Galois orbit of Artin representations 5.204004089.6t12.a

• Label: 5.204004089.6t12.a
• Dimension: $5$
• Group: $A_5$
• Conductor: $204004089$
• Indicator: $1$

Basic invariants

Dimension:	$5$
Group:	$A_5$
Conductor:	\(204004089\)\(\medspace = 3^{6} \cdot 23^{4} \)
Frobenius-Schur indicator:	$1$
Root number:	$1$
Artin number field:	Galois closure of 5.1.42849.1
Galois orbit size:	$1$
Smallest permutation container:	$\PSL(2,5)$
Parity:	even
Projective image:	$A_5$
Projective field:	Galois closure of 5.1.42849.1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 13 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 13 }$: \( x^{2} + 12x + 2 \)

Roots:

$r_{ 1 }$	$=$	\( 2 a + 9 + \left(a + 5\right)\cdot 13 + \left(3 a + 7\right)\cdot 13^{2} + \left(a + 7\right)\cdot 13^{3} + \left(4 a + 4\right)\cdot 13^{4} +O(13^{5})\)
$r_{ 2 }$	$=$	\( 11 a + 11 + \left(11 a + 4\right)\cdot 13 + \left(9 a + 9\right)\cdot 13^{2} + \left(11 a + 5\right)\cdot 13^{3} + \left(8 a + 7\right)\cdot 13^{4} +O(13^{5})\)
$r_{ 3 }$	$=$	\( 6 + 4\cdot 13 + 7\cdot 13^{2} + 8\cdot 13^{3} + 9\cdot 13^{4} +O(13^{5})\)
$r_{ 4 }$	$=$	\( 7 a + 10 + \left(2 a + 7\right)\cdot 13 + \left(6 a + 5\right)\cdot 13^{2} + \left(10 a + 6\right)\cdot 13^{3} + \left(3 a + 5\right)\cdot 13^{4} +O(13^{5})\)
$r_{ 5 }$	$=$	\( 6 a + 4 + \left(10 a + 3\right)\cdot 13 + \left(6 a + 9\right)\cdot 13^{2} + \left(2 a + 10\right)\cdot 13^{3} + \left(9 a + 11\right)\cdot 13^{4} +O(13^{5})\)

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

Cycle notation

$(1,2,3)$

$(3,4,5)$

Character values on conjugacy classes

Size	Order	Action on $r_1, \ldots, r_{ 5 }$	Character values
			$c1$
$1$	$1$	$()$	$5$
$15$	$2$	$(1,2)(3,4)$	$1$
$20$	$3$	$(1,2,3)$	$-1$
$12$	$5$	$(1,2,3,4,5)$	$0$
$12$	$5$	$(1,3,4,5,2)$	$0$

The blue line marks the conjugacy class containing complex conjugation.