Galois orbit of Artin representations 5.2e4_5e2_19e4.6t12.1

• Label: 5.2e4_5e2_19e4.6t12.1
• Dimension: 5
• Group: $A_5$
• Conductor: $ 2^{4} \cdot 5^{2} \cdot 19^{4}$
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$5$
Group:	$A_5$
Conductor:	$52128400= 2^{4} \cdot 5^{2} \cdot 19^{4} $
Artin number field:	Splitting field of $f= x^{5} - x^{4} - 11 x^{3} + 6 x^{2} + 64 x - 74 $ 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 $\Q_{ 557 }$ to precision 5.

Roots:

$r_{ 1 }$	$=$	$ 14 + 251\cdot 557 + 288\cdot 557^{2} + 375\cdot 557^{3} + 218\cdot 557^{4} +O\left(557^{ 5 }\right)$
$r_{ 2 }$	$=$	$ 235 + 435\cdot 557 + 247\cdot 557^{2} + 321\cdot 557^{3} + 239\cdot 557^{4} +O\left(557^{ 5 }\right)$
$r_{ 3 }$	$=$	$ 375 + 224\cdot 557 + 116\cdot 557^{2} + 75\cdot 557^{3} + 92\cdot 557^{4} +O\left(557^{ 5 }\right)$
$r_{ 4 }$	$=$	$ 506 + 262\cdot 557 + 255\cdot 557^{2} + 489\cdot 557^{3} + 304\cdot 557^{4} +O\left(557^{ 5 }\right)$
$r_{ 5 }$	$=$	$ 542 + 496\cdot 557 + 205\cdot 557^{2} + 409\cdot 557^{3} + 258\cdot 557^{4} +O\left(557^{ 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.