Galois orbit of Artin representations 2.2e6_7e2_107.6t3.2

• Label: 2.2e6_7e2_107.6t3.2
• Dimension: 2
• Group: $D_{6}$
• Conductor: $ 2^{6} \cdot 7^{2} \cdot 107 $
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$2$
Group:	$D_{6}$
Conductor:	$335552= 2^{6} \cdot 7^{2} \cdot 107 $
Artin number field:	Splitting field of $f= x^{6} - 2 x^{5} + 49 x^{4} - 66 x^{3} + 629 x^{2} - 320 x + 1838 $ over $\Q$
Size of Galois orbit:	1
Smallest containing permutation representation:	$D_{6}$
Parity:	Odd

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 59 }$ to precision 7.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 59 }$: $ x^{2} + 58 x + 2 $

Roots:

$r_{ 1 }$	$=$	$ 35 a + 41 + \left(45 a + 5\right)\cdot 59 + \left(44 a + 49\right)\cdot 59^{2} + \left(13 a + 13\right)\cdot 59^{3} + \left(11 a + 30\right)\cdot 59^{4} + \left(37 a + 53\right)\cdot 59^{5} + \left(20 a + 19\right)\cdot 59^{6} +O\left(59^{ 7 }\right)$
$r_{ 2 }$	$=$	$ 35 a + 30 + \left(45 a + 40\right)\cdot 59 + \left(44 a + 29\right)\cdot 59^{2} + \left(13 a + 15\right)\cdot 59^{3} + \left(11 a + 32\right)\cdot 59^{4} + \left(37 a + 40\right)\cdot 59^{5} + \left(20 a + 12\right)\cdot 59^{6} +O\left(59^{ 7 }\right)$
$r_{ 3 }$	$=$	$ 24 a + 6 + \left(13 a + 51\right)\cdot 59 + \left(14 a + 28\right)\cdot 59^{2} + \left(45 a + 43\right)\cdot 59^{3} + \left(47 a + 29\right)\cdot 59^{4} + \left(21 a + 7\right)\cdot 59^{5} + \left(38 a + 55\right)\cdot 59^{6} +O\left(59^{ 7 }\right)$
$r_{ 4 }$	$=$	$ 48 + 43\cdot 59 + 49\cdot 59^{2} + 57\cdot 59^{4} + 33\cdot 59^{5} + 46\cdot 59^{6} +O\left(59^{ 7 }\right)$
$r_{ 5 }$	$=$	$ 37 + 19\cdot 59 + 30\cdot 59^{2} + 2\cdot 59^{3} + 21\cdot 59^{5} + 39\cdot 59^{6} +O\left(59^{ 7 }\right)$
$r_{ 6 }$	$=$	$ 24 a + 17 + \left(13 a + 16\right)\cdot 59 + \left(14 a + 48\right)\cdot 59^{2} + \left(45 a + 41\right)\cdot 59^{3} + \left(47 a + 27\right)\cdot 59^{4} + \left(21 a + 20\right)\cdot 59^{5} + \left(38 a + 3\right)\cdot 59^{6} +O\left(59^{ 7 }\right)$

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

Cycle notation

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

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.