Galois orbit of Artin representations 1.2e2_17.8t1.1

• Label: 1.2e2_17.8t1.1
• Dimension: 1
• Group: $C_8$
• Conductor: $ 2^{2} \cdot 17 $
• Frobenius-Schur indicator: 0

Basic invariants

Dimension:	$1$
Group:	$C_8$
Conductor:	$68= 2^{2} \cdot 17 $
Artin number field:	Splitting field of $f= x^{8} + 17 x^{6} + 68 x^{4} + 85 x^{2} + 17 $ 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:

$r_{ 1 }$	$=$	$ 1 + 16\cdot 47 + 42\cdot 47^{2} + 9\cdot 47^{3} + 41\cdot 47^{4} +O\left(47^{ 5 }\right)$
$r_{ 2 }$	$=$	$ 7 + 8\cdot 47 + 44\cdot 47^{2} + 39\cdot 47^{3} + 36\cdot 47^{4} +O\left(47^{ 5 }\right)$
$r_{ 3 }$	$=$	$ 10 + 24\cdot 47 + 18\cdot 47^{2} + 23\cdot 47^{3} + 16\cdot 47^{4} +O\left(47^{ 5 }\right)$
$r_{ 4 }$	$=$	$ 16 + 14\cdot 47 + 4\cdot 47^{2} + 35\cdot 47^{3} + 23\cdot 47^{4} +O\left(47^{ 5 }\right)$
$r_{ 5 }$	$=$	$ 31 + 32\cdot 47 + 42\cdot 47^{2} + 11\cdot 47^{3} + 23\cdot 47^{4} +O\left(47^{ 5 }\right)$
$r_{ 6 }$	$=$	$ 37 + 22\cdot 47 + 28\cdot 47^{2} + 23\cdot 47^{3} + 30\cdot 47^{4} +O\left(47^{ 5 }\right)$
$r_{ 7 }$	$=$	$ 40 + 38\cdot 47 + 2\cdot 47^{2} + 7\cdot 47^{3} + 10\cdot 47^{4} +O\left(47^{ 5 }\right)$
$r_{ 8 }$	$=$	$ 46 + 30\cdot 47 + 4\cdot 47^{2} + 37\cdot 47^{3} + 5\cdot 47^{4} +O\left(47^{ 5 }\right)$

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

Cycle notation

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

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.