Galois orbit of Artin representations 5.2e6_7e2_11e4.6t12.1

• Label: 5.2e6_7e2_11e4.6t12.1
• Dimension: 5
• Group: $A_5$
• Conductor: $ 2^{6} \cdot 7^{2} \cdot 11^{4}$
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$5$
Group:	$A_5$
Conductor:	$45914176= 2^{6} \cdot 7^{2} \cdot 11^{4} $
Artin number field:	Splitting field of $f= x^{5} - x^{4} + 4 x^{3} - 8 x^{2} + x + 7 $ over $\Q$
Size of Galois orbit:	1
Smallest containing permutation representation:	$\PSL(2,5)$
Parity:	Even

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 503 }$ to precision 5.

Roots:

$r_{ 1 }$	$=$	$ 125 + 151\cdot 503 + 114\cdot 503^{2} + 427\cdot 503^{3} + 483\cdot 503^{4} +O\left(503^{ 5 }\right)$
$r_{ 2 }$	$=$	$ 162 + 31\cdot 503 + 202\cdot 503^{2} + 473\cdot 503^{3} + 305\cdot 503^{4} +O\left(503^{ 5 }\right)$
$r_{ 3 }$	$=$	$ 190 + 102\cdot 503 + 88\cdot 503^{2} + 398\cdot 503^{3} + 441\cdot 503^{4} +O\left(503^{ 5 }\right)$
$r_{ 4 }$	$=$	$ 244 + 130\cdot 503 + 186\cdot 503^{2} + 54\cdot 503^{3} + 435\cdot 503^{4} +O\left(503^{ 5 }\right)$
$r_{ 5 }$	$=$	$ 286 + 87\cdot 503 + 415\cdot 503^{2} + 155\cdot 503^{3} + 345\cdot 503^{4} +O\left(503^{ 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 values
			$c1$
$1$	$1$	$()$	$5$
$15$	$2$	$(1,2)(3,4)$	$1$
$20$	$3$	$(1,2,3)$	$-1$
$12$	$5$	$(1,2,3,4,5)$	$0$
$12$	$5$	$(1,3,4,5,2)$	$0$

The blue line marks the conjugacy class containing complex conjugation.