Galois orbit of Artin representations 2.2e10_7.8t7.2

• Label: 2.2e10_7.8t7.2
• Dimension: 2
• Group: $C_8:C_2$
• Conductor: $ 2^{10} \cdot 7 $
• Frobenius-Schur indicator: 0

Basic invariants

Dimension:	$2$
Group:	$C_8:C_2$
Conductor:	$7168= 2^{10} \cdot 7 $
Artin number field:	Splitting field of $f= x^{8} - 8 x^{6} - 92 x^{4} - 112 x^{2} + 98 $ over $\Q$
Size of Galois orbit:	2
Smallest containing permutation representation:	$C_8:C_2$
Parity:	Odd

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 113 }$ to precision 8.

Roots:

$r_{ 1 }$	$=$	$ 1 + 86\cdot 113 + 65\cdot 113^{2} + 112\cdot 113^{3} + 30\cdot 113^{4} + 18\cdot 113^{5} + 71\cdot 113^{6} + 12\cdot 113^{7} +O\left(113^{ 8 }\right)$
$r_{ 2 }$	$=$	$ 32 + 15\cdot 113 + 28\cdot 113^{2} + 98\cdot 113^{3} + 106\cdot 113^{4} + 80\cdot 113^{5} + 60\cdot 113^{6} + 9\cdot 113^{7} +O\left(113^{ 8 }\right)$
$r_{ 3 }$	$=$	$ 33 + 109\cdot 113 + 81\cdot 113^{2} + 92\cdot 113^{3} + 111\cdot 113^{4} + 34\cdot 113^{5} + 13\cdot 113^{6} + 111\cdot 113^{7} +O\left(113^{ 8 }\right)$
$r_{ 4 }$	$=$	$ 43 + 46\cdot 113 + 82\cdot 113^{2} + 73\cdot 113^{3} + 34\cdot 113^{4} + 21\cdot 113^{5} + 46\cdot 113^{6} + 82\cdot 113^{7} +O\left(113^{ 8 }\right)$
$r_{ 5 }$	$=$	$ 70 + 66\cdot 113 + 30\cdot 113^{2} + 39\cdot 113^{3} + 78\cdot 113^{4} + 91\cdot 113^{5} + 66\cdot 113^{6} + 30\cdot 113^{7} +O\left(113^{ 8 }\right)$
$r_{ 6 }$	$=$	$ 80 + 3\cdot 113 + 31\cdot 113^{2} + 20\cdot 113^{3} + 113^{4} + 78\cdot 113^{5} + 99\cdot 113^{6} + 113^{7} +O\left(113^{ 8 }\right)$
$r_{ 7 }$	$=$	$ 81 + 97\cdot 113 + 84\cdot 113^{2} + 14\cdot 113^{3} + 6\cdot 113^{4} + 32\cdot 113^{5} + 52\cdot 113^{6} + 103\cdot 113^{7} +O\left(113^{ 8 }\right)$
$r_{ 8 }$	$=$	$ 112 + 26\cdot 113 + 47\cdot 113^{2} + 82\cdot 113^{4} + 94\cdot 113^{5} + 41\cdot 113^{6} + 100\cdot 113^{7} +O\left(113^{ 8 }\right)$

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

Cycle notation

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

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

$(2,7)(4,5)$

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.