Galois orbit of Artin representations 3.2e3_7_107e2.6t11.2

• Label: 3.2e3_7_107e2.6t11.2
• Dimension: 3
• Group: $S_4\times C_2$
• Conductor: $ 2^{3} \cdot 7 \cdot 107^{2}$
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$3$
Group:	$S_4\times C_2$
Conductor:	$641144= 2^{3} \cdot 7 \cdot 107^{2} $
Artin number field:	Splitting field of $f= x^{6} - x^{5} + x^{4} + 2 x^{3} + 5 x^{2} - x + 1 $ 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 $\Q_{ 113 }$ to precision 7.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 113 }$: $ x^{2} + 101 x + 3 $

Roots:

$r_{ 1 }$	$=$	$ 43 a + 91 + \left(91 a + 36\right)\cdot 113 + \left(38 a + 94\right)\cdot 113^{2} + \left(70 a + 45\right)\cdot 113^{3} + \left(59 a + 67\right)\cdot 113^{4} + \left(70 a + 23\right)\cdot 113^{5} + \left(93 a + 92\right)\cdot 113^{6} +O\left(113^{ 7 }\right)$
$r_{ 2 }$	$=$	$ 36 a + 100 + \left(33 a + 13\right)\cdot 113 + \left(71 a + 60\right)\cdot 113^{2} + \left(38 a + 76\right)\cdot 113^{3} + \left(26 a + 102\right)\cdot 113^{4} + \left(55 a + 51\right)\cdot 113^{5} + \left(47 a + 36\right)\cdot 113^{6} +O\left(113^{ 7 }\right)$
$r_{ 3 }$	$=$	$ 70 a + 42 + \left(21 a + 73\right)\cdot 113 + \left(74 a + 16\right)\cdot 113^{2} + \left(42 a + 60\right)\cdot 113^{3} + \left(53 a + 34\right)\cdot 113^{4} + \left(42 a + 19\right)\cdot 113^{5} + \left(19 a + 15\right)\cdot 113^{6} +O\left(113^{ 7 }\right)$
$r_{ 4 }$	$=$	$ 77 a + 80 + \left(79 a + 38\right)\cdot 113 + \left(41 a + 91\right)\cdot 113^{2} + \left(74 a + 16\right)\cdot 113^{3} + \left(86 a + 41\right)\cdot 113^{4} + \left(57 a + 10\right)\cdot 113^{5} + \left(65 a + 99\right)\cdot 113^{6} +O\left(113^{ 7 }\right)$
$r_{ 5 }$	$=$	$ 107 + 77\cdot 113 + 101\cdot 113^{2} + 93\cdot 113^{3} + 31\cdot 113^{4} + 75\cdot 113^{5} + 77\cdot 113^{6} +O\left(113^{ 7 }\right)$
$r_{ 6 }$	$=$	$ 33 + 98\cdot 113 + 87\cdot 113^{2} + 45\cdot 113^{3} + 61\cdot 113^{4} + 45\cdot 113^{5} + 18\cdot 113^{6} +O\left(113^{ 7 }\right)$

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

Cycle notation

$(1,4)$

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.