Artin representation 3.2e4_101e2.6t8.2c1

• Label: 3.2e4_101e2.6t8.2c1
• Dimension: 3
• Group: $S_4$
• Conductor: $ 2^{4} \cdot 101^{2}$
• Root number: 1
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$3$
Group:	$S_4$
Conductor:	$163216= 2^{4} \cdot 101^{2} $
Artin number field:	Splitting field of $f= x^{4} - 2 x^{3} + 52 x^{2} + 50 x + 19 $ over $\Q$
Size of Galois orbit:	1
Smallest containing permutation representation:	$S_4$
Parity:	Even
Determinant:	1.1.1t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 233 }$ to precision 5.

Roots:

$r_{ 1 }$	$=$	$ 64 + 66\cdot 233 + 10\cdot 233^{2} + 144\cdot 233^{3} + 31\cdot 233^{4} +O\left(233^{ 5 }\right)$
$r_{ 2 }$	$=$	$ 115 + 41\cdot 233 + 9\cdot 233^{2} + 208\cdot 233^{3} + 233^{4} +O\left(233^{ 5 }\right)$
$r_{ 3 }$	$=$	$ 144 + 18\cdot 233 + 117\cdot 233^{2} + 216\cdot 233^{3} + 62\cdot 233^{4} +O\left(233^{ 5 }\right)$
$r_{ 4 }$	$=$	$ 145 + 106\cdot 233 + 96\cdot 233^{2} + 130\cdot 233^{3} + 136\cdot 233^{4} +O\left(233^{ 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 value
$1$	$1$	$()$	$3$
$3$	$2$	$(1,2)(3,4)$	$-1$
$6$	$2$	$(1,2)$	$-1$
$8$	$3$	$(1,2,3)$	$0$
$6$	$4$	$(1,2,3,4)$	$1$

The blue line marks the conjugacy class containing complex conjugation.