Artin representation 5.5e2_71e4.6t12.1c1

• Label: 5.5e2_71e4.6t12.1c1
• Dimension: 5
• Group: $A_5$
• Conductor: $ 5^{2} \cdot 71^{4}$
• Root number: 1
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$5$
Group:	$A_5$
Conductor:	$635292025= 5^{2} \cdot 71^{4} $
Artin number field:	Splitting field of $f= x^{5} - x^{4} + 3 x^{3} - 4 x^{2} + 5 x - 1 $ over $\Q$
Size of Galois orbit:	1
Smallest containing permutation representation:	$\PSL(2,5)$
Parity:	Even
Determinant:	1.1.1t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 457 }$ to precision 5.

Roots:

$r_{ 1 }$	$=$	$ 232 + 411\cdot 457 + 331\cdot 457^{2} + 218\cdot 457^{3} + 88\cdot 457^{4} +O\left(457^{ 5 }\right)$
$r_{ 2 }$	$=$	$ 265 + 412\cdot 457 + 390\cdot 457^{2} + 376\cdot 457^{3} + 347\cdot 457^{4} +O\left(457^{ 5 }\right)$
$r_{ 3 }$	$=$	$ 434 + 441\cdot 457 + 117\cdot 457^{2} + 448\cdot 457^{3} + 125\cdot 457^{4} +O\left(457^{ 5 }\right)$
$r_{ 4 }$	$=$	$ 444 + 338\cdot 457 + 281\cdot 457^{2} + 19\cdot 457^{3} + 138\cdot 457^{4} +O\left(457^{ 5 }\right)$
$r_{ 5 }$	$=$	$ 454 + 222\cdot 457 + 248\cdot 457^{2} + 307\cdot 457^{3} + 213\cdot 457^{4} +O\left(457^{ 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
$1$	$1$	$()$	$5$
$15$	$2$	$(1,2)(3,4)$	$1$
$20$	$3$	$(1,2,3)$	$-1$
$12$	$5$	$(1,2,3,4,5)$	$0$
$12$	$5$	$(1,3,4,5,2)$	$0$

The blue line marks the conjugacy class containing complex conjugation.