Galois orbit of Artin representations 3.2e4_11_47e2.4t5.2

• Label: 3.2e4_11_47e2.4t5.2
• Dimension: 3
• Group: $S_4$
• Conductor: $ 2^{4} \cdot 11 \cdot 47^{2}$
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$3$
Group:	$S_4$
Conductor:	$388784= 2^{4} \cdot 11 \cdot 47^{2} $
Artin number field:	Splitting field of $f= x^{4} - 2 x^{3} + 8 x^{2} + 40 x + 24 $ over $\Q$
Size of Galois orbit:	1
Smallest containing permutation representation:	$S_4$
Parity:	Odd

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 163 }$ to precision 5.

Roots:

$r_{ 1 }$	$=$	$ 51 + 2\cdot 163 + 159\cdot 163^{2} + 115\cdot 163^{3} + 151\cdot 163^{4} +O\left(163^{ 5 }\right)$
$r_{ 2 }$	$=$	$ 66 + 143\cdot 163 + 72\cdot 163^{2} + 135\cdot 163^{3} + 75\cdot 163^{4} +O\left(163^{ 5 }\right)$
$r_{ 3 }$	$=$	$ 76 + 163 + 86\cdot 163^{2} + 69\cdot 163^{3} + 141\cdot 163^{4} +O\left(163^{ 5 }\right)$
$r_{ 4 }$	$=$	$ 135 + 15\cdot 163 + 8\cdot 163^{2} + 5\cdot 163^{3} + 120\cdot 163^{4} +O\left(163^{ 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.