Artin representation 4.5e2_89e2.5t4.1c1

• Label: 4.5e2_89e2.5t4.1c1
• Dimension: 4
• Group: $A_5$
• Conductor: $ 5^{2} \cdot 89^{2}$
• Root number: 1
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$4$
Group:	$A_5$
Conductor:	$198025= 5^{2} \cdot 89^{2} $
Artin number field:	Splitting field of $f= x^{5} - x^{4} + 5 x^{3} - x^{2} + 6 x + 1 $ over $\Q$
Size of Galois orbit:	1
Smallest containing permutation representation:	$A_5$
Parity:	Even
Determinant:	1.1.1t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 137 }$ to precision 5.

Roots:

$r_{ 1 }$	$=$	$ 19 + 60\cdot 137 + 117\cdot 137^{2} + 111\cdot 137^{3} + 42\cdot 137^{4} +O\left(137^{ 5 }\right)$
$r_{ 2 }$	$=$	$ 24 + 13\cdot 137 + 60\cdot 137^{2} + 118\cdot 137^{3} + 89\cdot 137^{4} +O\left(137^{ 5 }\right)$
$r_{ 3 }$	$=$	$ 74 + 9\cdot 137 + 52\cdot 137^{2} + 86\cdot 137^{3} + 25\cdot 137^{4} +O\left(137^{ 5 }\right)$
$r_{ 4 }$	$=$	$ 78 + 131\cdot 137 + 16\cdot 137^{2} + 132\cdot 137^{3} + 40\cdot 137^{4} +O\left(137^{ 5 }\right)$
$r_{ 5 }$	$=$	$ 80 + 59\cdot 137 + 27\cdot 137^{2} + 99\cdot 137^{3} + 74\cdot 137^{4} +O\left(137^{ 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 value
$1$	$1$	$()$	$4$
$15$	$2$	$(1,2)(3,4)$	$0$
$20$	$3$	$(1,2,3)$	$1$
$12$	$5$	$(1,2,3,4,5)$	$-1$
$12$	$5$	$(1,3,4,5,2)$	$-1$

The blue line marks the conjugacy class containing complex conjugation.