Galois orbit of Artin representations 3.3_13_31.6t11.2

• Label: 3.3_13_31.6t11.2
• Dimension: 3
• Group: $S_4\times C_2$
• Conductor: $ 3 \cdot 13 \cdot 31 $
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$3$
Group:	$S_4\times C_2$
Conductor:	$1209= 3 \cdot 13 \cdot 31 $
Artin number field:	Splitting field of $f= x^{6} - x^{5} + 3 x^{4} - 3 x^{3} + 3 x^{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 $\Q_{ 47 }$ to precision 7.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 47 }$: $ x^{2} + 45 x + 5 $

Roots:

$r_{ 1 }$	$=$	$ 14 + 45\cdot 47 + 15\cdot 47^{2} + 5\cdot 47^{3} + 43\cdot 47^{4} + 13\cdot 47^{5} + 36\cdot 47^{6} +O\left(47^{ 7 }\right)$
$r_{ 2 }$	$=$	$ 18 a + 13 + \left(3 a + 37\right)\cdot 47 + \left(23 a + 13\right)\cdot 47^{2} + \left(11 a + 11\right)\cdot 47^{3} + \left(11 a + 24\right)\cdot 47^{4} + \left(25 a + 40\right)\cdot 47^{5} + \left(25 a + 2\right)\cdot 47^{6} +O\left(47^{ 7 }\right)$
$r_{ 3 }$	$=$	$ 11 a + 27 + \left(24 a + 6\right)\cdot 47 + \left(7 a + 19\right)\cdot 47^{2} + \left(28 a + 6\right)\cdot 47^{3} + \left(34 a + 6\right)\cdot 47^{4} + \left(18 a + 6\right)\cdot 47^{5} + \left(2 a + 9\right)\cdot 47^{6} +O\left(47^{ 7 }\right)$
$r_{ 4 }$	$=$	$ 36 a + 2 + \left(22 a + 44\right)\cdot 47 + \left(39 a + 9\right)\cdot 47^{2} + \left(18 a + 8\right)\cdot 47^{3} + 12 a\cdot 47^{4} + \left(28 a + 9\right)\cdot 47^{5} + \left(44 a + 42\right)\cdot 47^{6} +O\left(47^{ 7 }\right)$
$r_{ 5 }$	$=$	$ 29 a + 2 + \left(43 a + 26\right)\cdot 47 + \left(23 a + 9\right)\cdot 47^{2} + \left(35 a + 11\right)\cdot 47^{3} + \left(35 a + 35\right)\cdot 47^{4} + \left(21 a + 32\right)\cdot 47^{5} + \left(21 a + 28\right)\cdot 47^{6} +O\left(47^{ 7 }\right)$
$r_{ 6 }$	$=$	$ 37 + 28\cdot 47 + 25\cdot 47^{2} + 4\cdot 47^{3} + 32\cdot 47^{4} + 38\cdot 47^{5} + 21\cdot 47^{6} +O\left(47^{ 7 }\right)$

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

Cycle notation

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

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

$(3,4)$

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.