Artin representation 3.3_7e2_229.4t5.1c1

• Label: 3.3_7e2_229.4t5.1c1
• Dimension: 3
• Group: $S_4$
• Conductor: $ 3 \cdot 7^{2} \cdot 229 $
• Root number: 1
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$3$
Group:	$S_4$
Conductor:	$33663= 3 \cdot 7^{2} \cdot 229 $
Artin number field:	Splitting field of $f= x^{4} - 2 x^{3} - x^{2} - 5 x + 1 $ over $\Q$
Size of Galois orbit:	1
Smallest containing permutation representation:	$S_4$
Parity:	Odd
Determinant:	1.3_229.2t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 419 }$ to precision 5.

Roots:

$r_{ 1 }$	$=$	$ 15 + 96\cdot 419 + 77\cdot 419^{2} + 191\cdot 419^{3} + 315\cdot 419^{4} +O\left(419^{ 5 }\right)$
$r_{ 2 }$	$=$	$ 191 + 256\cdot 419 + 241\cdot 419^{2} + 9\cdot 419^{3} + 107\cdot 419^{4} +O\left(419^{ 5 }\right)$
$r_{ 3 }$	$=$	$ 293 + 207\cdot 419 + 203\cdot 419^{2} + 312\cdot 419^{3} + 409\cdot 419^{4} +O\left(419^{ 5 }\right)$
$r_{ 4 }$	$=$	$ 341 + 277\cdot 419 + 315\cdot 419^{2} + 324\cdot 419^{3} + 5\cdot 419^{4} +O\left(419^{ 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.