Artin representation 5.2e6_3e4_5e4.12t183.2c1

• Label: 5.2e6_3e4_5e4.12t183.2c1
• Dimension: 5
• Group: $S_6$
• Conductor: $ 2^{6} \cdot 3^{4} \cdot 5^{4}$
• Root number: 1
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$5$
Group:	$S_6$
Conductor:	$3240000= 2^{6} \cdot 3^{4} \cdot 5^{4} $
Artin number field:	Splitting field of $f= x^{6} + 2 x^{4} - 2 x^{3} - 3 x^{2} - 6 x - 3 $ over $\Q$
Size of Galois orbit:	1
Smallest containing permutation representation:	12T183
Parity:	Even
Determinant:	1.1.1t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 137 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 137 }$: $ x^{2} + 131 x + 3 $

Roots:

$r_{ 1 }$	$=$	$ 41 a + 70 + \left(43 a + 31\right)\cdot 137 + \left(73 a + 12\right)\cdot 137^{2} + \left(136 a + 9\right)\cdot 137^{3} + \left(94 a + 98\right)\cdot 137^{4} +O\left(137^{ 5 }\right)$
$r_{ 2 }$	$=$	$ 116 a + 57 + \left(90 a + 11\right)\cdot 137 + \left(14 a + 57\right)\cdot 137^{2} + \left(99 a + 99\right)\cdot 137^{3} + \left(66 a + 19\right)\cdot 137^{4} +O\left(137^{ 5 }\right)$
$r_{ 3 }$	$=$	$ 102 a + 55 + \left(101 a + 132\right)\cdot 137 + \left(21 a + 61\right)\cdot 137^{2} + \left(60 a + 86\right)\cdot 137^{3} + \left(44 a + 96\right)\cdot 137^{4} +O\left(137^{ 5 }\right)$
$r_{ 4 }$	$=$	$ 35 a + 119 + \left(35 a + 92\right)\cdot 137 + \left(115 a + 90\right)\cdot 137^{2} + \left(76 a + 14\right)\cdot 137^{3} + \left(92 a + 29\right)\cdot 137^{4} +O\left(137^{ 5 }\right)$
$r_{ 5 }$	$=$	$ 96 a + 42 + \left(93 a + 113\right)\cdot 137 + \left(63 a + 134\right)\cdot 137^{2} + 69\cdot 137^{3} + \left(42 a + 120\right)\cdot 137^{4} +O\left(137^{ 5 }\right)$
$r_{ 6 }$	$=$	$ 21 a + 68 + \left(46 a + 29\right)\cdot 137 + \left(122 a + 54\right)\cdot 137^{2} + \left(37 a + 131\right)\cdot 137^{3} + \left(70 a + 46\right)\cdot 137^{4} +O\left(137^{ 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 value
$1$	$1$	$()$	$5$
$15$	$2$	$(1,2)(3,4)(5,6)$	$1$
$15$	$2$	$(1,2)$	$-3$
$45$	$2$	$(1,2)(3,4)$	$1$
$40$	$3$	$(1,2,3)(4,5,6)$	$-1$
$40$	$3$	$(1,2,3)$	$2$
$90$	$4$	$(1,2,3,4)(5,6)$	$-1$
$90$	$4$	$(1,2,3,4)$	$-1$
$144$	$5$	$(1,2,3,4,5)$	$0$
$120$	$6$	$(1,2,3,4,5,6)$	$1$
$120$	$6$	$(1,2,3)(4,5)$	$0$

The blue line marks the conjugacy class containing complex conjugation.