Artin representation 3.2e6_3e4_19e2.4t4.1c1

• Label: 3.2e6_3e4_19e2.4t4.1c1
• Dimension: 3
• Group: $A_4$
• Conductor: $ 2^{6} \cdot 3^{4} \cdot 19^{2}$
• Root number: 1
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$3$
Group:	$A_4$
Conductor:	$1871424= 2^{6} \cdot 3^{4} \cdot 19^{2} $
Artin number field:	Splitting field of $f= x^{4} - 2 x^{3} + 6 x^{2} + 14 x + 11 $ over $\Q$
Size of Galois orbit:	1
Smallest containing permutation representation:	$A_4$
Parity:	Even
Determinant:	1.1.1t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 73 }$ to precision 5.

Roots:

$r_{ 1 }$	$=$	$ 12 + 41\cdot 73 + 25\cdot 73^{2} + 18\cdot 73^{3} + 44\cdot 73^{4} +O\left(73^{ 5 }\right)$
$r_{ 2 }$	$=$	$ 27 + 72\cdot 73 + 16\cdot 73^{2} + 9\cdot 73^{3} + 68\cdot 73^{4} +O\left(73^{ 5 }\right)$
$r_{ 3 }$	$=$	$ 50 + 33\cdot 73 + 53\cdot 73^{2} + 26\cdot 73^{3} + 16\cdot 73^{4} +O\left(73^{ 5 }\right)$
$r_{ 4 }$	$=$	$ 59 + 71\cdot 73 + 49\cdot 73^{2} + 18\cdot 73^{3} + 17\cdot 73^{4} +O\left(73^{ 5 }\right)$

Generators of the action on the roots $r_1, \ldots, r_{ 4 }$

Cycle notation

$(1,2,3)$

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

Character values on conjugacy classes

Size	Order	Action on $r_1, \ldots, r_{ 4 }$	Character value
$1$	$1$	$()$	$3$
$3$	$2$	$(1,2)(3,4)$	$-1$
$4$	$3$	$(1,2,3)$	$0$
$4$	$3$	$(1,3,2)$	$0$

The blue line marks the conjugacy class containing complex conjugation.