Galois orbit of Artin representations 5.2e4_167e4.6t12.1

• Label: 5.2e4_167e4.6t12.1
• Dimension: 5
• Group: $A_5$
• Conductor: $ 2^{4} \cdot 167^{4}$
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$5$
Group:	$A_5$
Conductor:	$12444741136= 2^{4} \cdot 167^{4} $
Artin number field:	Splitting field of $f= x^{5} - x^{4} - 5 x^{3} + 5 x^{2} + 5 x - 9 $ over $\Q$
Size of Galois orbit:	1
Smallest containing permutation representation:	$\PSL(2,5)$
Parity:	Even

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 311 }$ to precision 5.

Roots:

$r_{ 1 }$	$=$	$ 103 + 309\cdot 311 + 174\cdot 311^{2} + 194\cdot 311^{3} + 9\cdot 311^{4} +O\left(311^{ 5 }\right)$
$r_{ 2 }$	$=$	$ 134 + 140\cdot 311 + 113\cdot 311^{2} + 174\cdot 311^{3} + 176\cdot 311^{4} +O\left(311^{ 5 }\right)$
$r_{ 3 }$	$=$	$ 155 + 198\cdot 311 + 201\cdot 311^{2} + 265\cdot 311^{3} + 4\cdot 311^{4} +O\left(311^{ 5 }\right)$
$r_{ 4 }$	$=$	$ 255 + 185\cdot 311 + 282\cdot 311^{2} + 94\cdot 311^{3} + 219\cdot 311^{4} +O\left(311^{ 5 }\right)$
$r_{ 5 }$	$=$	$ 287 + 98\cdot 311 + 160\cdot 311^{2} + 203\cdot 311^{3} + 211\cdot 311^{4} +O\left(311^{ 5 }\right)$

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.