Galois orbit of Artin representations 1.5_17.8t1.1

• Label: 1.5_17.8t1.1
• Dimension: 1
• Group: $C_8$
• Conductor: $ 5 \cdot 17 $
• Frobenius-Schur indicator: 0

Basic invariants

Dimension:	$1$
Group:	$C_8$
Conductor:	$85= 5 \cdot 17 $
Artin number field:	Splitting field of $f= x^{8} - x^{7} - 24 x^{6} + 23 x^{5} + 151 x^{4} - 197 x^{3} - 214 x^{2} + 429 x - 169 $ 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 $\Q_{ 101 }$ to precision 5.

Roots:

$r_{ 1 }$	$=$	$ 20 + 45\cdot 101 + 52\cdot 101^{2} + 19\cdot 101^{3} + 26\cdot 101^{4} +O\left(101^{ 5 }\right)$
$r_{ 2 }$	$=$	$ 21 + 49\cdot 101 + 84\cdot 101^{2} + 10\cdot 101^{3} + 75\cdot 101^{4} +O\left(101^{ 5 }\right)$
$r_{ 3 }$	$=$	$ 32 + 101 + 101^{2} + 48\cdot 101^{3} + 55\cdot 101^{4} +O\left(101^{ 5 }\right)$
$r_{ 4 }$	$=$	$ 35 + 23\cdot 101 + 29\cdot 101^{2} + 55\cdot 101^{3} + 101^{4} +O\left(101^{ 5 }\right)$
$r_{ 5 }$	$=$	$ 46 + 80\cdot 101 + 45\cdot 101^{2} + 83\cdot 101^{3} + 7\cdot 101^{4} +O\left(101^{ 5 }\right)$
$r_{ 6 }$	$=$	$ 70 + 45\cdot 101 + 89\cdot 101^{2} + 30\cdot 101^{3} + 44\cdot 101^{4} +O\left(101^{ 5 }\right)$
$r_{ 7 }$	$=$	$ 87 + 35\cdot 101 + 54\cdot 101^{2} + 81\cdot 101^{3} + 14\cdot 101^{4} +O\left(101^{ 5 }\right)$
$r_{ 8 }$	$=$	$ 94 + 21\cdot 101 + 47\cdot 101^{2} + 74\cdot 101^{3} + 77\cdot 101^{4} +O\left(101^{ 5 }\right)$

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

Cycle notation

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

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.