Galois orbit of Artin representations 3.31e2_53.6t11.1

• Label: 3.31e2_53.6t11.1
• Dimension: 3
• Group: $S_4\times C_2$
• Conductor: $ 31^{2} \cdot 53 $
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$3$
Group:	$S_4\times C_2$
Conductor:	$50933= 31^{2} \cdot 53 $
Artin number field:	Splitting field of $f= x^{6} - 2 x^{5} - x^{4} + 4 x^{3} - 3 x^{2} + 3 x - 1 $ over $\Q$
Size of Galois orbit:	1
Smallest containing permutation representation:	$S_4\times C_2$
Parity:	Even

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 37 }$ to precision 7.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 37 }$: $ x^{2} + 33 x + 2 $

Roots:

$r_{ 1 }$	$=$	$ 15 + 35\cdot 37 + 23\cdot 37^{2} + 2\cdot 37^{3} + 14\cdot 37^{4} + 25\cdot 37^{5} + 7\cdot 37^{6} +O\left(37^{ 7 }\right)$
$r_{ 2 }$	$=$	$ 21 a + 20 + \left(10 a + 11\right)\cdot 37 + \left(14 a + 12\right)\cdot 37^{2} + \left(13 a + 1\right)\cdot 37^{3} + \left(9 a + 29\right)\cdot 37^{4} + \left(6 a + 21\right)\cdot 37^{5} + \left(22 a + 7\right)\cdot 37^{6} +O\left(37^{ 7 }\right)$
$r_{ 3 }$	$=$	$ 5 a + 7 + \left(35 a + 31\right)\cdot 37 + \left(11 a + 25\right)\cdot 37^{2} + \left(8 a + 16\right)\cdot 37^{3} + \left(33 a + 17\right)\cdot 37^{4} + \left(2 a + 14\right)\cdot 37^{5} + \left(23 a + 26\right)\cdot 37^{6} +O\left(37^{ 7 }\right)$
$r_{ 4 }$	$=$	$ 16 a + 30 + \left(26 a + 32\right)\cdot 37 + \left(22 a + 21\right)\cdot 37^{2} + \left(23 a + 3\right)\cdot 37^{3} + \left(27 a + 16\right)\cdot 37^{4} + 30 a\cdot 37^{5} + \left(14 a + 16\right)\cdot 37^{6} +O\left(37^{ 7 }\right)$
$r_{ 5 }$	$=$	$ 32 a + 27 + \left(a + 18\right)\cdot 37 + \left(25 a + 1\right)\cdot 37^{2} + \left(28 a + 1\right)\cdot 37^{3} + \left(3 a + 31\right)\cdot 37^{4} + \left(34 a + 29\right)\cdot 37^{5} + \left(13 a + 4\right)\cdot 37^{6} +O\left(37^{ 7 }\right)$
$r_{ 6 }$	$=$	$ 14 + 18\cdot 37 + 25\cdot 37^{2} + 11\cdot 37^{3} + 3\cdot 37^{4} + 19\cdot 37^{5} + 11\cdot 37^{6} +O\left(37^{ 7 }\right)$

Generators of the action on the roots $r_1, \ldots, r_{ 6 }$

Cycle notation

$(1,2,4)(3,5,6)$

$(2,3)$

$(2,4)(3,5)$

Character values on conjugacy classes

Size	Order	Action on $r_1, \ldots, r_{ 6 }$	Character values
			$c1$
$1$	$1$	$()$	$3$
$1$	$2$	$(1,6)(2,3)(4,5)$	$-3$
$3$	$2$	$(1,6)$	$1$
$3$	$2$	$(1,6)(2,3)$	$-1$
$6$	$2$	$(2,4)(3,5)$	$-1$
$6$	$2$	$(1,6)(2,4)(3,5)$	$1$
$8$	$3$	$(1,2,4)(3,5,6)$	$0$
$6$	$4$	$(1,3,6,2)$	$-1$
$6$	$4$	$(1,5,6,4)(2,3)$	$1$
$8$	$6$	$(1,3,5,6,2,4)$	$0$

The blue line marks the conjugacy class containing complex conjugation.