Galois orbit of Artin representations 5.2e6_3e2_59e2.6t12.1

• Label: 5.2e6_3e2_59e2.6t12.1
• Dimension: 5
• Group: $A_5$
• Conductor: $ 2^{6} \cdot 3^{2} \cdot 59^{2}$
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$5$
Group:	$A_5$
Conductor:	$2005056= 2^{6} \cdot 3^{2} \cdot 59^{2} $
Artin number field:	Splitting field of $f= x^{5} - x^{4} + 2 x^{3} - 2 x^{2} + 25 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 $\Q_{ 409 }$ to precision 5.

Roots:

$r_{ 1 }$	$=$	$ 238 + 165\cdot 409 + 359\cdot 409^{2} + 111\cdot 409^{3} + 367\cdot 409^{4} +O\left(409^{ 5 }\right)$
$r_{ 2 }$	$=$	$ 294 + 378\cdot 409 + 316\cdot 409^{2} + 407\cdot 409^{3} + 351\cdot 409^{4} +O\left(409^{ 5 }\right)$
$r_{ 3 }$	$=$	$ 352 + 50\cdot 409 + 2\cdot 409^{2} + 255\cdot 409^{3} + 121\cdot 409^{4} +O\left(409^{ 5 }\right)$
$r_{ 4 }$	$=$	$ 356 + 109\cdot 409 + 310\cdot 409^{2} + 341\cdot 409^{3} + 45\cdot 409^{4} +O\left(409^{ 5 }\right)$
$r_{ 5 }$	$=$	$ 397 + 112\cdot 409 + 238\cdot 409^{2} + 110\cdot 409^{3} + 340\cdot 409^{4} +O\left(409^{ 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.