Galois orbit of Artin representations 1.2e2_41.8t1.1

• Label: 1.2e2_41.8t1.1
• Dimension: 1
• Group: $C_8$
• Conductor: $ 2^{2} \cdot 41 $
• Frobenius-Schur indicator: 0

Basic invariants

Dimension:	$1$
Group:	$C_8$
Conductor:	$164= 2^{2} \cdot 41 $
Artin number field:	Splitting field of $f= x^{8} - 41 x^{6} + 533 x^{4} - 2296 x^{2} + 656 $ 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 $\Q_{ 107 }$ to precision 5.

Roots:

$r_{ 1 }$	$=$	$ 7 + 80\cdot 107 + 20\cdot 107^{2} + 38\cdot 107^{3} + 8\cdot 107^{4} +O\left(107^{ 5 }\right)$
$r_{ 2 }$	$=$	$ 18 + 30\cdot 107 + 24\cdot 107^{2} + 67\cdot 107^{3} + 45\cdot 107^{4} +O\left(107^{ 5 }\right)$
$r_{ 3 }$	$=$	$ 28 + 36\cdot 107 + 33\cdot 107^{2} + 94\cdot 107^{3} + 23\cdot 107^{4} +O\left(107^{ 5 }\right)$
$r_{ 4 }$	$=$	$ 32 + 101\cdot 107 + 91\cdot 107^{2} + 51\cdot 107^{3} + 45\cdot 107^{4} +O\left(107^{ 5 }\right)$
$r_{ 5 }$	$=$	$ 75 + 5\cdot 107 + 15\cdot 107^{2} + 55\cdot 107^{3} + 61\cdot 107^{4} +O\left(107^{ 5 }\right)$
$r_{ 6 }$	$=$	$ 79 + 70\cdot 107 + 73\cdot 107^{2} + 12\cdot 107^{3} + 83\cdot 107^{4} +O\left(107^{ 5 }\right)$
$r_{ 7 }$	$=$	$ 89 + 76\cdot 107 + 82\cdot 107^{2} + 39\cdot 107^{3} + 61\cdot 107^{4} +O\left(107^{ 5 }\right)$
$r_{ 8 }$	$=$	$ 100 + 26\cdot 107 + 86\cdot 107^{2} + 68\cdot 107^{3} + 98\cdot 107^{4} +O\left(107^{ 5 }\right)$

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

Cycle notation

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

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.