Galois orbit of Artin representations 3.2e2_3e4_5e3.4t5.1

• Label: 3.2e2_3e4_5e3.4t5.1
• Dimension: 3
• Group: $S_4$
• Conductor: $ 2^{2} \cdot 3^{4} \cdot 5^{3}$
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$3$
Group:	$S_4$
Conductor:	$40500= 2^{2} \cdot 3^{4} \cdot 5^{3} $
Artin number field:	Splitting field of $f= x^{4} - x^{3} + 6 x^{2} - 6 x + 6 $ 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 $\Q_{ 379 }$ to precision 5.

Roots:

$r_{ 1 }$	$=$	$ 116 + 12\cdot 379 + 223\cdot 379^{2} + 58\cdot 379^{3} + 317\cdot 379^{4} +O\left(379^{ 5 }\right)$
$r_{ 2 }$	$=$	$ 201 + 248\cdot 379 + 268\cdot 379^{2} + 154\cdot 379^{3} + 9\cdot 379^{4} +O\left(379^{ 5 }\right)$
$r_{ 3 }$	$=$	$ 216 + 284\cdot 379 + 21\cdot 379^{2} + 155\cdot 379^{3} + 153\cdot 379^{4} +O\left(379^{ 5 }\right)$
$r_{ 4 }$	$=$	$ 226 + 212\cdot 379 + 244\cdot 379^{2} + 10\cdot 379^{3} + 278\cdot 379^{4} +O\left(379^{ 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.