^{2} + 89\cdot 137^{3} + 86\cdot 137^{4} +O\left(137^{ 5 }\right)$
$r_{ 4 }$	$=$	$ 72 + 73\cdot 137 + 36\cdot 137^{2} + 119\cdot 137^{3} + 94\cdot 137^{4} +O\left(137^{ 5 }\right)$
$r_{ 5 }$	$=$	$ 75 + 115\cdot 137 + 119\cdot 137^{2} + 84\cdot 137^{3} + 39\cdot 137^{4} +O\left(137^{ 5 }\right)$
$r_{ 6 }$	$=$	$ 94 + 91\cdot 137 + 95\cdot 137^{2} + 34\cdot 137^{3} + 53\cdot 137^{4} +O\left(137^{ 5 }\right)$
$r_{ 7 }$	$=$	$ 105 + 55\cdot 137 + 110\cdot 137^{2} + 104\cdot 137^{3} + 70\cdot 137^{4} +O\left(137^{ 5 }\right)$
$r_{ 8 }$	$=$	$ 106 + 58\cdot 137 + 73\cdot 137^{2} + 57\cdot 137^{3} + 109\cdot 137^{4} +O\left(137^{ 5 }\right)$

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

Cycle notation

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

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

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

Character values on conjugacy classes

Size	Order	Action on $r_1, \ldots, r_{ 8 }$	Character values
			$c1$	$c2$	$c3$	$c4$
$1$	$1$	$()$	$1$	$1$	$1$	$1$
$1$	$2$	$(1,5)(2,3)(4,6)(7,8)$	$-1$	$-1$	$-1$	$-1$
$1$	$4$	$(1,7,5,8)(2,4,3,6)$	$\zeta_{8}^{2}$	$-\zeta_{8}^{2}$	$\zeta_{8}^{2}$	$-\zeta_{8}^{2}$
$1$	$4$	$(1,8,5,7)(2,6,3,4)$	$-\zeta_{8}^{2}$	$\zeta_{8}^{2}$	$-\zeta_{8}^{2}$	$\zeta_{8}^{2}$
$1$	$8$	$(1,3,7,6,5,2,8,4)$	$\zeta_{8}$	$\zeta_{8}^{3}$	$-\zeta_{8}$	$-\zeta_{8}^{3}$
$1$	$8$	$(1,6,8,3,5,4,7,2)$	$\zeta_{8}^{3}$	$\zeta_{8}$	$-\zeta_{8}^{3}$	$-\zeta_{8}$
$1$	$8$	$(1,2,7,4,5,3,8,6)$	$-\zeta_{8}$	$-\zeta_{8}^{3}$	$\zeta_{8}$	$\zeta_{8}^{3}$
$1$	$8$	$(1,4,8,2,5,6,7,3)$	$-\zeta_{8}^{3}$	$-\zeta_{8}$	$\zeta_{8}^{3}$	$\zeta_{8}$

The blue line marks the conjugacy class containing complex conjugation.