Artin representation 5.3e6_5e4_13e4.6t12.1c1

• Label: 5.3e6_5e4_13e4.6t12.1c1
• Dimension: 5
• Group: $A_5$
• Conductor: $ 3^{6} \cdot 5^{4} \cdot 13^{4}$
• Root number: 1
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$5$
Group:	$A_5$
Conductor:	$13013105625= 3^{6} \cdot 5^{4} \cdot 13^{4} $
Artin number field:	Splitting field of $f= x^{5} - x^{4} + x^{3} + 3 x^{2} - 9 $ 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 $\Q_{ 251 }$ to precision 5.

Roots:

$r_{ 1 }$	$=$	$ 8 + 237\cdot 251 + 142\cdot 251^{2} + 15\cdot 251^{3} + 89\cdot 251^{4} +O\left(251^{ 5 }\right)$
$r_{ 2 }$	$=$	$ 18 + 171\cdot 251 + 30\cdot 251^{2} + 109\cdot 251^{3} + 211\cdot 251^{4} +O\left(251^{ 5 }\right)$
$r_{ 3 }$	$=$	$ 72 + 35\cdot 251 + 195\cdot 251^{2} + 102\cdot 251^{3} + 159\cdot 251^{4} +O\left(251^{ 5 }\right)$
$r_{ 4 }$	$=$	$ 191 + 209\cdot 251 + 64\cdot 251^{2} + 3\cdot 251^{3} + 63\cdot 251^{4} +O\left(251^{ 5 }\right)$
$r_{ 5 }$	$=$	$ 214 + 99\cdot 251 + 68\cdot 251^{2} + 20\cdot 251^{3} + 230\cdot 251^{4} +O\left(251^{ 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.