Galois orbit of Artin representations 3.2e6_3e4_7.4t5.1

• Label: 3.2e6_3e4_7.4t5.1
• Dimension: 3
• Group: $S_4$
• Conductor: $ 2^{6} \cdot 3^{4} \cdot 7 $
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$3$
Group:	$S_4$
Conductor:	$36288= 2^{6} \cdot 3^{4} \cdot 7 $
Artin number field:	Splitting field of $f= x^{4} - 2 x^{3} + 6 x^{2} + 2 x - 1 $ 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 $\Q_{ 317 }$ to precision 5.

Roots:

$r_{ 1 }$	$=$	$ 202 + 214\cdot 317 + 272\cdot 317^{2} + 209\cdot 317^{3} + 301\cdot 317^{4} +O\left(317^{ 5 }\right)$
$r_{ 2 }$	$=$	$ 207 + 174\cdot 317 + 272\cdot 317^{2} + 61\cdot 317^{3} + 223\cdot 317^{4} +O\left(317^{ 5 }\right)$
$r_{ 3 }$	$=$	$ 265 + 238\cdot 317 + 95\cdot 317^{2} + 153\cdot 317^{3} + 175\cdot 317^{4} +O\left(317^{ 5 }\right)$
$r_{ 4 }$	$=$	$ 279 + 5\cdot 317 + 310\cdot 317^{2} + 208\cdot 317^{3} + 250\cdot 317^{4} +O\left(317^{ 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.