Artin representation 5.2e6_193e2.6t12.1c1

• Label: 5.2e6_193e2.6t12.1c1
• Dimension: 5
• Group: $A_5$
• Conductor: $ 2^{6} \cdot 193^{2}$
• Root number: 1
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$5$
Group:	$A_5$
Conductor:	$2383936= 2^{6} \cdot 193^{2} $
Artin number field:	Splitting field of $f= x^{5} - x^{4} + 8 x^{3} - 6 x^{2} + 14 x - 6 $ 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 $\Q_{ 499 }$ to precision 5.

Roots:

$r_{ 1 }$	$=$	$ 60 + 473\cdot 499 + 371\cdot 499^{2} + 218\cdot 499^{3} + 133\cdot 499^{4} +O\left(499^{ 5 }\right)$
$r_{ 2 }$	$=$	$ 249 + 388\cdot 499 + 127\cdot 499^{2} + 194\cdot 499^{3} + 137\cdot 499^{4} +O\left(499^{ 5 }\right)$
$r_{ 3 }$	$=$	$ 300 + 105\cdot 499 + 388\cdot 499^{2} + 356\cdot 499^{3} + 143\cdot 499^{4} +O\left(499^{ 5 }\right)$
$r_{ 4 }$	$=$	$ 402 + 461\cdot 499 + 494\cdot 499^{2} + 69\cdot 499^{3} + 22\cdot 499^{4} +O\left(499^{ 5 }\right)$
$r_{ 5 }$	$=$	$ 487 + 67\cdot 499 + 114\cdot 499^{2} + 158\cdot 499^{3} + 62\cdot 499^{4} +O\left(499^{ 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.