Artin representation 3.11e2_31_37e2.4t5.1c1

• Label: 3.11e2_31_37e2.4t5.1c1
• Dimension: 3
• Group: $S_4$
• Conductor: $ 11^{2} \cdot 31 \cdot 37^{2}$
• Root number: 1
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$3$
Group:	$S_4$
Conductor:	$5135119= 11^{2} \cdot 31 \cdot 37^{2} $
Artin number field:	Splitting field of $f= x^{4} - x^{3} - 20 x^{2} + 61 x + 58 $ over $\Q$
Size of Galois orbit:	1
Smallest containing permutation representation:	$S_4$
Parity:	Odd
Determinant:	1.31.2t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 227 }$ to precision 5.

Roots:

$r_{ 1 }$	$=$	$ 13 + 23\cdot 227 + 120\cdot 227^{2} + 215\cdot 227^{3} + 28\cdot 227^{4} +O\left(227^{ 5 }\right)$
$r_{ 2 }$	$=$	$ 74 + 108\cdot 227 + 30\cdot 227^{2} + 157\cdot 227^{3} + 27\cdot 227^{4} +O\left(227^{ 5 }\right)$
$r_{ 3 }$	$=$	$ 152 + 155\cdot 227 + 127\cdot 227^{2} + 158\cdot 227^{3} + 13\cdot 227^{4} +O\left(227^{ 5 }\right)$
$r_{ 4 }$	$=$	$ 216 + 166\cdot 227 + 175\cdot 227^{2} + 149\cdot 227^{3} + 156\cdot 227^{4} +O\left(227^{ 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.