Artin representation 5.2e6_5e2_41e2.6t12.1c1

• Label: 5.2e6_5e2_41e2.6t12.1c1
• Dimension: 5
• Group: $A_5$
• Conductor: $ 2^{6} \cdot 5^{2} \cdot 41^{2}$
• Root number: 1
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$5$
Group:	$A_5$
Conductor:	$2689600= 2^{6} \cdot 5^{2} \cdot 41^{2} $
Artin number field:	Splitting field of $f= x^{5} - x^{4} - 2 x^{3} - 22 x^{2} - 7 x - 33 $ 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 $\Q_{ 521 }$ to precision 5.

Roots:

$r_{ 1 }$	$=$	$ 20 + 449\cdot 521 + 265\cdot 521^{2} + 394\cdot 521^{3} + 354\cdot 521^{4} +O\left(521^{ 5 }\right)$
$r_{ 2 }$	$=$	$ 66 + 273\cdot 521 + 76\cdot 521^{2} + 123\cdot 521^{3} + 480\cdot 521^{4} +O\left(521^{ 5 }\right)$
$r_{ 3 }$	$=$	$ 284 + 355\cdot 521 + 142\cdot 521^{2} + 404\cdot 521^{3} + 191\cdot 521^{4} +O\left(521^{ 5 }\right)$
$r_{ 4 }$	$=$	$ 299 + 190\cdot 521 + 288\cdot 521^{2} + 312\cdot 521^{3} + 264\cdot 521^{4} +O\left(521^{ 5 }\right)$
$r_{ 5 }$	$=$	$ 374 + 294\cdot 521 + 268\cdot 521^{2} + 328\cdot 521^{3} + 271\cdot 521^{4} +O\left(521^{ 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.