Galois orbit of Artin representations 1.2e5.8t1.2

• Label: 1.2e5.8t1.2
• Dimension: 1
• Group: $C_8$
• Conductor: $ 2^{5}$
• Frobenius-Schur indicator: 0

Basic invariants

Dimension:	$1$
Group:	$C_8$
Conductor:	$32= 2^{5} $
Artin number field:	Splitting field of $f= x^{8} + 8 x^{6} + 20 x^{4} + 16 x^{2} + 2 $ 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 $\Q_{ 47 }$ to precision 5.

Roots:

$r_{ 1 }$	$=$	$ 1 + 40\cdot 47 + 5\cdot 47^{2} + 20\cdot 47^{4} +O\left(47^{ 5 }\right)$
$r_{ 2 }$	$=$	$ 4 + 5\cdot 47 + 41\cdot 47^{2} + 19\cdot 47^{3} + 16\cdot 47^{4} +O\left(47^{ 5 }\right)$
$r_{ 3 }$	$=$	$ 11 + 13\cdot 47 + 8\cdot 47^{2} + 22\cdot 47^{3} + 20\cdot 47^{4} +O\left(47^{ 5 }\right)$
$r_{ 4 }$	$=$	$ 18 + 25\cdot 47 + 45\cdot 47^{3} + 35\cdot 47^{4} +O\left(47^{ 5 }\right)$
$r_{ 5 }$	$=$	$ 29 + 21\cdot 47 + 46\cdot 47^{2} + 47^{3} + 11\cdot 47^{4} +O\left(47^{ 5 }\right)$
$r_{ 6 }$	$=$	$ 36 + 33\cdot 47 + 38\cdot 47^{2} + 24\cdot 47^{3} + 26\cdot 47^{4} +O\left(47^{ 5 }\right)$
$r_{ 7 }$	$=$	$ 43 + 41\cdot 47 + 5\cdot 47^{2} + 27\cdot 47^{3} + 30\cdot 47^{4} +O\left(47^{ 5 }\right)$
$r_{ 8 }$	$=$	$ 46 + 6\cdot 47 + 41\cdot 47^{2} + 46\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

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

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.