Artin representation 5.2e4_23e3_563e3.6t14.1c1

• Label: 5.2e4_23e3_563e3.6t14.1c1
• Dimension: 5
• Group: $S_5$
• Conductor: $ 2^{4} \cdot 23^{3} \cdot 563^{3}$
• Root number: 1
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$5$
Group:	$S_5$
Conductor:	$34739908901584= 2^{4} \cdot 23^{3} \cdot 563^{3} $
Artin number field:	Splitting field of $f= x^{5} - x^{4} - 6 x^{3} + 2 x^{2} + 7 x - 1 $ over $\Q$
Size of Galois orbit:	1
Smallest containing permutation representation:	$\PGL(2,5)$
Parity:	Even
Determinant:	1.23_563.2t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 283 }$ to precision 5.

Roots:

$r_{ 1 }$	$=$	$ 13 + 163\cdot 283 + 163\cdot 283^{2} + 114\cdot 283^{3} + 118\cdot 283^{4} +O\left(283^{ 5 }\right)$
$r_{ 2 }$	$=$	$ 16 + 43\cdot 283 + 34\cdot 283^{2} + 11\cdot 283^{3} + 28\cdot 283^{4} +O\left(283^{ 5 }\right)$
$r_{ 3 }$	$=$	$ 57 + 189\cdot 283 + 126\cdot 283^{2} + 169\cdot 283^{3} + 185\cdot 283^{4} +O\left(283^{ 5 }\right)$
$r_{ 4 }$	$=$	$ 234 + 201\cdot 283 + 246\cdot 283^{2} + 142\cdot 283^{3} + 31\cdot 283^{4} +O\left(283^{ 5 }\right)$
$r_{ 5 }$	$=$	$ 247 + 251\cdot 283 + 277\cdot 283^{2} + 127\cdot 283^{3} + 202\cdot 283^{4} +O\left(283^{ 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.