Galois orbit of Artin representations 1.2e3_17.8t1.1

• Label: 1.2e3_17.8t1.1
• Dimension: 1
• Group: $C_8$
• Conductor: $ 2^{3} \cdot 17 $
• Frobenius-Schur indicator: 0

Basic invariants

Dimension:	$1$
Group:	$C_8$
Conductor:	$136= 2^{3} \cdot 17 $
Artin number field:	Splitting field of $f= x^{8} - 34 x^{6} + 272 x^{4} - 680 x^{2} + 272 $ over $\Q$
Size of Galois orbit:	4
Smallest containing permutation representation:	$C_8$
Parity:	Even

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 103 }$ to precision 5.

Roots:

$r_{ 1 }$	$=$	$ 7 + 51\cdot 103 + 85\cdot 103^{2} + 74\cdot 103^{3} + 59\cdot 103^{4} +O\left(103^{ 5 }\right)$
$r_{ 2 }$	$=$	$ 40 + 26\cdot 103 + 42\cdot 103^{2} + 34\cdot 103^{3} + 99\cdot 103^{4} +O\left(103^{ 5 }\right)$
$r_{ 3 }$	$=$	$ 43 + 83\cdot 103 + 26\cdot 103^{2} + 63\cdot 103^{3} + 42\cdot 103^{4} +O\left(103^{ 5 }\right)$
$r_{ 4 }$	$=$	$ 48 + 29\cdot 103 + 14\cdot 103^{2} + 89\cdot 103^{3} + 59\cdot 103^{4} +O\left(103^{ 5 }\right)$
$r_{ 5 }$	$=$	$ 55 + 73\cdot 103 + 88\cdot 103^{2} + 13\cdot 103^{3} + 43\cdot 103^{4} +O\left(103^{ 5 }\right)$
$r_{ 6 }$	$=$	$ 60 + 19\cdot 103 + 76\cdot 103^{2} + 39\cdot 103^{3} + 60\cdot 103^{4} +O\left(103^{ 5 }\right)$
$r_{ 7 }$	$=$	$ 63 + 76\cdot 103 + 60\cdot 103^{2} + 68\cdot 103^{3} + 3\cdot 103^{4} +O\left(103^{ 5 }\right)$
$r_{ 8 }$	$=$	$ 96 + 51\cdot 103 + 17\cdot 103^{2} + 28\cdot 103^{3} + 43\cdot 103^{4} +O\left(103^{ 5 }\right)$

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

Cycle notation

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

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.