Galois orbit of Artin representations 4.5e3_19e4.8t21.1

• Label: 4.5e3_19e4.8t21.1
• Dimension: 4
• Group: $C_2^3: C_4$
• Conductor: $ 5^{3} \cdot 19^{4}$
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$4$
Group:	$C_2^3: C_4$
Conductor:	$16290125= 5^{3} \cdot 19^{4} $
Artin number field:	Splitting field of $f= x^{8} - 3 x^{7} - 13 x^{6} + 50 x^{5} - 29 x^{4} + 110 x^{3} - 582 x^{2} + 796 x - 319 $ over $\Q$
Size of Galois orbit:	1
Smallest containing permutation representation:	$C_2^3: C_4$
Parity:	Even

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 191 }$ to precision 7.

Roots:

$r_{ 1 }$	$=$	$ 41 + 52\cdot 191 + 86\cdot 191^{2} + 124\cdot 191^{3} + 18\cdot 191^{4} + 132\cdot 191^{5} + 183\cdot 191^{6} +O\left(191^{ 7 }\right)$
$r_{ 2 }$	$=$	$ 43 + 8\cdot 191 + 110\cdot 191^{2} + 33\cdot 191^{3} + 101\cdot 191^{4} + 75\cdot 191^{5} + 27\cdot 191^{6} +O\left(191^{ 7 }\right)$
$r_{ 3 }$	$=$	$ 76 + 79\cdot 191 + 103\cdot 191^{2} + 127\cdot 191^{3} + 130\cdot 191^{4} + 164\cdot 191^{5} + 91\cdot 191^{6} +O\left(191^{ 7 }\right)$
$r_{ 4 }$	$=$	$ 79 + 94\cdot 191 + 48\cdot 191^{2} + 65\cdot 191^{3} + 168\cdot 191^{4} + 29\cdot 191^{5} + 108\cdot 191^{6} +O\left(191^{ 7 }\right)$
$r_{ 5 }$	$=$	$ 87 + 161\cdot 191 + 109\cdot 191^{3} + 60\cdot 191^{4} + 96\cdot 191^{5} + 98\cdot 191^{6} +O\left(191^{ 7 }\right)$
$r_{ 6 }$	$=$	$ 132 + 115\cdot 191 + 135\cdot 191^{2} + 113\cdot 191^{3} + 136\cdot 191^{4} + 65\cdot 191^{5} + 43\cdot 191^{6} +O\left(191^{ 7 }\right)$
$r_{ 7 }$	$=$	$ 135 + 88\cdot 191 + 186\cdot 191^{2} + 179\cdot 191^{3} + 111\cdot 191^{4} + 50\cdot 191^{5} + 13\cdot 191^{6} +O\left(191^{ 7 }\right)$
$r_{ 8 }$	$=$	$ 174 + 163\cdot 191 + 92\cdot 191^{2} + 10\cdot 191^{3} + 36\cdot 191^{4} + 149\cdot 191^{5} + 6\cdot 191^{6} +O\left(191^{ 7 }\right)$

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

Cycle notation

$(1,2)(5,6)$

$(4,8)(5,6)$

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

$(3,7)(4,8)$

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.