Artin representation 4.2e4_3e2_5e2_11e2.8t22.7c1

• Label: 4.2e4_3e2_5e2_11e2.8t22.7c1
• Dimension: 4
• Group: $C_2^3 : D_4 $
• Conductor: $ 2^{4} \cdot 3^{2} \cdot 5^{2} \cdot 11^{2}$
• Root number: 1
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$4$
Group:	$C_2^3 : D_4 $
Conductor:	$435600= 2^{4} \cdot 3^{2} \cdot 5^{2} \cdot 11^{2} $
Artin number field:	Splitting field of $f= x^{8} - 4 x^{7} + 5 x^{6} + 4 x^{5} - 12 x^{4} - 2 x^{3} + 24 x^{2} - 20 x + 5 $ over $\Q$
Size of Galois orbit:	1
Smallest containing permutation representation:	$C_2^3 : D_4 $
Parity:	Even
Determinant:	1.1.1t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 181 }$ to precision 5.

Roots:

$r_{ 1 }$	$=$	$ 17 + 3\cdot 181 + 159\cdot 181^{2} + 49\cdot 181^{3} + 114\cdot 181^{4} +O\left(181^{ 5 }\right)$
$r_{ 2 }$	$=$	$ 23 + 99\cdot 181 + 52\cdot 181^{2} + 35\cdot 181^{3} + 50\cdot 181^{4} +O\left(181^{ 5 }\right)$
$r_{ 3 }$	$=$	$ 35 + 50\cdot 181 + 71\cdot 181^{2} + 130\cdot 181^{3} + 81\cdot 181^{4} +O\left(181^{ 5 }\right)$
$r_{ 4 }$	$=$	$ 37 + 21\cdot 181 + 113\cdot 181^{2} + 133\cdot 181^{3} + 172\cdot 181^{4} +O\left(181^{ 5 }\right)$
$r_{ 5 }$	$=$	$ 108 + 28\cdot 181 + 79\cdot 181^{2} + 146\cdot 181^{3} + 115\cdot 181^{4} +O\left(181^{ 5 }\right)$
$r_{ 6 }$	$=$	$ 156 + 162\cdot 181 + 89\cdot 181^{2} + 132\cdot 181^{3} + 179\cdot 181^{4} +O\left(181^{ 5 }\right)$
$r_{ 7 }$	$=$	$ 175 + 164\cdot 181 + 160\cdot 181^{2} + 96\cdot 181^{3} + 137\cdot 181^{4} +O\left(181^{ 5 }\right)$
$r_{ 8 }$	$=$	$ 177 + 12\cdot 181 + 179\cdot 181^{2} + 179\cdot 181^{3} + 52\cdot 181^{4} +O\left(181^{ 5 }\right)$

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

Cycle notation

$(2,5)(4,8)$

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

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

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

$(1,3)(2,5)$

$(2,5)(6,7)$

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.