Galois orbit of Artin representations 5.25747.6t16.a

• Label: 5.25747.6t16.a
• Dimension: $5$
• Group: $S_6$
• Conductor: $25747$
• Indicator: $1$

Basic invariants

Dimension:	$5$
Group:	$S_6$
Conductor:	\(25747\)
Frobenius-Schur indicator:	$1$
Root number:	$1$
Artin number field:	Galois closure of 6.0.25747.1
Galois orbit size:	$1$
Smallest permutation container:	$S_6$
Parity:	odd
Projective image:	$S_6$
Projective field:	Galois closure of 6.0.25747.1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 41 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 41 }$: \( x^{2} + 38x + 6 \)

Roots:

$r_{ 1 }$	$=$	\( 3 a + 14 + \left(20 a + 3\right)\cdot 41 + \left(13 a + 39\right)\cdot 41^{2} + \left(38 a + 40\right)\cdot 41^{3} + \left(16 a + 22\right)\cdot 41^{4} +O(41^{5})\)
$r_{ 2 }$	$=$	\( 18 + 27\cdot 41 + 29\cdot 41^{2} + 3\cdot 41^{3} + 32\cdot 41^{4} +O(41^{5})\)
$r_{ 3 }$	$=$	\( 35 a + \left(2 a + 30\right)\cdot 41 + \left(38 a + 35\right)\cdot 41^{2} + \left(40 a + 35\right)\cdot 41^{3} + \left(35 a + 25\right)\cdot 41^{4} +O(41^{5})\)
$r_{ 4 }$	$=$	\( 6 a + 23 + \left(38 a + 3\right)\cdot 41 + \left(2 a + 24\right)\cdot 41^{2} + 38\cdot 41^{3} + \left(5 a + 10\right)\cdot 41^{4} +O(41^{5})\)
$r_{ 5 }$	$=$	\( 5 + 39\cdot 41 + 16\cdot 41^{2} + 25\cdot 41^{3} + 36\cdot 41^{4} +O(41^{5})\)
$r_{ 6 }$	$=$	\( 38 a + 23 + \left(20 a + 19\right)\cdot 41 + \left(27 a + 18\right)\cdot 41^{2} + \left(2 a + 19\right)\cdot 41^{3} + \left(24 a + 35\right)\cdot 41^{4} +O(41^{5})\)

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
			$c1$
$1$	$1$	$()$	$5$
$15$	$2$	$(1,2)(3,4)(5,6)$	$-1$
$15$	$2$	$(1,2)$	$3$
$45$	$2$	$(1,2)(3,4)$	$1$
$40$	$3$	$(1,2,3)(4,5,6)$	$-1$
$40$	$3$	$(1,2,3)$	$2$
$90$	$4$	$(1,2,3,4)(5,6)$	$-1$
$90$	$4$	$(1,2,3,4)$	$1$
$144$	$5$	$(1,2,3,4,5)$	$0$
$120$	$6$	$(1,2,3,4,5,6)$	$-1$
$120$	$6$	$(1,2,3)(4,5)$	$0$

The blue line marks the conjugacy class containing complex conjugation.