Galois orbit of Artin representations 5.2e8_41e4.6t12.1

• Label: 5.2e8_41e4.6t12.1
• Dimension: 5
• Group: $A_5$
• Conductor: $ 2^{8} \cdot 41^{4}$
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$5$
Group:	$A_5$
Conductor:	$723394816= 2^{8} \cdot 41^{4} $
Artin number field:	Splitting field of $f= x^{5} - 2 x^{4} + 8 x^{3} - 12 x^{2} + 17 x - 14 $ 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 $\Q_{ 397 }$ to precision 5.

Roots:

$r_{ 1 }$	$=$	$ 66 + 125\cdot 397 + 103\cdot 397^{2} + 244\cdot 397^{3} + 31\cdot 397^{4} +O\left(397^{ 5 }\right)$
$r_{ 2 }$	$=$	$ 209 + 189\cdot 397 + 133\cdot 397^{2} + 339\cdot 397^{3} + 207\cdot 397^{4} +O\left(397^{ 5 }\right)$
$r_{ 3 }$	$=$	$ 247 + 339\cdot 397 + 276\cdot 397^{2} + 291\cdot 397^{3} + 42\cdot 397^{4} +O\left(397^{ 5 }\right)$
$r_{ 4 }$	$=$	$ 277 + 145\cdot 397 + 196\cdot 397^{2} + 258\cdot 397^{3} + 247\cdot 397^{4} +O\left(397^{ 5 }\right)$
$r_{ 5 }$	$=$	$ 394 + 390\cdot 397 + 83\cdot 397^{2} + 57\cdot 397^{3} + 264\cdot 397^{4} +O\left(397^{ 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.