Galois orbit of Artin representations 2.2e9.8t17.2

• Label: 2.2e9.8t17.2
• 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:

$r_{ 1 }$	$=$	$ 2 + 4\cdot 113 + 54\cdot 113^{2} + 102\cdot 113^{3} + 51\cdot 113^{4} + 95\cdot 113^{5} +O\left(113^{ 6 }\right)$
$r_{ 2 }$	$=$	$ 30 + 90\cdot 113 + 11\cdot 113^{2} + 27\cdot 113^{3} + 56\cdot 113^{4} + 36\cdot 113^{5} +O\left(113^{ 6 }\right)$
$r_{ 3 }$	$=$	$ 33 + 47\cdot 113 + 88\cdot 113^{2} + 57\cdot 113^{3} + 49\cdot 113^{4} + 43\cdot 113^{5} +O\left(113^{ 6 }\right)$
$r_{ 4 }$	$=$	$ 43 + 74\cdot 113 + 9\cdot 113^{2} + 31\cdot 113^{3} + 9\cdot 113^{4} + 59\cdot 113^{5} +O\left(113^{ 6 }\right)$
$r_{ 5 }$	$=$	$ 70 + 38\cdot 113 + 103\cdot 113^{2} + 81\cdot 113^{3} + 103\cdot 113^{4} + 53\cdot 113^{5} +O\left(113^{ 6 }\right)$
$r_{ 6 }$	$=$	$ 80 + 65\cdot 113 + 24\cdot 113^{2} + 55\cdot 113^{3} + 63\cdot 113^{4} + 69\cdot 113^{5} +O\left(113^{ 6 }\right)$
$r_{ 7 }$	$=$	$ 83 + 22\cdot 113 + 101\cdot 113^{2} + 85\cdot 113^{3} + 56\cdot 113^{4} + 76\cdot 113^{5} +O\left(113^{ 6 }\right)$
$r_{ 8 }$	$=$	$ 111 + 108\cdot 113 + 58\cdot 113^{2} + 10\cdot 113^{3} + 61\cdot 113^{4} + 17\cdot 113^{5} +O\left(113^{ 6 }\right)$

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

Cycle notation

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

$(3,4,6,5)$

$(3,6)(4,5)$

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.