Artin representation 5.3e6_67e2.6t12.1c1

• Label: 5.3e6_67e2.6t12.1c1
• Dimension: 5
• Group: $A_5$
• Conductor: $ 3^{6} \cdot 67^{2}$
• Root number: 1
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$5$
Group:	$A_5$
Conductor:	$3272481= 3^{6} \cdot 67^{2} $
Artin number field:	Splitting field of $f= x^{5} - 3 x^{3} - 3 x^{2} + 9 x - 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 $\Q_{ 389 }$ to precision 5.

Roots:

$r_{ 1 }$	$=$	$ 90 + 167\cdot 389 + 17\cdot 389^{2} + 311\cdot 389^{3} + 244\cdot 389^{4} +O\left(389^{ 5 }\right)$
$r_{ 2 }$	$=$	$ 225 + 260\cdot 389 + 182\cdot 389^{2} + 139\cdot 389^{3} + 112\cdot 389^{4} +O\left(389^{ 5 }\right)$
$r_{ 3 }$	$=$	$ 239 + 194\cdot 389 + 168\cdot 389^{2} + 65\cdot 389^{3} + 181\cdot 389^{4} +O\left(389^{ 5 }\right)$
$r_{ 4 }$	$=$	$ 300 + 378\cdot 389 + 265\cdot 389^{2} + 301\cdot 389^{3} + 247\cdot 389^{4} +O\left(389^{ 5 }\right)$
$r_{ 5 }$	$=$	$ 313 + 165\cdot 389 + 143\cdot 389^{2} + 349\cdot 389^{3} + 380\cdot 389^{4} +O\left(389^{ 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.