Galois orbit of Artin representations 5.2e8_167e2.6t12.1

• Label: 5.2e8_167e2.6t12.1
• Dimension: 5
• Group: $A_5$
• Conductor: $ 2^{8} \cdot 167^{2}$
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$5$
Group:	$A_5$
Conductor:	$7139584= 2^{8} \cdot 167^{2} $
Artin number field:	Splitting field of $f= x^{5} - x^{4} + 8 x^{2} + 9 x - 1 $ 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 $\Q_{ 479 }$ to precision 5.

Roots:

$r_{ 1 }$	$=$	$ 105 + 299\cdot 479 + 184\cdot 479^{2} + 429\cdot 479^{3} + 253\cdot 479^{4} +O\left(479^{ 5 }\right)$
$r_{ 2 }$	$=$	$ 193 + 272\cdot 479 + 152\cdot 479^{2} + 427\cdot 479^{3} + 275\cdot 479^{4} +O\left(479^{ 5 }\right)$
$r_{ 3 }$	$=$	$ 340 + 208\cdot 479 + 4\cdot 479^{2} + 33\cdot 479^{3} + 387\cdot 479^{4} +O\left(479^{ 5 }\right)$
$r_{ 4 }$	$=$	$ 366 + 379\cdot 479 + 360\cdot 479^{2} + 441\cdot 479^{3} + 181\cdot 479^{4} +O\left(479^{ 5 }\right)$
$r_{ 5 }$	$=$	$ 434 + 276\cdot 479 + 255\cdot 479^{2} + 105\cdot 479^{3} + 338\cdot 479^{4} +O\left(479^{ 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.