Galois orbit of Artin representations 1.2e3_17.8t1.2

• Label: 1.2e3_17.8t1.2
• 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:	Odd

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 47 }$ to precision 5.

Roots:

$r_{ 1 }$	$=$	$ 2 + 33\cdot 47 + 33\cdot 47^{2} + 46\cdot 47^{3} + 8\cdot 47^{4} +O\left(47^{ 5 }\right)$
$r_{ 2 }$	$=$	$ 7 + 28\cdot 47 + 29\cdot 47^{2} + 14\cdot 47^{3} + 29\cdot 47^{4} +O\left(47^{ 5 }\right)$
$r_{ 3 }$	$=$	$ 18 + 25\cdot 47 + 15\cdot 47^{2} + 46\cdot 47^{3} +O\left(47^{ 5 }\right)$
$r_{ 4 }$	$=$	$ 23 + 34\cdot 47 + 2\cdot 47^{2} + 13\cdot 47^{3} + 30\cdot 47^{4} +O\left(47^{ 5 }\right)$
$r_{ 5 }$	$=$	$ 24 + 12\cdot 47 + 44\cdot 47^{2} + 33\cdot 47^{3} + 16\cdot 47^{4} +O\left(47^{ 5 }\right)$
$r_{ 6 }$	$=$	$ 29 + 21\cdot 47 + 31\cdot 47^{2} + 46\cdot 47^{4} +O\left(47^{ 5 }\right)$
$r_{ 7 }$	$=$	$ 40 + 18\cdot 47 + 17\cdot 47^{2} + 32\cdot 47^{3} + 17\cdot 47^{4} +O\left(47^{ 5 }\right)$
$r_{ 8 }$	$=$	$ 45 + 13\cdot 47 + 13\cdot 47^{2} + 38\cdot 47^{4} +O\left(47^{ 5 }\right)$

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

Cycle notation

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

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.