Galois orbit of Artin representations 3.2e6_7e2_23.4t5.2

• Label: 3.2e6_7e2_23.4t5.2
• Dimension: 3
• Group: $S_4$
• Conductor: $ 2^{6} \cdot 7^{2} \cdot 23 $
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$3$
Group:	$S_4$
Conductor:	$72128= 2^{6} \cdot 7^{2} \cdot 23 $
Artin number field:	Splitting field of $f= x^{4} - 2 x^{3} + 2 x^{2} + 6 x - 5 $ 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 $\Q_{ 271 }$ to precision 5.

Roots:

$r_{ 1 }$	$=$	$ 90 + 57\cdot 271 + 25\cdot 271^{2} + 7\cdot 271^{3} + 78\cdot 271^{4} +O\left(271^{ 5 }\right)$
$r_{ 2 }$	$=$	$ 130 + 25\cdot 271 + 44\cdot 271^{2} + 199\cdot 271^{3} + 53\cdot 271^{4} +O\left(271^{ 5 }\right)$
$r_{ 3 }$	$=$	$ 159 + 59\cdot 271 + 28\cdot 271^{2} + 140\cdot 271^{3} + 186\cdot 271^{4} +O\left(271^{ 5 }\right)$
$r_{ 4 }$	$=$	$ 165 + 128\cdot 271 + 173\cdot 271^{2} + 195\cdot 271^{3} + 223\cdot 271^{4} +O\left(271^{ 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.