Galois orbit of Artin representations 2.704.6t3.d

• Label: 2.704.6t3.d
• Dimension: $2$
• Group: $D_{6}$
• Conductor: $704$
• Indicator: $1$

Basic invariants

Dimension:	$2$
Group:	$D_{6}$
Conductor:	$704$$\medspace = 2^{6} \cdot 11$
Frobenius-Schur indicator:	$1$
Root number:	$1$
Artin number field:	Galois closure of 6.2.247808.1
Galois orbit size:	$1$
Smallest permutation container:	$D_{6}$
Parity:	odd
Projective image:	$S_3$
Projective field:	Galois closure of 3.1.44.1

Galois action

Roots of defining polynomial

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

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 7 }$: $x^{2} + 6x + 3$

Roots:

$r_{ 1 }$	$=$	$5 a + 4 + \left(a + 5\right)\cdot 7 + \left(2 a + 2\right)\cdot 7^{2} + \left(a + 5\right)\cdot 7^{3} + \left(6 a + 1\right)\cdot 7^{4} + \left(5 a + 4\right)\cdot 7^{5} + \left(3 a + 5\right)\cdot 7^{6} +O(7^{7})$
$r_{ 2 }$	$=$	$3 + 6\cdot 7 + 3\cdot 7^{2} + 3\cdot 7^{3} + 6\cdot 7^{4} + 5\cdot 7^{5} +O(7^{7})$
$r_{ 3 }$	$=$	$2 a + 2 + \left(5 a + 2\right)\cdot 7 + \left(4 a + 3\right)\cdot 7^{2} + \left(5 a + 4\right)\cdot 7^{3} + 6\cdot 7^{4} + \left(a + 3\right)\cdot 7^{5} + \left(3 a + 3\right)\cdot 7^{6} +O(7^{7})$
$r_{ 4 }$	$=$	$2 a + 3 + \left(5 a + 1\right)\cdot 7 + \left(4 a + 4\right)\cdot 7^{2} + \left(5 a + 1\right)\cdot 7^{3} + 5\cdot 7^{4} + \left(a + 2\right)\cdot 7^{5} + \left(3 a + 1\right)\cdot 7^{6} +O(7^{7})$
$r_{ 5 }$	$=$	$4 + 3\cdot 7^{2} + 3\cdot 7^{3} + 7^{5} + 6\cdot 7^{6} +O(7^{7})$
$r_{ 6 }$	$=$	$5 a + 5 + \left(a + 4\right)\cdot 7 + \left(2 a + 3\right)\cdot 7^{2} + \left(a + 2\right)\cdot 7^{3} + 6 a\cdot 7^{4} + \left(5 a + 3\right)\cdot 7^{5} + \left(3 a + 3\right)\cdot 7^{6} +O(7^{7})$

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

Cycle notation

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.