Galois orbit of Artin representations 2.1728.6t3.c

• Label: 2.1728.6t3.c
• Dimension: $2$
• Group: $D_{6}$
• Conductor: $1728$
• Indicator: $1$

Basic invariants

Dimension:	$2$
Group:	$D_{6}$
Conductor:	\(1728\)\(\medspace = 2^{6} \cdot 3^{3} \)
Frobenius-Schur indicator:	$1$
Root number:	$1$
Artin number field:	Galois closure of 6.2.1492992.4
Galois orbit size:	$1$
Smallest permutation container:	$D_{6}$
Parity:	odd
Projective image:	$S_3$
Projective field:	Galois closure of 3.1.108.1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 17 }$ to precision 5.

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

Roots:

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

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.