Galois orbit of Artin representations 3.3e2_109.6t11.2

• Label: 3.3e2_109.6t11.2
• Dimension: 3
• Group: $S_4\times C_2$
• Conductor: $ 3^{2} \cdot 109 $
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$3$
Group:	$S_4\times C_2$
Conductor:	$981= 3^{2} \cdot 109 $
Artin number field:	Splitting field of $f= x^{6} - 2 x^{5} + 2 x^{4} + x^{3} + 2 x^{2} - 2 x + 1 $ over $\Q$
Size of Galois orbit:	1
Smallest containing permutation representation:	$S_4\times C_2$
Parity:	Even

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 67 }$ to precision 7.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 67 }$: $ x^{2} + 63 x + 2 $

Roots:

$r_{ 1 }$	$=$	$ 7 a + 36 + \left(40 a + 13\right)\cdot 67 + \left(31 a + 15\right)\cdot 67^{2} + \left(16 a + 39\right)\cdot 67^{3} + \left(40 a + 23\right)\cdot 67^{4} + \left(43 a + 4\right)\cdot 67^{5} + \left(38 a + 13\right)\cdot 67^{6} +O\left(67^{ 7 }\right)$
$r_{ 2 }$	$=$	$ 6 a + 50 + \left(24 a + 40\right)\cdot 67 + \left(24 a + 63\right)\cdot 67^{2} + \left(57 a + 15\right)\cdot 67^{3} + \left(57 a + 21\right)\cdot 67^{4} + \left(30 a + 36\right)\cdot 67^{5} + \left(7 a + 33\right)\cdot 67^{6} +O\left(67^{ 7 }\right)$
$r_{ 3 }$	$=$	$ 50 + 54\cdot 67 + 59\cdot 67^{2} + 62\cdot 67^{3} + 60\cdot 67^{4} + 33\cdot 67^{5} + 26\cdot 67^{6} +O\left(67^{ 7 }\right)$
$r_{ 4 }$	$=$	$ 63 + 61\cdot 67 + 24\cdot 67^{2} + 56\cdot 67^{3} + 66\cdot 67^{4} + 19\cdot 67^{5} + 38\cdot 67^{6} +O\left(67^{ 7 }\right)$
$r_{ 5 }$	$=$	$ 60 a + 64 + \left(26 a + 32\right)\cdot 67 + \left(35 a + 34\right)\cdot 67^{2} + \left(50 a + 6\right)\cdot 67^{3} + \left(26 a + 34\right)\cdot 67^{4} + \left(23 a + 4\right)\cdot 67^{5} + \left(28 a + 57\right)\cdot 67^{6} +O\left(67^{ 7 }\right)$
$r_{ 6 }$	$=$	$ 61 a + 7 + \left(42 a + 64\right)\cdot 67 + \left(42 a + 2\right)\cdot 67^{2} + \left(9 a + 20\right)\cdot 67^{3} + \left(9 a + 61\right)\cdot 67^{4} + \left(36 a + 34\right)\cdot 67^{5} + \left(59 a + 32\right)\cdot 67^{6} +O\left(67^{ 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
			$c1$
$1$	$1$	$()$	$3$
$1$	$2$	$(1,6)(2,5)(3,4)$	$-3$
$3$	$2$	$(1,6)$	$1$
$3$	$2$	$(1,6)(2,5)$	$-1$
$6$	$2$	$(2,3)(4,5)$	$1$
$6$	$2$	$(1,6)(2,3)(4,5)$	$-1$
$8$	$3$	$(1,2,3)(4,6,5)$	$0$
$6$	$4$	$(1,5,6,2)$	$1$
$6$	$4$	$(1,6)(2,4,5,3)$	$-1$
$8$	$6$	$(1,5,4,6,2,3)$	$0$

The blue line marks the conjugacy class containing complex conjugation.