Galois orbit of Artin representations 2.728.6t3.b

• Label: 2.728.6t3.b
• Dimension: $2$
• Group: $D_{6}$
• Conductor: $728$
• Indicator: $1$

Basic invariants

Dimension:	$2$
Group:	$D_{6}$
Conductor:	$728$$\medspace = 2^{3} \cdot 7 \cdot 13$
Frobenius-Schur indicator:	$1$
Root number:	$1$
Artin number field:	Galois closure of 6.2.4239872.1
Galois orbit size:	$1$
Smallest permutation container:	$D_{6}$
Parity:	odd
Projective image:	$S_3$
Projective field:	Galois closure of 3.1.728.1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 17 }$ to precision 7.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 17 }$: $x^{2} + 16x + 3$

Roots:

$r_{ 1 }$	$=$	$9 + 12\cdot 17 + 8\cdot 17^{2} + 3\cdot 17^{3} + 10\cdot 17^{4} + 9\cdot 17^{5} + 17^{6} +O(17^{7})$
$r_{ 2 }$	$=$	$7 a + 1 + \left(16 a + 6\right)\cdot 17 + \left(8 a + 16\right)\cdot 17^{2} + \left(12 a + 4\right)\cdot 17^{3} + \left(4 a + 7\right)\cdot 17^{4} + \left(7 a + 2\right)\cdot 17^{5} + \left(10 a + 6\right)\cdot 17^{6} +O(17^{7})$
$r_{ 3 }$	$=$	$13 + 11\cdot 17 + 6\cdot 17^{2} + 3\cdot 17^{4} + 10\cdot 17^{5} + 6\cdot 17^{6} +O(17^{7})$
$r_{ 4 }$	$=$	$11 a + 14 + \left(12 a + 1\right)\cdot 17 + \left(6 a + 8\right)\cdot 17^{2} + \left(10 a + 6\right)\cdot 17^{3} + \left(7 a + 8\right)\cdot 17^{4} + \left(5 a + 4\right)\cdot 17^{5} + \left(9 a + 3\right)\cdot 17^{6} +O(17^{7})$
$r_{ 5 }$	$=$	$10 a + 8 + 15\cdot 17 + \left(8 a + 8\right)\cdot 17^{2} + \left(4 a + 8\right)\cdot 17^{3} + \left(12 a + 16\right)\cdot 17^{4} + \left(9 a + 4\right)\cdot 17^{5} + \left(6 a + 9\right)\cdot 17^{6} +O(17^{7})$
$r_{ 6 }$	$=$	$6 a + 8 + \left(4 a + 3\right)\cdot 17 + \left(10 a + 2\right)\cdot 17^{2} + \left(6 a + 10\right)\cdot 17^{3} + \left(9 a + 5\right)\cdot 17^{4} + \left(11 a + 2\right)\cdot 17^{5} + \left(7 a + 7\right)\cdot 17^{6} +O(17^{7})$

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

Cycle notation

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

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.