Artin representation 2.2e10_7.8t7.4c2

• Label: 2.2e10_7.8t7.4c2
• Dimension: 2
• Group: $C_8:C_2$
• Conductor: $ 2^{10} \cdot 7 $
• Root number: not computed
• 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} + 24 x^{6} + 180 x^{4} + 448 x^{2} + 98 $ over $\Q$
Size of Galois orbit:	2
Smallest containing permutation representation:	$C_8:C_2$
Parity:	Even
Determinant:	1.2e4_7.4t1.1c2

Galois action

Roots of defining polynomial

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

Roots:

$r_{ 1 }$	$=$	$ 4 + 48\cdot 113 + 30\cdot 113^{2} + 22\cdot 113^{3} + 57\cdot 113^{4} + 84\cdot 113^{5} + 45\cdot 113^{6} + 3\cdot 113^{7} +O\left(113^{ 8 }\right)$
$r_{ 2 }$	$=$	$ 5 + 85\cdot 113 + 62\cdot 113^{2} + 48\cdot 113^{3} + 83\cdot 113^{4} + 8\cdot 113^{5} + 51\cdot 113^{6} + 78\cdot 113^{7} +O\left(113^{ 8 }\right)$
$r_{ 3 }$	$=$	$ 17 + 14\cdot 113 + 82\cdot 113^{3} + 6\cdot 113^{4} + 17\cdot 113^{5} + 57\cdot 113^{6} + 26\cdot 113^{7} +O\left(113^{ 8 }\right)$
$r_{ 4 }$	$=$	$ 18 + 103\cdot 113 + 44\cdot 113^{2} + 84\cdot 113^{3} + 41\cdot 113^{4} + 12\cdot 113^{5} + 51\cdot 113^{6} + 75\cdot 113^{7} +O\left(113^{ 8 }\right)$
$r_{ 5 }$	$=$	$ 95 + 9\cdot 113 + 68\cdot 113^{2} + 28\cdot 113^{3} + 71\cdot 113^{4} + 100\cdot 113^{5} + 61\cdot 113^{6} + 37\cdot 113^{7} +O\left(113^{ 8 }\right)$
$r_{ 6 }$	$=$	$ 96 + 98\cdot 113 + 112\cdot 113^{2} + 30\cdot 113^{3} + 106\cdot 113^{4} + 95\cdot 113^{5} + 55\cdot 113^{6} + 86\cdot 113^{7} +O\left(113^{ 8 }\right)$
$r_{ 7 }$	$=$	$ 108 + 27\cdot 113 + 50\cdot 113^{2} + 64\cdot 113^{3} + 29\cdot 113^{4} + 104\cdot 113^{5} + 61\cdot 113^{6} + 34\cdot 113^{7} +O\left(113^{ 8 }\right)$
$r_{ 8 }$	$=$	$ 109 + 64\cdot 113 + 82\cdot 113^{2} + 90\cdot 113^{3} + 55\cdot 113^{4} + 28\cdot 113^{5} + 67\cdot 113^{6} + 109\cdot 113^{7} +O\left(113^{ 8 }\right)$

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

Cycle notation

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

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

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.