Galois orbit of Artin representations 3.2e2_3e2_271.6t11.2

• Label: 3.2e2_3e2_271.6t11.2
• Dimension: 3
• Group: $S_4\times C_2$
• Conductor: $ 2^{2} \cdot 3^{2} \cdot 271 $
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$3$
Group:	$S_4\times C_2$
Conductor:	$9756= 2^{2} \cdot 3^{2} \cdot 271 $
Artin number field:	Splitting field of $f= x^{6} - x^{5} - 3 x^{3} - 9 x + 27 $ 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 $\Q_{ 43 }$ to precision 7.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 43 }$: $ x^{2} + 42 x + 3 $

Roots:

$r_{ 1 }$	$=$	$ 10 + 7\cdot 43 + 28\cdot 43^{2} + 5\cdot 43^{3} + 33\cdot 43^{4} + 38\cdot 43^{5} + 12\cdot 43^{6} +O\left(43^{ 7 }\right)$
$r_{ 2 }$	$=$	$ 20 a + 23 + \left(2 a + 8\right)\cdot 43 + \left(35 a + 3\right)\cdot 43^{2} + \left(28 a + 24\right)\cdot 43^{3} + \left(41 a + 8\right)\cdot 43^{4} + \left(3 a + 2\right)\cdot 43^{5} + 15 a\cdot 43^{6} +O\left(43^{ 7 }\right)$
$r_{ 3 }$	$=$	$ 23 a + \left(40 a + 34\right)\cdot 43 + \left(7 a + 35\right)\cdot 43^{2} + \left(14 a + 17\right)\cdot 43^{3} + \left(a + 21\right)\cdot 43^{4} + \left(39 a + 7\right)\cdot 43^{5} + \left(27 a + 11\right)\cdot 43^{6} +O\left(43^{ 7 }\right)$
$r_{ 4 }$	$=$	$ 28 a + 15 + \left(16 a + 7\right)\cdot 43 + \left(12 a + 13\right)\cdot 43^{2} + \left(3 a + 33\right)\cdot 43^{3} + \left(26 a + 37\right)\cdot 43^{4} + \left(4 a + 30\right)\cdot 43^{5} + \left(23 a + 16\right)\cdot 43^{6} +O\left(43^{ 7 }\right)$
$r_{ 5 }$	$=$	$ 15 a + \left(26 a + 39\right)\cdot 43 + \left(30 a + 8\right)\cdot 43^{2} + \left(39 a + 24\right)\cdot 43^{3} + \left(16 a + 17\right)\cdot 43^{4} + \left(38 a + 9\right)\cdot 43^{5} + \left(19 a + 35\right)\cdot 43^{6} +O\left(43^{ 7 }\right)$
$r_{ 6 }$	$=$	$ 39 + 32\cdot 43 + 39\cdot 43^{2} + 23\cdot 43^{3} + 10\cdot 43^{4} + 40\cdot 43^{5} + 9\cdot 43^{6} +O\left(43^{ 7 }\right)$

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

Cycle notation

$(1,6)$

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.