Artin representation 4.2e6_5e2_17e2.5t4.1c1

• Label: 4.2e6_5e2_17e2.5t4.1c1
• Dimension: 4
• Group: $A_5$
• Conductor: $ 2^{6} \cdot 5^{2} \cdot 17^{2}$
• Root number: 1
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$4$
Group:	$A_5$
Conductor:	$462400= 2^{6} \cdot 5^{2} \cdot 17^{2} $
Artin number field:	Splitting field of $f= x^{5} - x^{4} - x^{3} - 5 x^{2} + 9 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 $\Q_{ 173 }$ to precision 5.

Roots:

$r_{ 1 }$	$=$	$ 38 + 92\cdot 173 + 122\cdot 173^{2} + 130\cdot 173^{3} + 51\cdot 173^{4} +O\left(173^{ 5 }\right)$
$r_{ 2 }$	$=$	$ 57 + 110\cdot 173 + 50\cdot 173^{2} + 24\cdot 173^{3} + 162\cdot 173^{4} +O\left(173^{ 5 }\right)$
$r_{ 3 }$	$=$	$ 112 + 133\cdot 173 + 152\cdot 173^{2} + 104\cdot 173^{3} + 88\cdot 173^{4} +O\left(173^{ 5 }\right)$
$r_{ 4 }$	$=$	$ 152 + 127\cdot 173 + 62\cdot 173^{2} + 131\cdot 173^{3} + 153\cdot 173^{4} +O\left(173^{ 5 }\right)$
$r_{ 5 }$	$=$	$ 161 + 54\cdot 173 + 130\cdot 173^{2} + 127\cdot 173^{3} + 62\cdot 173^{4} +O\left(173^{ 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.