Artin representation 3.3e2_13e2_41e2.6t8.1c1

• Label: 3.3e2_13e2_41e2.6t8.1c1
• Dimension: 3
• Group: $S_4$
• Conductor: $ 3^{2} \cdot 13^{2} \cdot 41^{2}$
• Root number: 1
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$3$
Group:	$S_4$
Conductor:	$2556801= 3^{2} \cdot 13^{2} \cdot 41^{2} $
Artin number field:	Splitting field of $f= x^{4} - x^{3} + 5 x - 2 $ 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 $\Q_{ 311 }$ to precision 5.

Roots:

$r_{ 1 }$	$=$	$ 45 + 192\cdot 311 + 87\cdot 311^{2} + 202\cdot 311^{3} + 126\cdot 311^{4} +O\left(311^{ 5 }\right)$
$r_{ 2 }$	$=$	$ 115 + 133\cdot 311 + 48\cdot 311^{2} + 241\cdot 311^{3} + 212\cdot 311^{4} +O\left(311^{ 5 }\right)$
$r_{ 3 }$	$=$	$ 170 + 9\cdot 311 + 280\cdot 311^{2} + 293\cdot 311^{3} + 93\cdot 311^{4} +O\left(311^{ 5 }\right)$
$r_{ 4 }$	$=$	$ 293 + 286\cdot 311 + 205\cdot 311^{2} + 195\cdot 311^{3} + 188\cdot 311^{4} +O\left(311^{ 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.