Galois orbit of Artin representations 5.840243610690147.6t16.a

• Label: 5.840...147.6t16.a
• Dimension: $5$
• Group: $S_6$
• Conductor: $8.402\times 10^{14}$
• Indicator: $1$

Basic invariants

Dimension:	$5$
Group:	$S_6$
Conductor:	\(840243610690147\)\(\medspace = 197^{3} \cdot 479^{3} \)
Frobenius-Schur indicator:	$1$
Root number:	$1$
Artin number field:	Galois closure of 6.4.94363.1
Galois orbit size:	$1$
Smallest permutation container:	$S_6$
Parity:	odd
Projective image:	$S_6$
Projective field:	Galois closure of 6.4.94363.1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 23 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 23 }$: \( x^{2} + 21x + 5 \)

Roots:

$r_{ 1 }$	$=$	\( 2 + 3\cdot 23 + 3\cdot 23^{2} + 18\cdot 23^{3} + 12\cdot 23^{4} +O(23^{5})\)
$r_{ 2 }$	$=$	\( 16 a + 11 + \left(21 a + 8\right)\cdot 23 + \left(6 a + 13\right)\cdot 23^{2} + \left(22 a + 21\right)\cdot 23^{3} + \left(8 a + 22\right)\cdot 23^{4} +O(23^{5})\)
$r_{ 3 }$	$=$	\( 11 + 18\cdot 23 + 5\cdot 23^{3} + 8\cdot 23^{4} +O(23^{5})\)
$r_{ 4 }$	$=$	\( 5 a + 19 + \left(15 a + 11\right)\cdot 23 + \left(15 a + 3\right)\cdot 23^{2} + \left(9 a + 15\right)\cdot 23^{3} + \left(21 a + 9\right)\cdot 23^{4} +O(23^{5})\)
$r_{ 5 }$	$=$	\( 7 a + 20 + \left(a + 12\right)\cdot 23 + \left(16 a + 5\right)\cdot 23^{2} + 13\cdot 23^{3} + \left(14 a + 18\right)\cdot 23^{4} +O(23^{5})\)
$r_{ 6 }$	$=$	\( 18 a + 6 + \left(7 a + 14\right)\cdot 23 + \left(7 a + 19\right)\cdot 23^{2} + \left(13 a + 18\right)\cdot 23^{3} + \left(a + 19\right)\cdot 23^{4} +O(23^{5})\)

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

Cycle notation

$(1,2)$

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

Character values on conjugacy classes

Size	Order	Action on $r_1, \ldots, r_{ 6 }$	Character values
			$c1$
$1$	$1$	$()$	$5$
$15$	$2$	$(1,2)(3,4)(5,6)$	$3$
$15$	$2$	$(1,2)$	$-1$
$45$	$2$	$(1,2)(3,4)$	$1$
$40$	$3$	$(1,2,3)(4,5,6)$	$2$
$40$	$3$	$(1,2,3)$	$-1$
$90$	$4$	$(1,2,3,4)(5,6)$	$-1$
$90$	$4$	$(1,2,3,4)$	$1$
$144$	$5$	$(1,2,3,4,5)$	$0$
$120$	$6$	$(1,2,3,4,5,6)$	$0$
$120$	$6$	$(1,2,3)(4,5)$	$-1$

The blue line marks the conjugacy class containing complex conjugation.