Galois orbit of Artin representations 5.17e4_101e3.6t16.1

• Label: 5.17e4_101e3.6t16.1
• Dimension: 5
• Group: $S_6$
• Conductor: $ 17^{4} \cdot 101^{3}$
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$5$
Group:	$S_6$
Conductor:	$86051769821= 17^{4} \cdot 101^{3} $
Artin number field:	Splitting field of $f= x^{6} - 2 x^{5} + x^{4} + x^{3} - 2 x^{2} + x + 1 $ over $\Q$
Size of Galois orbit:	1
Smallest containing permutation representation:	$S_6$
Parity:	Even

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 109 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 109 }$: $ x^{2} + 108 x + 6 $

Roots:

$r_{ 1 }$	$=$	$ 97 a + 59 + \left(40 a + 17\right)\cdot 109 + \left(86 a + 40\right)\cdot 109^{2} + \left(2 a + 24\right)\cdot 109^{3} + \left(15 a + 40\right)\cdot 109^{4} +O\left(109^{ 5 }\right)$
$r_{ 2 }$	$=$	$ 5 a + 80 + \left(a + 76\right)\cdot 109 + \left(92 a + 36\right)\cdot 109^{2} + \left(3 a + 18\right)\cdot 109^{3} + \left(82 a + 75\right)\cdot 109^{4} +O\left(109^{ 5 }\right)$
$r_{ 3 }$	$=$	$ 12 a + 47 + \left(68 a + 70\right)\cdot 109 + \left(22 a + 85\right)\cdot 109^{2} + \left(106 a + 49\right)\cdot 109^{3} + \left(93 a + 52\right)\cdot 109^{4} +O\left(109^{ 5 }\right)$
$r_{ 4 }$	$=$	$ 93 a + 37 + \left(33 a + 74\right)\cdot 109 + \left(3 a + 33\right)\cdot 109^{2} + \left(82 a + 58\right)\cdot 109^{3} + \left(51 a + 72\right)\cdot 109^{4} +O\left(109^{ 5 }\right)$
$r_{ 5 }$	$=$	$ 104 a + 85 + \left(107 a + 72\right)\cdot 109 + \left(16 a + 18\right)\cdot 109^{2} + \left(105 a + 39\right)\cdot 109^{3} + \left(26 a + 44\right)\cdot 109^{4} +O\left(109^{ 5 }\right)$
$r_{ 6 }$	$=$	$ 16 a + 21 + \left(75 a + 15\right)\cdot 109 + \left(105 a + 3\right)\cdot 109^{2} + \left(26 a + 28\right)\cdot 109^{3} + \left(57 a + 42\right)\cdot 109^{4} +O\left(109^{ 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
			$c1$
$1$	$1$	$()$	$5$
$15$	$2$	$(1,2)(3,4)(5,6)$	$3$
$15$	$2$	$(1,2)$	$-1$
$45$	$2$	$(1,2)(3,4)$	$1$
$40$	$3$	$(1,2,3)(4,5,6)$	$2$
$40$	$3$	$(1,2,3)$	$-1$
$90$	$4$	$(1,2,3,4)(5,6)$	$-1$
$90$	$4$	$(1,2,3,4)$	$1$
$144$	$5$	$(1,2,3,4,5)$	$0$
$120$	$6$	$(1,2,3,4,5,6)$	$0$
$120$	$6$	$(1,2,3)(4,5)$	$-1$

The blue line marks the conjugacy class containing complex conjugation.