Galois orbit of Artin representations 1.7_17.8t1.1

• Label: 1.7_17.8t1.1
• Dimension: 1
• Group: $C_8$
• Conductor: $ 7 \cdot 17 $
• Frobenius-Schur indicator: 0

Basic invariants

Dimension:	$1$
Group:	$C_8$
Conductor:	$119= 7 \cdot 17 $
Artin number field:	Splitting field of $f= x^{8} - x^{7} + 27 x^{6} - 28 x^{5} + 151 x^{4} - 350 x^{3} + 500 x^{2} - 846 x + 1157 $ 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 + 46\cdot 47 + 8\cdot 47^{2} + 15\cdot 47^{3} + 13\cdot 47^{4} +O\left(47^{ 5 }\right)$
$r_{ 2 }$	$=$	$ 5 + 40\cdot 47 + 29\cdot 47^{2} + 25\cdot 47^{3} + 41\cdot 47^{4} +O\left(47^{ 5 }\right)$
$r_{ 3 }$	$=$	$ 20 + 43\cdot 47 + 3\cdot 47^{2} + 23\cdot 47^{3} + 44\cdot 47^{4} +O\left(47^{ 5 }\right)$
$r_{ 4 }$	$=$	$ 26 + 39\cdot 47 + 4\cdot 47^{2} + 34\cdot 47^{3} + 32\cdot 47^{4} +O\left(47^{ 5 }\right)$
$r_{ 5 }$	$=$	$ 27 + 7\cdot 47 + 6\cdot 47^{2} + 22\cdot 47^{3} + 20\cdot 47^{4} +O\left(47^{ 5 }\right)$
$r_{ 6 }$	$=$	$ 35 + 16\cdot 47 + 15\cdot 47^{2} + 35\cdot 47^{3} + 18\cdot 47^{4} +O\left(47^{ 5 }\right)$
$r_{ 7 }$	$=$	$ 37 + 34\cdot 47 + 27\cdot 47^{2} + 10\cdot 47^{3} + 15\cdot 47^{4} +O\left(47^{ 5 }\right)$
$r_{ 8 }$	$=$	$ 38 + 6\cdot 47 + 44\cdot 47^{2} + 21\cdot 47^{3} + 47^{4} +O\left(47^{ 5 }\right)$

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

Cycle notation

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

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.