Artin representation 3.2e6_3e2_61e2.6t8.1c1

• Label: 3.2e6_3e2_61e2.6t8.1c1
• Dimension: 3
• Group: $S_4$
• Conductor: $ 2^{6} \cdot 3^{2} \cdot 61^{2}$
• Root number: 1
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$3$
Group:	$S_4$
Conductor:	$2143296= 2^{6} \cdot 3^{2} \cdot 61^{2} $
Artin number field:	Splitting field of $f= x^{4} - x^{3} + 3 x^{2} + 2 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 $\Q_{ 233 }$ to precision 5.

Roots:

$r_{ 1 }$	$=$	$ 105 + 25\cdot 233 + 20\cdot 233^{2} + 215\cdot 233^{3} + 124\cdot 233^{4} +O\left(233^{ 5 }\right)$
$r_{ 2 }$	$=$	$ 165 + 42\cdot 233 + 169\cdot 233^{2} + 5\cdot 233^{3} + 153\cdot 233^{4} +O\left(233^{ 5 }\right)$
$r_{ 3 }$	$=$	$ 213 + 205\cdot 233 + 67\cdot 233^{2} + 8\cdot 233^{3} + 233^{4} +O\left(233^{ 5 }\right)$
$r_{ 4 }$	$=$	$ 217 + 191\cdot 233 + 208\cdot 233^{2} + 3\cdot 233^{3} + 187\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.