Galois orbit of Artin representations 3.2e6_5e2_23e2.6t8.5

• Label: 3.2e6_5e2_23e2.6t8.5
• Dimension: 3
• Group: $S_4$
• Conductor: $ 2^{6} \cdot 5^{2} \cdot 23^{2}$
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$3$
Group:	$S_4$
Conductor:	$846400= 2^{6} \cdot 5^{2} \cdot 23^{2} $
Artin number field:	Splitting field of $f= x^{4} - 2 x^{3} - 2 x^{2} - 2 x + 11 $ 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 $\Q_{ 211 }$ to precision 5.

Roots:

$r_{ 1 }$	$=$	$ 32 + 187\cdot 211 + 11\cdot 211^{2} + 51\cdot 211^{3} + 57\cdot 211^{4} +O\left(211^{ 5 }\right)$
$r_{ 2 }$	$=$	$ 85 + 169\cdot 211 + 72\cdot 211^{2} + 209\cdot 211^{3} + 24\cdot 211^{4} +O\left(211^{ 5 }\right)$
$r_{ 3 }$	$=$	$ 134 + 30\cdot 211 + 87\cdot 211^{2} + 191\cdot 211^{3} + 33\cdot 211^{4} +O\left(211^{ 5 }\right)$
$r_{ 4 }$	$=$	$ 173 + 34\cdot 211 + 39\cdot 211^{2} + 181\cdot 211^{3} + 94\cdot 211^{4} +O\left(211^{ 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.