Artin representation 4.3e4_61e3.8t21.1c1

• Label: 4.3e4_61e3.8t21.1c1
• Dimension: 4
• Group: $C_2^3 : C_4 $
• Conductor: $ 3^{4} \cdot 61^{3}$
• Root number: 1
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$4$
Group:	$C_2^3 : C_4 $
Conductor:	$18385461= 3^{4} \cdot 61^{3} $
Artin number field:	Splitting field of $f= x^{8} - x^{7} - 6 x^{6} + 20 x^{5} - 32 x^{4} - 51 x^{3} + 228 x^{2} - 240 x + 129 $ over $\Q$
Size of Galois orbit:	1
Smallest containing permutation representation:	$C_2^3: C_4$
Parity:	Even
Determinant:	1.61.2t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 199 }$ to precision 6.

Roots:

$r_{ 1 }$	$=$	$ 4 + 104\cdot 199 + 84\cdot 199^{2} + 11\cdot 199^{3} + 122\cdot 199^{4} + 25\cdot 199^{5} +O\left(199^{ 6 }\right)$
$r_{ 2 }$	$=$	$ 20 + 135\cdot 199 + 141\cdot 199^{2} + 152\cdot 199^{3} + 43\cdot 199^{4} + 79\cdot 199^{5} +O\left(199^{ 6 }\right)$
$r_{ 3 }$	$=$	$ 56 + 142\cdot 199 + 150\cdot 199^{2} + 39\cdot 199^{3} + 135\cdot 199^{4} + 185\cdot 199^{5} +O\left(199^{ 6 }\right)$
$r_{ 4 }$	$=$	$ 59 + 126\cdot 199 + 37\cdot 199^{2} + 78\cdot 199^{3} + 29\cdot 199^{4} + 67\cdot 199^{5} +O\left(199^{ 6 }\right)$
$r_{ 5 }$	$=$	$ 69 + 199 + 3\cdot 199^{2} + 195\cdot 199^{3} + 188\cdot 199^{4} + 71\cdot 199^{5} +O\left(199^{ 6 }\right)$
$r_{ 6 }$	$=$	$ 87 + 81\cdot 199 + 72\cdot 199^{2} + 53\cdot 199^{3} + 155\cdot 199^{4} + 37\cdot 199^{5} +O\left(199^{ 6 }\right)$
$r_{ 7 }$	$=$	$ 129 + 170\cdot 199 + 173\cdot 199^{2} + 101\cdot 199^{3} + 92\cdot 199^{4} + 91\cdot 199^{5} +O\left(199^{ 6 }\right)$
$r_{ 8 }$	$=$	$ 174 + 34\cdot 199 + 132\cdot 199^{2} + 163\cdot 199^{3} + 28\cdot 199^{4} + 38\cdot 199^{5} +O\left(199^{ 6 }\right)$

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

Cycle notation

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

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

$(1,4)(5,8)$

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.