Galois orbit of Artin representations 2.2e9.8t17.3

• Label: 2.2e9.8t17.3
• Dimension: 2
• Group: $C_4\wr C_2$
• Conductor: $ 2^{9}$
• Frobenius-Schur indicator: 0

Basic invariants

Dimension:	$2$
Group:	$C_4\wr C_2$
Conductor:	$512= 2^{9} $
Artin number field:	Splitting field of $f= x^{8} + 2 x^{4} + 2 $ over $\Q$
Size of Galois orbit:	2
Smallest containing permutation representation:	$C_4\wr C_2$
Parity:	Odd

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 113 }$ to precision 6.

Roots:

$r_{ 1 }$	$=$	$ 17 + 95\cdot 113 + 27\cdot 113^{2} + 35\cdot 113^{3} + 25\cdot 113^{4} + 111\cdot 113^{5} +O\left(113^{ 6 }\right)$
$r_{ 2 }$	$=$	$ 25 + 85\cdot 113 + 44\cdot 113^{2} + 23\cdot 113^{3} + 14\cdot 113^{4} + 92\cdot 113^{5} +O\left(113^{ 6 }\right)$
$r_{ 3 }$	$=$	$ 29 + 100\cdot 113 + 23\cdot 113^{2} + 52\cdot 113^{3} + 102\cdot 113^{4} + 88\cdot 113^{5} +O\left(113^{ 6 }\right)$
$r_{ 4 }$	$=$	$ 36 + 71\cdot 113 + 82\cdot 113^{2} + 91\cdot 113^{3} + 8\cdot 113^{4} + 99\cdot 113^{5} +O\left(113^{ 6 }\right)$
$r_{ 5 }$	$=$	$ 77 + 41\cdot 113 + 30\cdot 113^{2} + 21\cdot 113^{3} + 104\cdot 113^{4} + 13\cdot 113^{5} +O\left(113^{ 6 }\right)$
$r_{ 6 }$	$=$	$ 84 + 12\cdot 113 + 89\cdot 113^{2} + 60\cdot 113^{3} + 10\cdot 113^{4} + 24\cdot 113^{5} +O\left(113^{ 6 }\right)$
$r_{ 7 }$	$=$	$ 88 + 27\cdot 113 + 68\cdot 113^{2} + 89\cdot 113^{3} + 98\cdot 113^{4} + 20\cdot 113^{5} +O\left(113^{ 6 }\right)$
$r_{ 8 }$	$=$	$ 96 + 17\cdot 113 + 85\cdot 113^{2} + 77\cdot 113^{3} + 87\cdot 113^{4} + 113^{5} +O\left(113^{ 6 }\right)$

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

Cycle notation

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

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

$(1,8)(3,6)$

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

$(1,3,8,6)$

Character values on conjugacy classes

Size	Order	Action on $r_1, \ldots, r_{ 8 }$	Character values
			$c1$	$c2$
$1$	$1$	$()$	$2$	$2$
$1$	$2$	$(1,8)(2,7)(3,6)(4,5)$	$-2$	$-2$
$2$	$2$	$(1,8)(3,6)$	$0$	$0$
$4$	$2$	$(1,4)(2,3)(5,8)(6,7)$	$0$	$0$
$1$	$4$	$(1,3,8,6)(2,5,7,4)$	$-2 \zeta_{4}$	$2 \zeta_{4}$
$1$	$4$	$(1,6,8,3)(2,4,7,5)$	$2 \zeta_{4}$	$-2 \zeta_{4}$
$2$	$4$	$(1,6,8,3)(2,5,7,4)$	$0$	$0$
$2$	$4$	$(1,3,8,6)$	$\zeta_{4} - 1$	$-\zeta_{4} - 1$
$2$	$4$	$(1,6,8,3)$	$-\zeta_{4} - 1$	$\zeta_{4} - 1$
$2$	$4$	$(1,6,8,3)(2,7)(4,5)$	$-\zeta_{4} + 1$	$\zeta_{4} + 1$
$2$	$4$	$(1,3,8,6)(2,7)(4,5)$	$\zeta_{4} + 1$	$-\zeta_{4} + 1$
$4$	$4$	$(1,4,8,5)(2,6,7,3)$	$0$	$0$
$4$	$8$	$(1,4,3,2,8,5,6,7)$	$0$	$0$
$4$	$8$	$(1,2,6,4,8,7,3,5)$	$0$	$0$

The blue line marks the conjugacy class containing complex conjugation.