Galois orbit of Artin representations 4.3e4_7e2_37e2.8t15.2

• Label: 4.3e4_7e2_37e2.8t15.2
• Dimension: 4
• Group: $Z_8 : Z_8^\times$
• Conductor: $ 3^{4} \cdot 7^{2} \cdot 37^{2}$
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$4$
Group:	$Z_8 : Z_8^\times$
Conductor:	$5433561= 3^{4} \cdot 7^{2} \cdot 37^{2} $
Artin number field:	Splitting field of $f= x^{8} - 3 x^{7} + 11 x^{6} - 33 x^{5} + 57 x^{4} - 108 x^{3} + 149 x^{2} - 90 x + 127 $ over $\Q$
Size of Galois orbit:	1
Smallest containing permutation representation:	$Z_8 : Z_8^\times$
Parity:	Even

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 127 }$ to precision 6.

Roots:

$r_{ 1 }$	$=$	$ 24\cdot 127 + 63\cdot 127^{2} + 31\cdot 127^{3} + 118\cdot 127^{4} + 99\cdot 127^{5} +O\left(127^{ 6 }\right)$
$r_{ 2 }$	$=$	$ 28 + 95\cdot 127 + 19\cdot 127^{2} + 36\cdot 127^{3} + 25\cdot 127^{4} + 117\cdot 127^{5} +O\left(127^{ 6 }\right)$
$r_{ 3 }$	$=$	$ 42 + 91\cdot 127 + 124\cdot 127^{2} + 121\cdot 127^{3} + 62\cdot 127^{4} + 73\cdot 127^{5} +O\left(127^{ 6 }\right)$
$r_{ 4 }$	$=$	$ 62 + 73\cdot 127 + 81\cdot 127^{2} + 105\cdot 127^{3} + 66\cdot 127^{4} + 107\cdot 127^{5} +O\left(127^{ 6 }\right)$
$r_{ 5 }$	$=$	$ 83 + 25\cdot 127 + 48\cdot 127^{2} + 16\cdot 127^{3} + 46\cdot 127^{4} + 96\cdot 127^{5} +O\left(127^{ 6 }\right)$
$r_{ 6 }$	$=$	$ 91 + 91\cdot 127 + 70\cdot 127^{2} + 90\cdot 127^{3} + 51\cdot 127^{4} + 105\cdot 127^{5} +O\left(127^{ 6 }\right)$
$r_{ 7 }$	$=$	$ 96 + 26\cdot 127 + 58\cdot 127^{2} + 19\cdot 127^{3} + 41\cdot 127^{4} + 61\cdot 127^{5} +O\left(127^{ 6 }\right)$
$r_{ 8 }$	$=$	$ 109 + 79\cdot 127 + 41\cdot 127^{2} + 86\cdot 127^{3} + 95\cdot 127^{4} + 100\cdot 127^{5} +O\left(127^{ 6 }\right)$

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

Cycle notation

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

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

$(1,5)(4,6)$

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

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

Character values on conjugacy classes

Size	Order	Action on $r_1, \ldots, r_{ 8 }$	Character values
			$c1$
$1$	$1$	$()$	$4$
$1$	$2$	$(1,5)(2,3)(4,6)(7,8)$	$-4$
$2$	$2$	$(1,5)(4,6)$	$0$
$4$	$2$	$(1,6)(4,5)(7,8)$	$0$
$4$	$2$	$(1,8)(2,6)(3,4)(5,7)$	$0$
$4$	$2$	$(2,8)(3,7)(4,6)$	$0$
$2$	$4$	$(1,6,5,4)(2,8,3,7)$	$0$
$2$	$4$	$(1,4,5,6)(2,8,3,7)$	$0$
$4$	$4$	$(1,8,5,7)(2,4,3,6)$	$0$
$4$	$8$	$(1,2,6,8,5,3,4,7)$	$0$
$4$	$8$	$(1,2,4,7,5,3,6,8)$	$0$

The blue line marks the conjugacy class containing complex conjugation.