Galois orbit of Artin representations 1.13_17.8t1.1

• Label: 1.13_17.8t1.1
• Dimension: 1
• Group: $C_8$
• Conductor: $ 13 \cdot 17 $
• Frobenius-Schur indicator: 0

Basic invariants

Dimension:	$1$
Group:	$C_8$
Conductor:	$221= 13 \cdot 17 $
Artin number field:	Splitting field of $f= x^{8} - x^{7} - 58 x^{6} + 57 x^{5} + 831 x^{4} - 1285 x^{3} - 3070 x^{2} + 6889 x - 2753 $ 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 $\Q_{ 47 }$ to precision 5.

Roots:

$r_{ 1 }$	$=$	$ 1 + 43\cdot 47 + 15\cdot 47^{2} + 16\cdot 47^{3} + 16\cdot 47^{4} +O\left(47^{ 5 }\right)$
$r_{ 2 }$	$=$	$ 7 + 7\cdot 47 + 47^{2} + 36\cdot 47^{3} + 9\cdot 47^{4} +O\left(47^{ 5 }\right)$
$r_{ 3 }$	$=$	$ 10 + 35\cdot 47 + 32\cdot 47^{2} + 43\cdot 47^{3} + 25\cdot 47^{4} +O\left(47^{ 5 }\right)$
$r_{ 4 }$	$=$	$ 27 + 5\cdot 47 + 33\cdot 47^{2} + 4\cdot 47^{3} + 41\cdot 47^{4} +O\left(47^{ 5 }\right)$
$r_{ 5 }$	$=$	$ 29 + 32\cdot 47 + 2\cdot 47^{2} + 20\cdot 47^{3} + 19\cdot 47^{4} +O\left(47^{ 5 }\right)$
$r_{ 6 }$	$=$	$ 31 + 44\cdot 47 + 14\cdot 47^{2} + 40\cdot 47^{3} + 4\cdot 47^{4} +O\left(47^{ 5 }\right)$
$r_{ 7 }$	$=$	$ 39 + 13\cdot 47 + 29\cdot 47^{2} + 44\cdot 47^{3} + 43\cdot 47^{4} +O\left(47^{ 5 }\right)$
$r_{ 8 }$	$=$	$ 45 + 5\cdot 47 + 11\cdot 47^{2} + 29\cdot 47^{3} + 26\cdot 47^{4} +O\left(47^{ 5 }\right)$

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

Cycle notation

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

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.