Galois orbit of Artin representations 3.2e6_3e4_5e2.6t8.3

• Label: 3.2e6_3e4_5e2.6t8.3
• Dimension: 3
• Group: $S_4$
• Conductor: $ 2^{6} \cdot 3^{4} \cdot 5^{2}$
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$3$
Group:	$S_4$
Conductor:	$129600= 2^{6} \cdot 3^{4} \cdot 5^{2} $
Artin number field:	Splitting field of $f= x^{4} - 2 x^{3} - 6 x^{2} - 8 x - 14 $ over $\Q$
Size of Galois orbit:	1
Smallest containing permutation representation:	$S_4$
Parity:	Even

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 199 }$ to precision 5.

Roots:

$r_{ 1 }$	$=$	$ 47 + 80\cdot 199 + 164\cdot 199^{2} + 109\cdot 199^{3} + 131\cdot 199^{4} +O\left(199^{ 5 }\right)$
$r_{ 2 }$	$=$	$ 72 + 33\cdot 199 + 58\cdot 199^{2} + 29\cdot 199^{3} + 58\cdot 199^{4} +O\left(199^{ 5 }\right)$
$r_{ 3 }$	$=$	$ 115 + 2\cdot 199 + 68\cdot 199^{2} + 64\cdot 199^{3} + 82\cdot 199^{4} +O\left(199^{ 5 }\right)$
$r_{ 4 }$	$=$	$ 166 + 82\cdot 199 + 107\cdot 199^{2} + 194\cdot 199^{3} + 125\cdot 199^{4} +O\left(199^{ 5 }\right)$

Generators of the action on the roots $r_1, \ldots, r_{ 4 }$

Cycle notation

$(1,2,3,4)$

$(1,2)$

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.