Galois orbit of Artin representations 1.5_17.8t1.2

• Label: 1.5_17.8t1.2
• 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} + 10 x^{6} - 79 x^{5} + 134 x^{4} + 41 x^{3} + 245 x^{2} - 846 x + 596 $ 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 $\Q_{ 83 }$ to precision 5.

Roots:

$r_{ 1 }$	$=$	$ 7 + 39\cdot 83 + 3\cdot 83^{2} + 39\cdot 83^{3} + 72\cdot 83^{4} +O\left(83^{ 5 }\right)$
$r_{ 2 }$	$=$	$ 15 + 71\cdot 83 + 51\cdot 83^{2} + 45\cdot 83^{3} + 21\cdot 83^{4} +O\left(83^{ 5 }\right)$
$r_{ 3 }$	$=$	$ 19 + 56\cdot 83 + 63\cdot 83^{2} + 34\cdot 83^{3} + 53\cdot 83^{4} +O\left(83^{ 5 }\right)$
$r_{ 4 }$	$=$	$ 42 + 24\cdot 83 + 21\cdot 83^{2} + 28\cdot 83^{3} + 81\cdot 83^{4} +O\left(83^{ 5 }\right)$
$r_{ 5 }$	$=$	$ 44 + 37\cdot 83 + 37\cdot 83^{2} + 13\cdot 83^{3} + 36\cdot 83^{4} +O\left(83^{ 5 }\right)$
$r_{ 6 }$	$=$	$ 52 + 67\cdot 83 + 45\cdot 83^{2} + 38\cdot 83^{3} + 14\cdot 83^{4} +O\left(83^{ 5 }\right)$
$r_{ 7 }$	$=$	$ 75 + 50\cdot 83 + 73\cdot 83^{2} + 25\cdot 83^{4} +O\left(83^{ 5 }\right)$
$r_{ 8 }$	$=$	$ 79 + 67\cdot 83 + 34\cdot 83^{2} + 48\cdot 83^{3} + 27\cdot 83^{4} +O\left(83^{ 5 }\right)$

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

Cycle notation

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

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.