Artin representation 4.2e4_73e3.8t21.1c1

• Label: 4.2e4_73e3.8t21.1c1
• Dimension: 4
• Group: $C_2^3 : C_4 $
• Conductor: $ 2^{4} \cdot 73^{3}$
• Root number: 1
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$4$
Group:	$C_2^3 : C_4 $
Conductor:	$6224272= 2^{4} \cdot 73^{3} $
Artin number field:	Splitting field of $f= x^{8} - 2 x^{7} + x^{6} + 12 x^{5} - 7 x^{4} - 12 x^{3} + 52 x^{2} + 24 x + 4 $ over $\Q$
Size of Galois orbit:	1
Smallest containing permutation representation:	$C_2^3: C_4$
Parity:	Even
Determinant:	1.73.2t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 137 }$ to precision 6.

Roots:

$r_{ 1 }$	$=$	$ 15 + 86\cdot 137 + 135\cdot 137^{2} + 49\cdot 137^{3} + 10\cdot 137^{4} + 60\cdot 137^{5} +O\left(137^{ 6 }\right)$
$r_{ 2 }$	$=$	$ 19 + 110\cdot 137^{2} + 105\cdot 137^{3} + 81\cdot 137^{4} + 2\cdot 137^{5} +O\left(137^{ 6 }\right)$
$r_{ 3 }$	$=$	$ 30 + 79\cdot 137 + 87\cdot 137^{2} + 80\cdot 137^{3} + 114\cdot 137^{4} + 129\cdot 137^{5} +O\left(137^{ 6 }\right)$
$r_{ 4 }$	$=$	$ 31 + 60\cdot 137 + 125\cdot 137^{2} + 16\cdot 137^{3} + 123\cdot 137^{4} + 78\cdot 137^{5} +O\left(137^{ 6 }\right)$
$r_{ 5 }$	$=$	$ 96 + 10\cdot 137 + 92\cdot 137^{2} + 26\cdot 137^{3} + 42\cdot 137^{4} + 33\cdot 137^{5} +O\left(137^{ 6 }\right)$
$r_{ 6 }$	$=$	$ 104 + 128\cdot 137 + 112\cdot 137^{2} + 21\cdot 137^{3} + 25\cdot 137^{4} + 70\cdot 137^{5} +O\left(137^{ 6 }\right)$
$r_{ 7 }$	$=$	$ 121 + 84\cdot 137 + 62\cdot 137^{2} + 129\cdot 137^{3} + 43\cdot 137^{4} + 122\cdot 137^{5} +O\left(137^{ 6 }\right)$
$r_{ 8 }$	$=$	$ 134 + 97\cdot 137 + 95\cdot 137^{2} + 116\cdot 137^{3} + 106\cdot 137^{4} + 50\cdot 137^{5} +O\left(137^{ 6 }\right)$

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

Cycle notation

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

$(1,3)(5,8)$

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

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.