Galois orbit of Artin representations 4.480077552.12t34.a

• Label: 4.480077552.12t34.a
• Dimension: $4$
• Group: $C_3^2:D_4$
• Conductor: $480077552$
• Indicator: $1$

Basic invariants

Dimension:	$4$
Group:	$C_3^2:D_4$
Conductor:	\(480077552\)\(\medspace = 2^{4} \cdot 17^{2} \cdot 47^{3} \)
Frobenius-Schur indicator:	$1$
Root number:	$1$
Artin number field:	Galois closure of 6.4.3694576.1
Galois orbit size:	$1$
Smallest permutation container:	12T34
Parity:	odd
Projective image:	$\SOPlus(4,2)$
Projective field:	Galois closure of 6.4.3694576.1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 59 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 59 }$: \( x^{2} + 58x + 2 \)

Roots:

$r_{ 1 }$	$=$	\( 20 + 47\cdot 59 + 56\cdot 59^{2} + 23\cdot 59^{3} + 23\cdot 59^{4} +O(59^{5})\)
$r_{ 2 }$	$=$	\( 56 a + 29 + \left(37 a + 23\right)\cdot 59 + \left(53 a + 51\right)\cdot 59^{2} + \left(18 a + 16\right)\cdot 59^{3} + \left(32 a + 1\right)\cdot 59^{4} +O(59^{5})\)
$r_{ 3 }$	$=$	\( 3 a + 26 + \left(21 a + 5\right)\cdot 59 + \left(5 a + 8\right)\cdot 59^{2} + \left(40 a + 41\right)\cdot 59^{3} + \left(26 a + 14\right)\cdot 59^{4} +O(59^{5})\)
$r_{ 4 }$	$=$	\( 53 a + 47 + \left(2 a + 51\right)\cdot 59 + \left(14 a + 20\right)\cdot 59^{2} + \left(34 a + 39\right)\cdot 59^{3} + \left(11 a + 40\right)\cdot 59^{4} +O(59^{5})\)
$r_{ 5 }$	$=$	\( 6 a + 41 + \left(56 a + 1\right)\cdot 59 + \left(44 a + 32\right)\cdot 59^{2} + 24 a\cdot 59^{3} + \left(47 a + 18\right)\cdot 59^{4} +O(59^{5})\)
$r_{ 6 }$	$=$	\( 15 + 47\cdot 59 + 7\cdot 59^{2} + 55\cdot 59^{3} + 19\cdot 59^{4} +O(59^{5})\)

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

Cycle notation

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

$(1,2)$

$(1,2,3)$

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.