Galois orbit of Artin representations 4.3e4_7e2_37e2.8t15.1

• Label: 4.3e4_7e2_37e2.8t15.1
• 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} + 14 x^{6} - 27 x^{5} + 66 x^{4} - 57 x^{3} + 116 x^{2} + 6 x + 13 $ 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:

$r_{ 1 }$	$=$	$ 14 + 70\cdot 127 + 92\cdot 127^{2} + 91\cdot 127^{3} + 6\cdot 127^{4} + 85\cdot 127^{5} +O\left(127^{ 6 }\right)$
$r_{ 2 }$	$=$	$ 24 + 127 + 35\cdot 127^{2} + 22\cdot 127^{3} + 81\cdot 127^{4} + 46\cdot 127^{5} +O\left(127^{ 6 }\right)$
$r_{ 3 }$	$=$	$ 28 + 69\cdot 127 + 93\cdot 127^{2} + 5\cdot 127^{3} + 34\cdot 127^{4} + 9\cdot 127^{5} +O\left(127^{ 6 }\right)$
$r_{ 4 }$	$=$	$ 34 + 54\cdot 127 + 93\cdot 127^{2} + 54\cdot 127^{3} + 116\cdot 127^{4} + 94\cdot 127^{5} +O\left(127^{ 6 }\right)$
$r_{ 5 }$	$=$	$ 85 + 15\cdot 127 + 97\cdot 127^{2} + 93\cdot 127^{3} + 67\cdot 127^{4} + 39\cdot 127^{5} +O\left(127^{ 6 }\right)$
$r_{ 6 }$	$=$	$ 102 + 111\cdot 127 + 65\cdot 127^{2} + 119\cdot 127^{3} + 108\cdot 127^{4} +O\left(127^{ 6 }\right)$
$r_{ 7 }$	$=$	$ 109 + 59\cdot 127 + 107\cdot 127^{2} + 52\cdot 127^{3} + 47\cdot 127^{4} + 21\cdot 127^{5} +O\left(127^{ 6 }\right)$
$r_{ 8 }$	$=$	$ 115 + 125\cdot 127 + 49\cdot 127^{2} + 67\cdot 127^{3} + 45\cdot 127^{4} + 83\cdot 127^{5} +O\left(127^{ 6 }\right)$

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

Cycle notation

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

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

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

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

$(1,3)(5,7)$

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.