Galois orbit of Artin representations 5.2e6_331e2.6t12.1

• Label: 5.2e6_331e2.6t12.1
• Dimension: 5
• Group: $A_5$
• Conductor: $ 2^{6} \cdot 331^{2}$
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$5$
Group:	$A_5$
Conductor:	$7011904= 2^{6} \cdot 331^{2} $
Artin number field:	Splitting field of $f= x^{5} - x^{4} + 4 x^{3} - 20 x^{2} + 16 x + 8 $ 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 $\Q_{ 97 }$ to precision 5.

Roots:

$r_{ 1 }$	$=$	$ 4 + 76\cdot 97 + 35\cdot 97^{2} + 47\cdot 97^{3} + 83\cdot 97^{4} +O\left(97^{ 5 }\right)$
$r_{ 2 }$	$=$	$ 27 + 16\cdot 97 + 33\cdot 97^{2} + 39\cdot 97^{3} + 55\cdot 97^{4} +O\left(97^{ 5 }\right)$
$r_{ 3 }$	$=$	$ 44 + 34\cdot 97 + 42\cdot 97^{2} + 57\cdot 97^{3} + 18\cdot 97^{4} +O\left(97^{ 5 }\right)$
$r_{ 4 }$	$=$	$ 50 + 93\cdot 97 + 62\cdot 97^{2} + 88\cdot 97^{3} + 77\cdot 97^{4} +O\left(97^{ 5 }\right)$
$r_{ 5 }$	$=$	$ 70 + 70\cdot 97 + 19\cdot 97^{2} + 58\cdot 97^{3} + 55\cdot 97^{4} +O\left(97^{ 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.