Artin representation 4.2e4_3e2_5e2_31e2.8t22.4c1

• Label: 4.2e4_3e2_5e2_31e2.8t22.4c1
• Dimension: 4
• Group: $C_2^3 : D_4 $
• Conductor: $ 2^{4} \cdot 3^{2} \cdot 5^{2} \cdot 31^{2}$
• Root number: 1
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$4$
Group:	$C_2^3 : D_4 $
Conductor:	$3459600= 2^{4} \cdot 3^{2} \cdot 5^{2} \cdot 31^{2} $
Artin number field:	Splitting field of $f= x^{8} - 3 x^{7} - 7 x^{6} + 20 x^{5} + 21 x^{4} - 55 x^{3} - 27 x^{2} + 98 x + 76 $ 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 $\Q_{ 109 }$ to precision 5.

Roots:

$r_{ 1 }$	$=$	$ 8 + 75\cdot 109 + 109^{2} + 41\cdot 109^{3} + 66\cdot 109^{4} +O\left(109^{ 5 }\right)$
$r_{ 2 }$	$=$	$ 31 + 37\cdot 109 + 8\cdot 109^{2} + 100\cdot 109^{3} + 50\cdot 109^{4} +O\left(109^{ 5 }\right)$
$r_{ 3 }$	$=$	$ 62 + 92\cdot 109 + 24\cdot 109^{2} + 106\cdot 109^{3} + 70\cdot 109^{4} +O\left(109^{ 5 }\right)$
$r_{ 4 }$	$=$	$ 64 + 103\cdot 109 + 9\cdot 109^{2} + 23\cdot 109^{3} + 86\cdot 109^{4} +O\left(109^{ 5 }\right)$
$r_{ 5 }$	$=$	$ 87 + 63\cdot 109 + 101\cdot 109^{2} + 49\cdot 109^{3} + 84\cdot 109^{4} +O\left(109^{ 5 }\right)$
$r_{ 6 }$	$=$	$ 88 + 50\cdot 109 + 101\cdot 109^{2} + 92\cdot 109^{3} + 30\cdot 109^{4} +O\left(109^{ 5 }\right)$
$r_{ 7 }$	$=$	$ 103 + 28\cdot 109 + 100\cdot 109^{2} + 25\cdot 109^{3} + 42\cdot 109^{4} +O\left(109^{ 5 }\right)$
$r_{ 8 }$	$=$	$ 105 + 92\cdot 109 + 87\cdot 109^{2} + 105\cdot 109^{3} + 3\cdot 109^{4} +O\left(109^{ 5 }\right)$

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

Cycle notation

$(1,6)(4,8)$

$(1,6)(2,7)$

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

$(1,6)(3,5)$

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

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.