Galois orbit of Artin representations 3.2e6_13e2_31e2.6t8.1

• Label: 3.2e6_13e2_31e2.6t8.1
• Dimension: 3
• Group: $S_4$
• Conductor: $ 2^{6} \cdot 13^{2} \cdot 31^{2}$
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$3$
Group:	$S_4$
Conductor:	$10394176= 2^{6} \cdot 13^{2} \cdot 31^{2} $
Artin number field:	Splitting field of $f= x^{4} - 2 x^{3} + 6 x^{2} + 34 x + 3 $ 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 $\Q_{ 349 }$ to precision 5.

Roots:

$r_{ 1 }$	$=$	$ 5 + 11\cdot 349 + 170\cdot 349^{2} + 27\cdot 349^{3} + 253\cdot 349^{4} +O\left(349^{ 5 }\right)$
$r_{ 2 }$	$=$	$ 34 + 223\cdot 349 + 170\cdot 349^{2} + 77\cdot 349^{3} + 153\cdot 349^{4} +O\left(349^{ 5 }\right)$
$r_{ 3 }$	$=$	$ 320 + 319\cdot 349 + 260\cdot 349^{2} + 133\cdot 349^{3} + 203\cdot 349^{4} +O\left(349^{ 5 }\right)$
$r_{ 4 }$	$=$	$ 341 + 143\cdot 349 + 96\cdot 349^{2} + 110\cdot 349^{3} + 88\cdot 349^{4} +O\left(349^{ 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.