Galois orbit of Artin representations 3.2e2_23_37.6t11.1

• Label: 3.2e2_23_37.6t11.1
• Dimension: 3
• Group: $S_4\times C_2$
• Conductor: $ 2^{2} \cdot 23 \cdot 37 $
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$3$
Group:	$S_4\times C_2$
Conductor:	$3404= 2^{2} \cdot 23 \cdot 37 $
Artin number field:	Splitting field of $f= x^{6} - 3 x^{5} + 9 x^{4} - 13 x^{3} + 14 x^{2} - 8 x + 2 $ over $\Q$
Size of Galois orbit:	1
Smallest containing permutation representation:	$S_4\times C_2$
Parity:	Odd

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 131 }$ to precision 7.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 131 }$: $ x^{2} + 127 x + 2 $

Roots:

$r_{ 1 }$	$=$	$ 46 a + 84 + \left(31 a + 91\right)\cdot 131 + \left(112 a + 29\right)\cdot 131^{2} + \left(9 a + 16\right)\cdot 131^{3} + \left(124 a + 32\right)\cdot 131^{4} + \left(80 a + 70\right)\cdot 131^{5} + \left(2 a + 56\right)\cdot 131^{6} +O\left(131^{ 7 }\right)$
$r_{ 2 }$	$=$	$ 85 a + 6 + \left(99 a + 40\right)\cdot 131 + \left(18 a + 54\right)\cdot 131^{2} + \left(121 a + 74\right)\cdot 131^{3} + \left(6 a + 125\right)\cdot 131^{4} + \left(50 a + 7\right)\cdot 131^{5} + \left(128 a + 117\right)\cdot 131^{6} +O\left(131^{ 7 }\right)$
$r_{ 3 }$	$=$	$ 45 + 54\cdot 131 + 46\cdot 131^{2} + 13\cdot 131^{3} + 66\cdot 131^{4} + 13\cdot 131^{5} + 72\cdot 131^{6} +O\left(131^{ 7 }\right)$
$r_{ 4 }$	$=$	$ 46 a + 126 + \left(31 a + 90\right)\cdot 131 + \left(112 a + 76\right)\cdot 131^{2} + \left(9 a + 56\right)\cdot 131^{3} + \left(124 a + 5\right)\cdot 131^{4} + \left(80 a + 123\right)\cdot 131^{5} + \left(2 a + 13\right)\cdot 131^{6} +O\left(131^{ 7 }\right)$
$r_{ 5 }$	$=$	$ 85 a + 48 + \left(99 a + 39\right)\cdot 131 + \left(18 a + 101\right)\cdot 131^{2} + \left(121 a + 114\right)\cdot 131^{3} + \left(6 a + 98\right)\cdot 131^{4} + \left(50 a + 60\right)\cdot 131^{5} + \left(128 a + 74\right)\cdot 131^{6} +O\left(131^{ 7 }\right)$
$r_{ 6 }$	$=$	$ 87 + 76\cdot 131 + 84\cdot 131^{2} + 117\cdot 131^{3} + 64\cdot 131^{4} + 117\cdot 131^{5} + 58\cdot 131^{6} +O\left(131^{ 7 }\right)$

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

Cycle notation

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

$(1,5)$

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.