Artin representation 4.2e6_3e2_11e4.5t4.1c1

• Label: 4.2e6_3e2_11e4.5t4.1c1
• Dimension: 4
• Group: $A_5$
• Conductor: $ 2^{6} \cdot 3^{2} \cdot 11^{4}$
• Root number: 1
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$4$
Group:	$A_5$
Conductor:	$8433216= 2^{6} \cdot 3^{2} \cdot 11^{4} $
Artin number field:	Splitting field of $f= x^{5} - x^{4} - 4 x^{3} - 8 x^{2} + 25 x - 1 $ over $\Q$
Size of Galois orbit:	1
Smallest containing permutation representation:	$A_5$
Parity:	Even
Determinant:	1.1.1t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 449 }$ to precision 5.

Roots:

$r_{ 1 }$	$=$	$ 107 + 396\cdot 449 + 197\cdot 449^{2} + 434\cdot 449^{3} + 397\cdot 449^{4} +O\left(449^{ 5 }\right)$
$r_{ 2 }$	$=$	$ 133 + 215\cdot 449 + 174\cdot 449^{2} + 346\cdot 449^{3} + 332\cdot 449^{4} +O\left(449^{ 5 }\right)$
$r_{ 3 }$	$=$	$ 140 + 24\cdot 449 + 378\cdot 449^{2} + 324\cdot 449^{3} + 233\cdot 449^{4} +O\left(449^{ 5 }\right)$
$r_{ 4 }$	$=$	$ 190 + 17\cdot 449 + 398\cdot 449^{2} + 425\cdot 449^{3} + 262\cdot 449^{4} +O\left(449^{ 5 }\right)$
$r_{ 5 }$	$=$	$ 329 + 244\cdot 449 + 198\cdot 449^{2} + 264\cdot 449^{3} + 119\cdot 449^{4} +O\left(449^{ 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
$1$	$1$	$()$	$4$
$15$	$2$	$(1,2)(3,4)$	$0$
$20$	$3$	$(1,2,3)$	$1$
$12$	$5$	$(1,2,3,4,5)$	$-1$
$12$	$5$	$(1,3,4,5,2)$	$-1$

The blue line marks the conjugacy class containing complex conjugation.