Galois orbit of Artin representations 10.2e6_3e12_149e4.30t176.1

• Label: 10.2e6_3e12_149e4.30t176.1
• Dimension: 10
• Group: $S_6$
• Conductor: $ 2^{6} \cdot 3^{12} \cdot 149^{4}$
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$10$
Group:	$S_6$
Conductor:	$16764094652917824= 2^{6} \cdot 3^{12} \cdot 149^{4} $
Artin number field:	Splitting field of $f= x^{6} - x^{5} + 2 x^{4} + x^{3} - 2 x^{2} + x + 1 $ over $\Q$
Size of Galois orbit:	1
Smallest containing permutation representation:	30T176
Parity:	Even

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 139 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 139 }$: $ x^{2} + 138 x + 2 $

Roots:

$r_{ 1 }$	$=$	$ 136 a + 125 + \left(110 a + 62\right)\cdot 139 + \left(135 a + 120\right)\cdot 139^{2} + \left(78 a + 121\right)\cdot 139^{3} + \left(43 a + 107\right)\cdot 139^{4} +O\left(139^{ 5 }\right)$
$r_{ 2 }$	$=$	$ 3 a + 122 + \left(28 a + 37\right)\cdot 139 + \left(3 a + 6\right)\cdot 139^{2} + \left(60 a + 65\right)\cdot 139^{3} + \left(95 a + 72\right)\cdot 139^{4} +O\left(139^{ 5 }\right)$
$r_{ 3 }$	$=$	$ 93 a + 72 + \left(12 a + 137\right)\cdot 139 + \left(7 a + 84\right)\cdot 139^{2} + \left(109 a + 52\right)\cdot 139^{3} + \left(55 a + 46\right)\cdot 139^{4} +O\left(139^{ 5 }\right)$
$r_{ 4 }$	$=$	$ 22 a + 95 + \left(133 a + 74\right)\cdot 139 + \left(116 a + 1\right)\cdot 139^{2} + \left(2 a + 138\right)\cdot 139^{3} + \left(55 a + 2\right)\cdot 139^{4} +O\left(139^{ 5 }\right)$
$r_{ 5 }$	$=$	$ 117 a + 117 + \left(5 a + 46\right)\cdot 139 + \left(22 a + 124\right)\cdot 139^{2} + \left(136 a + 23\right)\cdot 139^{3} + \left(83 a + 55\right)\cdot 139^{4} +O\left(139^{ 5 }\right)$
$r_{ 6 }$	$=$	$ 46 a + 26 + \left(126 a + 57\right)\cdot 139 + \left(131 a + 79\right)\cdot 139^{2} + \left(29 a + 15\right)\cdot 139^{3} + \left(83 a + 132\right)\cdot 139^{4} +O\left(139^{ 5 }\right)$

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

Cycle notation

$(1,2)$

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

Character values on conjugacy classes

Size	Order	Action on $r_1, \ldots, r_{ 6 }$	Character values
			$c1$
$1$	$1$	$()$	$10$
$15$	$2$	$(1,2)(3,4)(5,6)$	$-2$
$15$	$2$	$(1,2)$	$2$
$45$	$2$	$(1,2)(3,4)$	$-2$
$40$	$3$	$(1,2,3)(4,5,6)$	$1$
$40$	$3$	$(1,2,3)$	$1$
$90$	$4$	$(1,2,3,4)(5,6)$	$0$
$90$	$4$	$(1,2,3,4)$	$0$
$144$	$5$	$(1,2,3,4,5)$	$0$
$120$	$6$	$(1,2,3,4,5,6)$	$1$
$120$	$6$	$(1,2,3)(4,5)$	$-1$

The blue line marks the conjugacy class containing complex conjugation.