Galois orbit of Artin representations 3.2e2_5e2_13e3.4t5.1

• Label: 3.2e2_5e2_13e3.4t5.1
• Dimension: 3
• Group: $S_4$
• Conductor: $ 2^{2} \cdot 5^{2} \cdot 13^{3}$
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$3$
Group:	$S_4$
Conductor:	$219700= 2^{2} \cdot 5^{2} \cdot 13^{3} $
Artin number field:	Splitting field of $f= x^{4} - 2 x^{3} + 8 x^{2} + 6 x + 35 $ 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 $\Q_{ 173 }$ to precision 5.

Roots:

$r_{ 1 }$	$=$	$ 113 + 171\cdot 173 + 57\cdot 173^{2} + 148\cdot 173^{3} + 48\cdot 173^{4} +O\left(173^{ 5 }\right)$
$r_{ 2 }$	$=$	$ 114 + 36\cdot 173 + 32\cdot 173^{2} + 82\cdot 173^{3} + 66\cdot 173^{4} +O\left(173^{ 5 }\right)$
$r_{ 3 }$	$=$	$ 143 + 60\cdot 173 + 145\cdot 173^{2} + 45\cdot 173^{3} + 137\cdot 173^{4} +O\left(173^{ 5 }\right)$
$r_{ 4 }$	$=$	$ 151 + 76\cdot 173 + 110\cdot 173^{2} + 69\cdot 173^{3} + 93\cdot 173^{4} +O\left(173^{ 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.