Artin representation 5.2e6_67e4.6t12.1c1

• Label: 5.2e6_67e4.6t12.1c1
• Dimension: 5
• Group: $A_5$
• Conductor: $ 2^{6} \cdot 67^{4}$
• Root number: 1
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$5$
Group:	$A_5$
Conductor:	$1289671744= 2^{6} \cdot 67^{4} $
Artin number field:	Splitting field of $f= x^{5} - x^{4} - x^{3} + 7 x^{2} - 9 x + 5 $ 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 $\Q_{ 263 }$ to precision 5.

Roots:

$r_{ 1 }$	$=$	$ 53 + 203\cdot 263 + 22\cdot 263^{2} + 129\cdot 263^{3} + 229\cdot 263^{4} +O\left(263^{ 5 }\right)$
$r_{ 2 }$	$=$	$ 57 + 137\cdot 263 + 38\cdot 263^{2} + 116\cdot 263^{3} + 57\cdot 263^{4} +O\left(263^{ 5 }\right)$
$r_{ 3 }$	$=$	$ 80 + 215\cdot 263 + 78\cdot 263^{2} + 82\cdot 263^{3} + 224\cdot 263^{4} +O\left(263^{ 5 }\right)$
$r_{ 4 }$	$=$	$ 99 + 211\cdot 263 + 143\cdot 263^{2} + 253\cdot 263^{3} + 128\cdot 263^{4} +O\left(263^{ 5 }\right)$
$r_{ 5 }$	$=$	$ 238 + 21\cdot 263 + 242\cdot 263^{2} + 207\cdot 263^{3} + 148\cdot 263^{4} +O\left(263^{ 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.