Artin representation 4.2e4_13e2_17e2.8t15.4c1

• Label: 4.2e4_13e2_17e2.8t15.4c1
• Dimension: 4
• Group: $Z_8 : Z_8^\times$
• Conductor: $ 2^{4} \cdot 13^{2} \cdot 17^{2}$
• Root number: 1
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$4$
Group:	$Z_8 : Z_8^\times$
Conductor:	$781456= 2^{4} \cdot 13^{2} \cdot 17^{2} $
Artin number field:	Splitting field of $f= x^{8} - 4 x^{7} + x^{6} + 8 x^{5} + 5 x^{4} - 14 x^{3} - 13 x^{2} + 4 x + 17 $ over $\Q$
Size of Galois orbit:	1
Smallest containing permutation representation:	$Z_8 : Z_8^\times$
Parity:	Even
Determinant:	1.1.1t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 157 }$ to precision 6.

Roots:

$r_{ 1 }$	$=$	$ 31 + 21\cdot 157 + 94\cdot 157^{2} + 39\cdot 157^{3} + 106\cdot 157^{4} + 139\cdot 157^{5} +O\left(157^{ 6 }\right)$
$r_{ 2 }$	$=$	$ 42 + 31\cdot 157 + 65\cdot 157^{2} + 61\cdot 157^{3} + 146\cdot 157^{4} + 56\cdot 157^{5} +O\left(157^{ 6 }\right)$
$r_{ 3 }$	$=$	$ 49 + 117\cdot 157 + 155\cdot 157^{2} + 133\cdot 157^{3} + 129\cdot 157^{4} + 26\cdot 157^{5} +O\left(157^{ 6 }\right)$
$r_{ 4 }$	$=$	$ 113 + 106\cdot 157 + 64\cdot 157^{2} + 83\cdot 157^{3} + 116\cdot 157^{4} + 80\cdot 157^{5} +O\left(157^{ 6 }\right)$
$r_{ 5 }$	$=$	$ 132 + 100\cdot 157 + 30\cdot 157^{2} + 50\cdot 157^{3} + 13\cdot 157^{4} + 97\cdot 157^{5} +O\left(157^{ 6 }\right)$
$r_{ 6 }$	$=$	$ 139 + 73\cdot 157 + 151\cdot 157^{2} + 119\cdot 157^{3} + 46\cdot 157^{4} + 87\cdot 157^{5} +O\left(157^{ 6 }\right)$
$r_{ 7 }$	$=$	$ 141 + 103\cdot 157 + 127\cdot 157^{2} + 17\cdot 157^{3} + 76\cdot 157^{4} + 123\cdot 157^{5} +O\left(157^{ 6 }\right)$
$r_{ 8 }$	$=$	$ 142 + 72\cdot 157 + 95\cdot 157^{2} + 121\cdot 157^{3} + 149\cdot 157^{4} + 15\cdot 157^{5} +O\left(157^{ 6 }\right)$

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

Cycle notation

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

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

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

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

$(3,8)(4,7)$

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.