Galois orbit of Artin representations 3.3e2_11e2_43e2.6t8.1

• Label: 3.3e2_11e2_43e2.6t8.1
• Dimension: 3
• Group: $S_4$
• Conductor: $ 3^{2} \cdot 11^{2} \cdot 43^{2}$
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$3$
Group:	$S_4$
Conductor:	$2013561= 3^{2} \cdot 11^{2} \cdot 43^{2} $
Artin number field:	Splitting field of $f= x^{4} - x^{3} + 5 x^{2} - 2 x + 1 $ 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 $\Q_{ 277 }$ to precision 5.

Roots:

$r_{ 1 }$	$=$	$ 73 + 268\cdot 277 + 63\cdot 277^{2} + 65\cdot 277^{3} + 169\cdot 277^{4} +O\left(277^{ 5 }\right)$
$r_{ 2 }$	$=$	$ 90 + 140\cdot 277 + 247\cdot 277^{2} + 143\cdot 277^{3} + 246\cdot 277^{4} +O\left(277^{ 5 }\right)$
$r_{ 3 }$	$=$	$ 192 + 58\cdot 277 + 167\cdot 277^{2} + 230\cdot 277^{3} + 134\cdot 277^{4} +O\left(277^{ 5 }\right)$
$r_{ 4 }$	$=$	$ 200 + 86\cdot 277 + 75\cdot 277^{2} + 114\cdot 277^{3} + 3\cdot 277^{4} +O\left(277^{ 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.