Galois orbit of Artin representations 2.1027.6t5.b

• Label: 2.1027.6t5.b
• Dimension: $2$
• Group: $S_3\times C_3$
• Conductor: $1027$
• Indicator: $0$

Basic invariants

Dimension:	$2$
Group:	$S_3\times C_3$
Conductor:	$1027$$\medspace = 13 \cdot 79$
Artin number field:	Galois closure of 6.0.83323591.1
Galois orbit size:	$2$
Smallest permutation container:	$S_3\times C_3$
Parity:	odd
Projective image:	$S_3$
Projective field:	Galois closure of 3.1.13351.1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 47 }$ to precision 6.

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

Roots:

$r_{ 1 }$	$=$	$16 a + 24 + \left(34 a + 42\right)\cdot 47 + \left(34 a + 32\right)\cdot 47^{2} + \left(46 a + 9\right)\cdot 47^{3} + \left(20 a + 40\right)\cdot 47^{4} + \left(24 a + 38\right)\cdot 47^{5} +O(47^{6})$
$r_{ 2 }$	$=$	$24 a + 18 + \left(43 a + 29\right)\cdot 47 + \left(43 a + 2\right)\cdot 47^{2} + \left(38 a + 30\right)\cdot 47^{3} + \left(21 a + 3\right)\cdot 47^{4} + \left(27 a + 1\right)\cdot 47^{5} +O(47^{6})$
$r_{ 3 }$	$=$	$7 a + 6 + \left(16 a + 22\right)\cdot 47 + \left(15 a + 11\right)\cdot 47^{2} + \left(8 a + 7\right)\cdot 47^{3} + \left(4 a + 3\right)\cdot 47^{4} + \left(42 a + 7\right)\cdot 47^{5} +O(47^{6})$
$r_{ 4 }$	$=$	$40 a + 20 + 30 a\cdot 47 + \left(31 a + 26\right)\cdot 47^{2} + \left(38 a + 8\right)\cdot 47^{3} + \left(42 a + 3\right)\cdot 47^{4} + \left(4 a + 40\right)\cdot 47^{5} +O(47^{6})$
$r_{ 5 }$	$=$	$23 a + 19 + \left(3 a + 45\right)\cdot 47 + \left(3 a + 46\right)\cdot 47^{2} + \left(8 a + 16\right)\cdot 47^{3} + \left(25 a + 8\right)\cdot 47^{4} + \left(19 a + 34\right)\cdot 47^{5} +O(47^{6})$
$r_{ 6 }$	$=$	$31 a + 9 + \left(12 a + 1\right)\cdot 47 + \left(12 a + 21\right)\cdot 47^{2} + 21\cdot 47^{3} + \left(26 a + 35\right)\cdot 47^{4} + \left(22 a + 19\right)\cdot 47^{5} +O(47^{6})$

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

Cycle notation

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

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

$(4,6,5)$

Character values on conjugacy classes

Size	Order	Action on $r_1, \ldots, r_{ 6 }$	Character values
			$c1$	$c2$
$1$	$1$	$()$	$2$	$2$
$3$	$2$	$(1,4)(2,6)(3,5)$	$0$	$0$
$1$	$3$	$(1,2,3)(4,6,5)$	$2 \zeta_{3}$	$-2 \zeta_{3} - 2$
$1$	$3$	$(1,3,2)(4,5,6)$	$-2 \zeta_{3} - 2$	$2 \zeta_{3}$
$2$	$3$	$(1,3,2)(4,6,5)$	$-1$	$-1$
$2$	$3$	$(4,6,5)$	$\zeta_{3} + 1$	$-\zeta_{3}$
$2$	$3$	$(4,5,6)$	$-\zeta_{3}$	$\zeta_{3} + 1$
$3$	$6$	$(1,5,2,4,3,6)$	$0$	$0$
$3$	$6$	$(1,6,3,4,2,5)$	$0$	$0$

The blue line marks the conjugacy class containing complex conjugation.