Galois orbit of Artin representations 1.5_41.8t1.3

• Label: 1.5_41.8t1.3
• Dimension: 1
• Group: $C_8$
• Conductor: $ 5 \cdot 41 $
• Frobenius-Schur indicator: 0

Basic invariants

Dimension:	$1$
Group:	$C_8$
Conductor:	$205= 5 \cdot 41 $
Artin number field:	Splitting field of $f= x^{8} - x^{7} + 44 x^{6} - 52 x^{5} + 823 x^{4} + 94 x^{3} + 6016 x^{2} + 6843 x + 4651 $ 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:

$r_{ 1 }$	$=$	$ 3 + 24\cdot 59 + 53\cdot 59^{2} + 13\cdot 59^{3} + 7\cdot 59^{4} +O\left(59^{ 5 }\right)$
$r_{ 2 }$	$=$	$ 12 + 27\cdot 59 + 41\cdot 59^{2} + 22\cdot 59^{3} + 27\cdot 59^{4} +O\left(59^{ 5 }\right)$
$r_{ 3 }$	$=$	$ 22 + 46\cdot 59 + 47\cdot 59^{2} + 16\cdot 59^{3} + 57\cdot 59^{4} +O\left(59^{ 5 }\right)$
$r_{ 4 }$	$=$	$ 25 + 3\cdot 59 + 35\cdot 59^{2} + 29\cdot 59^{3} + 13\cdot 59^{4} +O\left(59^{ 5 }\right)$
$r_{ 5 }$	$=$	$ 27 + 20\cdot 59^{2} + 28\cdot 59^{3} + 20\cdot 59^{4} +O\left(59^{ 5 }\right)$
$r_{ 6 }$	$=$	$ 44 + 5\cdot 59 + 20\cdot 59^{3} + 39\cdot 59^{4} +O\left(59^{ 5 }\right)$
$r_{ 7 }$	$=$	$ 49 + 6\cdot 59 + 42\cdot 59^{2} + 8\cdot 59^{3} + 14\cdot 59^{4} +O\left(59^{ 5 }\right)$
$r_{ 8 }$	$=$	$ 55 + 3\cdot 59 + 55\cdot 59^{2} + 36\cdot 59^{3} + 56\cdot 59^{4} +O\left(59^{ 5 }\right)$

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

Cycle notation

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

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.