Artin representation 5.2e6_307e2.6t12.1c1

• Label: 5.2e6_307e2.6t12.1c1
• Dimension: 5
• Group: $A_5$
• Conductor: $ 2^{6} \cdot 307^{2}$
• Root number: 1
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$5$
Group:	$A_5$
Conductor:	$6031936= 2^{6} \cdot 307^{2} $
Artin number field:	Splitting field of $f= x^{5} - x^{4} + 12 x^{3} + 2 x^{2} - 14 x - 10 $ 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 $\Q_{ 547 }$ to precision 5.

Roots:

$r_{ 1 }$	$=$	$ 91 + 452\cdot 547 + 370\cdot 547^{2} + 433\cdot 547^{3} + 238\cdot 547^{4} +O\left(547^{ 5 }\right)$
$r_{ 2 }$	$=$	$ 351 + 192\cdot 547 + 518\cdot 547^{2} + 440\cdot 547^{3} + 467\cdot 547^{4} +O\left(547^{ 5 }\right)$
$r_{ 3 }$	$=$	$ 371 + 134\cdot 547 + 412\cdot 547^{2} + 141\cdot 547^{3} + 402\cdot 547^{4} +O\left(547^{ 5 }\right)$
$r_{ 4 }$	$=$	$ 400 + 62\cdot 547 + 211\cdot 547^{2} + 524\cdot 547^{3} + 435\cdot 547^{4} +O\left(547^{ 5 }\right)$
$r_{ 5 }$	$=$	$ 429 + 251\cdot 547 + 128\cdot 547^{2} + 100\cdot 547^{3} + 96\cdot 547^{4} +O\left(547^{ 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.