Galois orbit of Artin representations 3.2e6_3e2_7e2_11e2.6t8.3

• Label: 3.2e6_3e2_7e2_11e2.6t8.3
• Dimension: 3
• Group: $S_4$
• Conductor: $ 2^{6} \cdot 3^{2} \cdot 7^{2} \cdot 11^{2}$
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$3$
Group:	$S_4$
Conductor:	$3415104= 2^{6} \cdot 3^{2} \cdot 7^{2} \cdot 11^{2} $
Artin number field:	Splitting field of $f= x^{4} - 2 x^{3} + 6 x^{2} + 4 x - 2 $ over $\Q$
Size of Galois orbit:	1
Smallest containing permutation representation:	$S_4$
Parity:	Even

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 197 }$ to precision 5.

Roots:

$r_{ 1 }$	$=$	$ 50 + 196\cdot 197 + 181\cdot 197^{2} + 22\cdot 197^{3} + 56\cdot 197^{4} +O\left(197^{ 5 }\right)$
$r_{ 2 }$	$=$	$ 54 + 157\cdot 197 + 196\cdot 197^{2} + 136\cdot 197^{3} + 7\cdot 197^{4} +O\left(197^{ 5 }\right)$
$r_{ 3 }$	$=$	$ 126 + 90\cdot 197 + 25\cdot 197^{2} + 142\cdot 197^{3} + 79\cdot 197^{4} +O\left(197^{ 5 }\right)$
$r_{ 4 }$	$=$	$ 166 + 146\cdot 197 + 186\cdot 197^{2} + 91\cdot 197^{3} + 53\cdot 197^{4} +O\left(197^{ 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 values
			$c1$
$1$	$1$	$()$	$3$
$3$	$2$	$(1,2)(3,4)$	$-1$
$6$	$2$	$(1,2)$	$-1$
$8$	$3$	$(1,2,3)$	$0$
$6$	$4$	$(1,2,3,4)$	$1$

The blue line marks the conjugacy class containing complex conjugation.