Galois orbit of Artin representations 5.2e6_13e2_23e2.6t12.1

• Label: 5.2e6_13e2_23e2.6t12.1
• Dimension: 5
• Group: $A_5$
• Conductor: $ 2^{6} \cdot 13^{2} \cdot 23^{2}$
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$5$
Group:	$A_5$
Conductor:	$5721664= 2^{6} \cdot 13^{2} \cdot 23^{2} $
Artin number field:	Splitting field of $f= x^{5} + 8 x^{3} - 16 x^{2} + 4 x + 16 $ 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 $\Q_{ 367 }$ to precision 5.

Roots:

$r_{ 1 }$	$=$	$ 25 + 38\cdot 367 + 330\cdot 367^{2} + 114\cdot 367^{3} + 342\cdot 367^{4} +O\left(367^{ 5 }\right)$
$r_{ 2 }$	$=$	$ 35 + 241\cdot 367 + 115\cdot 367^{2} + 290\cdot 367^{3} + 290\cdot 367^{4} +O\left(367^{ 5 }\right)$
$r_{ 3 }$	$=$	$ 44 + 133\cdot 367 + 267\cdot 367^{2} + 38\cdot 367^{3} + 186\cdot 367^{4} +O\left(367^{ 5 }\right)$
$r_{ 4 }$	$=$	$ 285 + 233\cdot 367 + 138\cdot 367^{2} + 94\cdot 367^{3} + 46\cdot 367^{4} +O\left(367^{ 5 }\right)$
$r_{ 5 }$	$=$	$ 345 + 87\cdot 367 + 249\cdot 367^{2} + 195\cdot 367^{3} + 235\cdot 367^{4} +O\left(367^{ 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.