Galois orbit of Artin representations 4.5e4_11e2_19e2.8t16.11

• Label: 4.5e4_11e2_19e2.8t16.11
• Dimension: 4
• Group: $(C_8:C_2):C_2$
• Conductor: $ 5^{4} \cdot 11^{2} \cdot 19^{2}$
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$4$
Group:	$(C_8:C_2):C_2$
Conductor:	$27300625= 5^{4} \cdot 11^{2} \cdot 19^{2} $
Artin number field:	Splitting field of $f= x^{8} - x^{7} - 18 x^{6} + 7 x^{5} + 130 x^{4} - 52 x^{3} - 378 x^{2} + 431 x + 991 $ over $\Q$
Size of Galois orbit:	1
Smallest containing permutation representation:	$(C_8:C_2):C_2$
Parity:	Even

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 251 }$ to precision 6.

Roots:

$r_{ 1 }$	$=$	$ 19 + 171\cdot 251 + 225\cdot 251^{2} + 178\cdot 251^{3} + 16\cdot 251^{4} + 220\cdot 251^{5} +O\left(251^{ 6 }\right)$
$r_{ 2 }$	$=$	$ 40 + 75\cdot 251 + 239\cdot 251^{2} + 112\cdot 251^{3} + 176\cdot 251^{4} + 118\cdot 251^{5} +O\left(251^{ 6 }\right)$
$r_{ 3 }$	$=$	$ 75 + 14\cdot 251 + 250\cdot 251^{2} + 175\cdot 251^{3} + 174\cdot 251^{4} + 211\cdot 251^{5} +O\left(251^{ 6 }\right)$
$r_{ 4 }$	$=$	$ 119 + 213\cdot 251 + 91\cdot 251^{2} + 82\cdot 251^{3} + 13\cdot 251^{4} + 18\cdot 251^{5} +O\left(251^{ 6 }\right)$
$r_{ 5 }$	$=$	$ 136 + 214\cdot 251 + 194\cdot 251^{2} + 190\cdot 251^{3} + 18\cdot 251^{4} + 14\cdot 251^{5} +O\left(251^{ 6 }\right)$
$r_{ 6 }$	$=$	$ 156 + 28\cdot 251 + 140\cdot 251^{2} + 14\cdot 251^{3} + 91\cdot 251^{4} + 136\cdot 251^{5} +O\left(251^{ 6 }\right)$
$r_{ 7 }$	$=$	$ 217 + 248\cdot 251 + 8\cdot 251^{2} + 239\cdot 251^{3} + 2\cdot 251^{4} + 153\cdot 251^{5} +O\left(251^{ 6 }\right)$
$r_{ 8 }$	$=$	$ 243 + 37\cdot 251 + 104\cdot 251^{2} + 9\cdot 251^{3} + 8\cdot 251^{4} + 132\cdot 251^{5} +O\left(251^{ 6 }\right)$

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

Cycle notation

$(3,6)(5,7)$

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

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

$(1,4)(2,8)$

$(2,8)(5,7)$

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.