Galois orbit of Artin representations 4.1304625.6t13.a

• Label: 4.1304625.6t13.a
• Dimension: $4$
• Group: $C_3^2:D_4$
• Conductor: $1304625$
• Indicator: $1$

Basic invariants

Dimension:	$4$
Group:	$C_3^2:D_4$
Conductor:	\(1304625\)\(\medspace = 3 \cdot 5^{3} \cdot 7^{2} \cdot 71 \)
Frobenius-Schur indicator:	$1$
Root number:	$1$
Artin number field:	Galois closure of 6.4.42278236875.1
Galois orbit size:	$1$
Smallest permutation container:	$C_3^2:D_4$
Parity:	even
Projective image:	$\SOPlus(4,2)$
Projective field:	Galois closure of 6.4.42278236875.1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 11 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 11 }$: \( x^{2} + 7x + 2 \)

Roots:

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

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

Cycle notation

$(4,5,6)$

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

$(4,5)$

Character values on conjugacy classes

Size	Order	Action on $r_1, \ldots, r_{ 6 }$	Character values
			$c1$
$1$	$1$	$()$	$4$
$6$	$2$	$(1,4)(2,5)(3,6)$	$2$
$6$	$2$	$(4,5)$	$0$
$9$	$2$	$(1,2)(4,5)$	$0$
$4$	$3$	$(1,2,3)(4,5,6)$	$1$
$4$	$3$	$(1,2,3)$	$-2$
$18$	$4$	$(1,4,2,5)(3,6)$	$0$
$12$	$6$	$(1,5,2,6,3,4)$	$-1$
$12$	$6$	$(1,2,3)(4,5)$	$0$

The blue line marks the conjugacy class containing complex conjugation.