Galois orbit of Artin representations 2.680.6t3.a

• Label: 2.680.6t3.a
• Dimension: $2$
• Group: $D_{6}$
• Conductor: $680$
• Indicator: $1$

Basic invariants

Dimension:	$2$
Group:	$D_{6}$
Conductor:	\(680\)\(\medspace = 2^{3} \cdot 5 \cdot 17 \)
Frobenius-Schur indicator:	$1$
Root number:	$1$
Artin number field:	Galois closure of 6.2.7860800.1
Galois orbit size:	$1$
Smallest permutation container:	$D_{6}$
Parity:	odd
Projective image:	$S_3$
Projective field:	Galois closure of 3.1.680.1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 43 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 43 }$: \( x^{2} + 42x + 3 \)

Roots:

$r_{ 1 }$	$=$	\( 36 + 8\cdot 43 + 29\cdot 43^{2} + 39\cdot 43^{3} + 23\cdot 43^{4} +O(43^{5})\)
$r_{ 2 }$	$=$	\( 15 a + 1 + \left(23 a + 22\right)\cdot 43 + \left(31 a + 13\right)\cdot 43^{2} + \left(7 a + 9\right)\cdot 43^{3} + \left(26 a + 9\right)\cdot 43^{4} +O(43^{5})\)
$r_{ 3 }$	$=$	\( 28 a + 16 + \left(19 a + 30\right)\cdot 43 + \left(11 a + 21\right)\cdot 43^{2} + \left(35 a + 28\right)\cdot 43^{3} + \left(16 a + 27\right)\cdot 43^{4} +O(43^{5})\)
$r_{ 4 }$	$=$	\( 33 a + 28 + \left(26 a + 41\right)\cdot 43 + \left(26 a + 10\right)\cdot 43^{2} + \left(15 a + 7\right)\cdot 43^{3} + \left(13 a + 41\right)\cdot 43^{4} +O(43^{5})\)
$r_{ 5 }$	$=$	\( 10 a + 18 + \left(16 a + 35\right)\cdot 43 + \left(16 a + 10\right)\cdot 43^{2} + \left(27 a + 39\right)\cdot 43^{3} + \left(29 a + 38\right)\cdot 43^{4} +O(43^{5})\)
$r_{ 6 }$	$=$	\( 31 + 33\cdot 43 + 42\cdot 43^{2} + 4\cdot 43^{3} + 31\cdot 43^{4} +O(43^{5})\)

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

Cycle notation

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

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.