Galois orbit of Artin representations 3.2e6_3e2_5e2_7.4t5.1

• Label: 3.2e6_3e2_5e2_7.4t5.1
• Dimension: 3
• Group: $S_4$
• Conductor: $ 2^{6} \cdot 3^{2} \cdot 5^{2} \cdot 7 $
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$3$
Group:	$S_4$
Conductor:	$100800= 2^{6} \cdot 3^{2} \cdot 5^{2} \cdot 7 $
Artin number field:	Splitting field of $f= x^{4} - 4 x^{2} - 24 x - 8 $ over $\Q$
Size of Galois orbit:	1
Smallest containing permutation representation:	$S_4$
Parity:	Odd

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 191 }$ to precision 5.

Roots:

$r_{ 1 }$	$=$	$ 12 + 74\cdot 191 + 23\cdot 191^{2} + 29\cdot 191^{3} + 100\cdot 191^{4} +O\left(191^{ 5 }\right)$
$r_{ 2 }$	$=$	$ 34 + 71\cdot 191 + 13\cdot 191^{2} + 145\cdot 191^{3} + 111\cdot 191^{4} +O\left(191^{ 5 }\right)$
$r_{ 3 }$	$=$	$ 37 + 98\cdot 191 + 184\cdot 191^{2} + 11\cdot 191^{3} + 182\cdot 191^{4} +O\left(191^{ 5 }\right)$
$r_{ 4 }$	$=$	$ 108 + 138\cdot 191 + 160\cdot 191^{2} + 4\cdot 191^{3} + 179\cdot 191^{4} +O\left(191^{ 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.