Galois orbit of Artin representations 2.2592.6t3.c

• Label: 2.2592.6t3.c
• Dimension: $2$
• Group: $D_{6}$
• Conductor: $2592$
• Indicator: $1$

Basic invariants

Dimension:	$2$
Group:	$D_{6}$
Conductor:	\(2592\)\(\medspace = 2^{5} \cdot 3^{4} \)
Frobenius-Schur indicator:	$1$
Root number:	$1$
Artin number field:	Galois closure of 6.0.6718464.3
Galois orbit size:	$1$
Smallest permutation container:	$D_{6}$
Parity:	odd
Projective image:	$S_3$
Projective field:	Galois closure of 3.1.648.1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 13 }$ to precision 6.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 13 }$: \( x^{2} + 12x + 2 \)

Roots:

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

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

Cycle notation

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.