Galois orbit of Artin representations 3.3e6_7e2.6t8.1

• Label: 3.3e6_7e2.6t8.1
• Dimension: 3
• Group: $S_4$
• Conductor: $ 3^{6} \cdot 7^{2}$
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$3$
Group:	$S_4$
Conductor:	$35721= 3^{6} \cdot 7^{2} $
Artin number field:	Splitting field of $f= x^{4} - x^{3} - 3 x^{2} - 7 x + 1 $ 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 $\Q_{ 211 }$ to precision 5.

Roots:

$r_{ 1 }$	$=$	$ 11 + 155\cdot 211 + 23\cdot 211^{2} + 30\cdot 211^{3} + 195\cdot 211^{4} +O\left(211^{ 5 }\right)$
$r_{ 2 }$	$=$	$ 31 + 161\cdot 211 + 50\cdot 211^{2} + 194\cdot 211^{3} + 58\cdot 211^{4} +O\left(211^{ 5 }\right)$
$r_{ 3 }$	$=$	$ 66 + 112\cdot 211 + 167\cdot 211^{2} + 60\cdot 211^{3} + 148\cdot 211^{4} +O\left(211^{ 5 }\right)$
$r_{ 4 }$	$=$	$ 104 + 204\cdot 211 + 179\cdot 211^{2} + 136\cdot 211^{3} + 19\cdot 211^{4} +O\left(211^{ 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.