Artin representation 5.3e7_5e3_479e3.6t14.1c1

• Label: 5.3e7_5e3_479e3.6t14.1c1
• Dimension: 5
• Group: $S_5$
• Conductor: $ 3^{7} \cdot 5^{3} \cdot 479^{3}$
• Root number: 1
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$5$
Group:	$S_5$
Conductor:	$30044524586625= 3^{7} \cdot 5^{3} \cdot 479^{3} $
Artin number field:	Splitting field of $f= x^{5} - x^{4} + 2 x^{3} - 5 x^{2} + 5 x - 1 $ over $\Q$
Size of Galois orbit:	1
Smallest containing permutation representation:	$\PGL(2,5)$
Parity:	Even
Determinant:	1.3_5_479.2t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 269 }$ to precision 5.

Roots:

$r_{ 1 }$	$=$	$ 19 + 38\cdot 269 + 220\cdot 269^{2} + 266\cdot 269^{3} + 205\cdot 269^{4} +O\left(269^{ 5 }\right)$
$r_{ 2 }$	$=$	$ 56 + 234\cdot 269 + 195\cdot 269^{2} + 160\cdot 269^{3} + 264\cdot 269^{4} +O\left(269^{ 5 }\right)$
$r_{ 3 }$	$=$	$ 211 + 93\cdot 269 + 159\cdot 269^{2} + 5\cdot 269^{3} + 232\cdot 269^{4} +O\left(269^{ 5 }\right)$
$r_{ 4 }$	$=$	$ 260 + 228\cdot 269 + 179\cdot 269^{2} + 112\cdot 269^{3} + 219\cdot 269^{4} +O\left(269^{ 5 }\right)$
$r_{ 5 }$	$=$	$ 262 + 211\cdot 269 + 51\cdot 269^{2} + 261\cdot 269^{3} + 153\cdot 269^{4} +O\left(269^{ 5 }\right)$

Generators of the action on the roots $r_1, \ldots, r_{ 5 }$

Cycle notation

$(1,2)$

$(1,2,3,4,5)$

Character values on conjugacy classes

Size	Order	Action on $r_1, \ldots, r_{ 5 }$	Character value
$1$	$1$	$()$	$5$
$10$	$2$	$(1,2)$	$-1$
$15$	$2$	$(1,2)(3,4)$	$1$
$20$	$3$	$(1,2,3)$	$-1$
$30$	$4$	$(1,2,3,4)$	$1$
$24$	$5$	$(1,2,3,4,5)$	$0$
$20$	$6$	$(1,2,3)(4,5)$	$-1$

The blue line marks the conjugacy class containing complex conjugation.