Galois orbit of Artin representations 3.5e2_7e2_107e2.6t8.3

• Label: 3.5e2_7e2_107e2.6t8.3
• Dimension: 3
• Group: $S_4$
• Conductor: $ 5^{2} \cdot 7^{2} \cdot 107^{2}$
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$3$
Group:	$S_4$
Conductor:	$14025025= 5^{2} \cdot 7^{2} \cdot 107^{2} $
Artin number field:	Splitting field of $f= x^{4} - x^{3} + 7 x^{2} + 10 x - 7 $ 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 $\Q_{ 163 }$ to precision 5.

Roots:

$r_{ 1 }$	$=$	$ 56 + 89\cdot 163 + 107\cdot 163^{2} + 19\cdot 163^{3} + 102\cdot 163^{4} +O\left(163^{ 5 }\right)$
$r_{ 2 }$	$=$	$ 70 + 132\cdot 163 + 35\cdot 163^{2} + 30\cdot 163^{3} + 122\cdot 163^{4} +O\left(163^{ 5 }\right)$
$r_{ 3 }$	$=$	$ 93 + 120\cdot 163 + 66\cdot 163^{2} + 18\cdot 163^{3} + 123\cdot 163^{4} +O\left(163^{ 5 }\right)$
$r_{ 4 }$	$=$	$ 108 + 146\cdot 163 + 115\cdot 163^{2} + 94\cdot 163^{3} + 141\cdot 163^{4} +O\left(163^{ 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.