Galois orbit of Artin representations 5.14876449.10t13.a

• Label: 5.14876449.10t13.a
• Dimension: $5$
• Group: $S_5$
• Conductor: $14876449$
• Indicator: $1$

Basic invariants

Dimension:	$5$
Group:	$S_5$
Conductor:	\(14876449\)\(\medspace = 7^{2} \cdot 19^{2} \cdot 29^{2} \)
Frobenius-Schur indicator:	$1$
Root number:	$1$
Artin number field:	Galois closure of 5.1.3857.1
Galois orbit size:	$1$
Smallest permutation container:	$S_5$
Parity:	even
Projective image:	$S_5$
Projective field:	Galois closure of 5.1.3857.1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 61 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 61 }$: \( x^{2} + 60x + 2 \)

Roots:

$r_{ 1 }$	$=$	\( 10 a + 57 + \left(37 a + 10\right)\cdot 61 + \left(9 a + 57\right)\cdot 61^{2} + 33\cdot 61^{3} + \left(4 a + 4\right)\cdot 61^{4} +O(61^{5})\)
$r_{ 2 }$	$=$	\( 51 a + 6 + \left(23 a + 38\right)\cdot 61 + \left(51 a + 29\right)\cdot 61^{2} + \left(60 a + 24\right)\cdot 61^{3} + \left(56 a + 8\right)\cdot 61^{4} +O(61^{5})\)
$r_{ 3 }$	$=$	\( 31 a + 11 + \left(53 a + 28\right)\cdot 61 + \left(30 a + 43\right)\cdot 61^{2} + \left(8 a + 37\right)\cdot 61^{3} + \left(21 a + 3\right)\cdot 61^{4} +O(61^{5})\)
$r_{ 4 }$	$=$	\( 7 + 55\cdot 61 + 31\cdot 61^{2} + 10\cdot 61^{3} + 28\cdot 61^{4} +O(61^{5})\)
$r_{ 5 }$	$=$	\( 30 a + 42 + \left(7 a + 50\right)\cdot 61 + \left(30 a + 20\right)\cdot 61^{2} + \left(52 a + 15\right)\cdot 61^{3} + \left(39 a + 16\right)\cdot 61^{4} +O(61^{5})\)

Generators of the action on the roots $r_1, \ldots, r_{ 5 }$

Cycle notation

$(1,2)$

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

Character values on conjugacy classes

Size	Order	Action on $r_1, \ldots, r_{ 5 }$	Character values
			$c1$
$1$	$1$	$()$	$5$
$10$	$2$	$(1,2)$	$1$
$15$	$2$	$(1,2)(3,4)$	$1$
$20$	$3$	$(1,2,3)$	$-1$
$30$	$4$	$(1,2,3,4)$	$-1$
$24$	$5$	$(1,2,3,4,5)$	$0$
$20$	$6$	$(1,2,3)(4,5)$	$1$

The blue line marks the conjugacy class containing complex conjugation.