Artin representation 5.2e4_3e6_41e4.6t12.1c1

• Label: 5.2e4_3e6_41e4.6t12.1c1
• Dimension: 5
• Group: $A_5$
• Conductor: $ 2^{4} \cdot 3^{6} \cdot 41^{4}$
• Root number: 1
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$5$
Group:	$A_5$
Conductor:	$32959676304= 2^{4} \cdot 3^{6} \cdot 41^{4} $
Artin number field:	Splitting field of $f= x^{5} - 2 x^{4} + x^{3} + 5 x^{2} - 10 x - 13 $ 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 $\Q_{ 431 }$ to precision 5.

Roots:

$r_{ 1 }$	$=$	$ 182 + 402\cdot 431 + 162\cdot 431^{2} + 224\cdot 431^{3} + 405\cdot 431^{4} +O\left(431^{ 5 }\right)$
$r_{ 2 }$	$=$	$ 213 + 401\cdot 431 + 181\cdot 431^{2} + 331\cdot 431^{3} + 287\cdot 431^{4} +O\left(431^{ 5 }\right)$
$r_{ 3 }$	$=$	$ 252 + 404\cdot 431 + 138\cdot 431^{2} + 317\cdot 431^{3} + 276\cdot 431^{4} +O\left(431^{ 5 }\right)$
$r_{ 4 }$	$=$	$ 261 + 387\cdot 431 + 14\cdot 431^{2} + 76\cdot 431^{3} + 77\cdot 431^{4} +O\left(431^{ 5 }\right)$
$r_{ 5 }$	$=$	$ 387 + 127\cdot 431 + 363\cdot 431^{2} + 343\cdot 431^{3} + 245\cdot 431^{4} +O\left(431^{ 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.