Galois orbit of Artin representations 1.13_17.8t1.2

• Label: 1.13_17.8t1.2
• 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} + 27 x^{6} + 125 x^{5} - 308 x^{4} - 2628 x^{3} + 2336 x^{2} + 35840 x + 65536 $ 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 $\Q_{ 59 }$ to precision 5.

Roots:

$r_{ 1 }$	$=$	$ 4 + 29\cdot 59 + 9\cdot 59^{2} + 4\cdot 59^{3} + 22\cdot 59^{4} +O\left(59^{ 5 }\right)$
$r_{ 2 }$	$=$	$ 18 + 3\cdot 59 + 35\cdot 59^{2} + 51\cdot 59^{3} + 18\cdot 59^{4} +O\left(59^{ 5 }\right)$
$r_{ 3 }$	$=$	$ 21 + 22\cdot 59 + 11\cdot 59^{2} + 41\cdot 59^{3} + 37\cdot 59^{4} +O\left(59^{ 5 }\right)$
$r_{ 4 }$	$=$	$ 31 + 25\cdot 59 + 58\cdot 59^{2} + 21\cdot 59^{3} + 6\cdot 59^{4} +O\left(59^{ 5 }\right)$
$r_{ 5 }$	$=$	$ 32 + 21\cdot 59 + 59^{2} + 44\cdot 59^{3} + 19\cdot 59^{4} +O\left(59^{ 5 }\right)$
$r_{ 6 }$	$=$	$ 37 + 41\cdot 59 + 57\cdot 59^{2} + 39\cdot 59^{3} + 36\cdot 59^{4} +O\left(59^{ 5 }\right)$
$r_{ 7 }$	$=$	$ 45 + 26\cdot 59 + 45\cdot 59^{2} + 47\cdot 59^{3} + 34\cdot 59^{4} +O\left(59^{ 5 }\right)$
$r_{ 8 }$	$=$	$ 49 + 6\cdot 59 + 17\cdot 59^{2} + 44\cdot 59^{3} +O\left(59^{ 5 }\right)$

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

Cycle notation

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

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.