Galois orbit of Artin representations 3.637.6t6.a

• Label: 3.637.6t6.a
• Dimension: $3$
• Group: $A_4\times C_2$
• Conductor: $637$
• Indicator: $1$

Basic invariants

Dimension:	$3$
Group:	$A_4\times C_2$
Conductor:	\(637\)\(\medspace = 7^{2} \cdot 13 \)
Frobenius-Schur indicator:	$1$
Root number:	$1$
Artin number field:	Galois closure of 6.2.31213.1
Galois orbit size:	$1$
Smallest permutation container:	$A_4\times C_2$
Parity:	even
Projective image:	$A_4$
Projective field:	Galois closure of 4.0.8281.1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 29 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 29 }$: \( x^{2} + 24x + 2 \)

Roots:

$r_{ 1 }$	$=$	\( 8 + 2\cdot 29 + 20\cdot 29^{2} + 11\cdot 29^{3} + 2\cdot 29^{4} +O(29^{5})\)
$r_{ 2 }$	$=$	\( 16 a + 22 + \left(2 a + 14\right)\cdot 29 + \left(6 a + 8\right)\cdot 29^{2} + \left(7 a + 28\right)\cdot 29^{3} + \left(15 a + 9\right)\cdot 29^{4} +O(29^{5})\)
$r_{ 3 }$	$=$	\( 12 a + 1 + \left(9 a + 4\right)\cdot 29 + \left(11 a + 12\right)\cdot 29^{2} + \left(12 a + 11\right)\cdot 29^{3} + \left(17 a + 21\right)\cdot 29^{4} +O(29^{5})\)
$r_{ 4 }$	$=$	\( 17 a + 3 + \left(19 a + 10\right)\cdot 29 + \left(17 a + 1\right)\cdot 29^{2} + \left(16 a + 4\right)\cdot 29^{3} + \left(11 a + 9\right)\cdot 29^{4} +O(29^{5})\)
$r_{ 5 }$	$=$	\( 13 a + 15 + \left(26 a + 11\right)\cdot 29 + \left(22 a + 7\right)\cdot 29^{2} + 21 a\cdot 29^{3} + \left(13 a + 21\right)\cdot 29^{4} +O(29^{5})\)
$r_{ 6 }$	$=$	\( 11 + 15\cdot 29 + 8\cdot 29^{2} + 2\cdot 29^{3} + 23\cdot 29^{4} +O(29^{5})\)

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

Cycle notation

$(1,6)$

$(2,5)$

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

$(3,4)$

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.