Artin representation 3.2e2_3e2_7_53.4t5.1c1

• Label: 3.2e2_3e2_7_53.4t5.1c1
• Dimension: 3
• Group: $S_4$
• Conductor: $ 2^{2} \cdot 3^{2} \cdot 7 \cdot 53 $
• Root number: 1
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$3$
Group:	$S_4$
Conductor:	$13356= 2^{2} \cdot 3^{2} \cdot 7 \cdot 53 $
Artin number field:	Splitting field of $f= x^{4} - x^{3} - 7 x^{2} + 7 x - 2 $ over $\Q$
Size of Galois orbit:	1
Smallest containing permutation representation:	$S_4$
Parity:	Odd
Determinant:	1.7_53.2t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 317 }$ to precision 5.

Roots:

$r_{ 1 }$	$=$	$ 126 + 231\cdot 317 + 315\cdot 317^{3} + 37\cdot 317^{4} +O\left(317^{ 5 }\right)$
$r_{ 2 }$	$=$	$ 131 + 34\cdot 317 + 20\cdot 317^{2} + 63\cdot 317^{3} + 215\cdot 317^{4} +O\left(317^{ 5 }\right)$
$r_{ 3 }$	$=$	$ 157 + 135\cdot 317 + 8\cdot 317^{2} + 272\cdot 317^{3} + 143\cdot 317^{4} +O\left(317^{ 5 }\right)$
$r_{ 4 }$	$=$	$ 221 + 232\cdot 317 + 287\cdot 317^{2} + 300\cdot 317^{3} + 236\cdot 317^{4} +O\left(317^{ 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.