Galois orbit of Artin representations 3.2e2_19e2_23e2.6t8.1

• Label: 3.2e2_19e2_23e2.6t8.1
• Dimension: 3
• Group: $S_4$
• Conductor: $ 2^{2} \cdot 19^{2} \cdot 23^{2}$
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$3$
Group:	$S_4$
Conductor:	$763876= 2^{2} \cdot 19^{2} \cdot 23^{2} $
Artin number field:	Splitting field of $f= x^{4} - x^{3} - 7 x^{2} - 5 x + 2 $ 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 $\Q_{ 313 }$ to precision 5.

Roots:

$r_{ 1 }$	$=$	$ 73 + 71\cdot 313 + 308\cdot 313^{2} + 220\cdot 313^{3} + 64\cdot 313^{4} +O\left(313^{ 5 }\right)$
$r_{ 2 }$	$=$	$ 141 + 272\cdot 313 + 259\cdot 313^{2} + 91\cdot 313^{3} + 295\cdot 313^{4} +O\left(313^{ 5 }\right)$
$r_{ 3 }$	$=$	$ 158 + 31\cdot 313 + 41\cdot 313^{2} + 63\cdot 313^{3} + 137\cdot 313^{4} +O\left(313^{ 5 }\right)$
$r_{ 4 }$	$=$	$ 255 + 250\cdot 313 + 16\cdot 313^{2} + 250\cdot 313^{3} + 128\cdot 313^{4} +O\left(313^{ 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.