Artin representation 5.2e4_353e2.6t12.1c1

• Label: 5.2e4_353e2.6t12.1c1
• Dimension: 5
• Group: $A_5$
• Conductor: $ 2^{4} \cdot 353^{2}$
• Root number: 1
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$5$
Group:	$A_5$
Conductor:	$1993744= 2^{4} \cdot 353^{2} $
Artin number field:	Splitting field of $f= x^{5} + x^{3} - 5 x^{2} - 4 x - 7 $ 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 $\Q_{ 241 }$ to precision 5.

Roots:

$r_{ 1 }$	$=$	$ 7 + 118\cdot 241 + 117\cdot 241^{2} + 122\cdot 241^{3} + 121\cdot 241^{4} +O\left(241^{ 5 }\right)$
$r_{ 2 }$	$=$	$ 13 + 36\cdot 241 + 87\cdot 241^{2} + 225\cdot 241^{3} + 196\cdot 241^{4} +O\left(241^{ 5 }\right)$
$r_{ 3 }$	$=$	$ 139 + 146\cdot 241 + 17\cdot 241^{2} + 224\cdot 241^{3} + 23\cdot 241^{4} +O\left(241^{ 5 }\right)$
$r_{ 4 }$	$=$	$ 154 + 71\cdot 241 + 21\cdot 241^{2} + 205\cdot 241^{3} + 198\cdot 241^{4} +O\left(241^{ 5 }\right)$
$r_{ 5 }$	$=$	$ 169 + 109\cdot 241 + 238\cdot 241^{2} + 186\cdot 241^{3} + 181\cdot 241^{4} +O\left(241^{ 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.