Artin representation 5.2e6_3e8_5e4.6t12.1c1

• Label: 5.2e6_3e8_5e4.6t12.1c1
• Dimension: 5
• Group: $A_5$
• Conductor: $ 2^{6} \cdot 3^{8} \cdot 5^{4}$
• Root number: 1
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$5$
Group:	$A_5$
Conductor:	$262440000= 2^{6} \cdot 3^{8} \cdot 5^{4} $
Artin number field:	Splitting field of $f= x^{5} - 2 x^{3} - 4 x^{2} - 6 x - 4 $ 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 $\Q_{ 467 }$ to precision 5.

Roots:

$r_{ 1 }$	$=$	$ 49 + 50\cdot 467 + 28\cdot 467^{2} + 158\cdot 467^{3} + 139\cdot 467^{4} +O\left(467^{ 5 }\right)$
$r_{ 2 }$	$=$	$ 59 + 367\cdot 467 + 148\cdot 467^{2} + 411\cdot 467^{3} + 177\cdot 467^{4} +O\left(467^{ 5 }\right)$
$r_{ 3 }$	$=$	$ 102 + 228\cdot 467 + 80\cdot 467^{2} + 286\cdot 467^{3} + 364\cdot 467^{4} +O\left(467^{ 5 }\right)$
$r_{ 4 }$	$=$	$ 319 + 331\cdot 467 + 385\cdot 467^{2} + 388\cdot 467^{3} + 209\cdot 467^{4} +O\left(467^{ 5 }\right)$
$r_{ 5 }$	$=$	$ 405 + 423\cdot 467 + 290\cdot 467^{2} + 156\cdot 467^{3} + 42\cdot 467^{4} +O\left(467^{ 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.