Galois orbit of Artin representations 3.2e6_5_197.4t5.1

• Label: 3.2e6_5_197.4t5.1
• Dimension: 3
• Group: $S_4$
• Conductor: $ 2^{6} \cdot 5 \cdot 197 $
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$3$
Group:	$S_4$
Conductor:	$63040= 2^{6} \cdot 5 \cdot 197 $
Artin number field:	Splitting field of $f= x^{4} - 2 x^{3} - 6 x^{2} + 6 x + 7 $ 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 $\Q_{ 337 }$ to precision 5.

Roots:

$r_{ 1 }$	$=$	$ 95 + 181\cdot 337 + 212\cdot 337^{2} + 100\cdot 337^{3} + 141\cdot 337^{4} +O\left(337^{ 5 }\right)$
$r_{ 2 }$	$=$	$ 126 + 172\cdot 337 + 238\cdot 337^{2} + 259\cdot 337^{3} + 16\cdot 337^{4} +O\left(337^{ 5 }\right)$
$r_{ 3 }$	$=$	$ 133 + 21\cdot 337 + 230\cdot 337^{2} + 9\cdot 337^{3} + 47\cdot 337^{4} +O\left(337^{ 5 }\right)$
$r_{ 4 }$	$=$	$ 322 + 298\cdot 337 + 329\cdot 337^{2} + 303\cdot 337^{3} + 131\cdot 337^{4} +O\left(337^{ 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.