Artin representation 3.3e4_7e2_19e2.4t4.1c1

• Label: 3.3e4_7e2_19e2.4t4.1c1
• Dimension: 3
• Group: $A_4$
• Conductor: $ 3^{4} \cdot 7^{2} \cdot 19^{2}$
• Root number: 1
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$3$
Group:	$A_4$
Conductor:	$1432809= 3^{4} \cdot 7^{2} \cdot 19^{2} $
Artin number field:	Splitting field of $f= x^{4} - x^{3} - 30 x^{2} - 41 x + 35 $ 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 $\Q_{ 211 }$ to precision 5.

Roots:

$r_{ 1 }$	$=$	$ 6 + 148\cdot 211 + 190\cdot 211^{2} + 2\cdot 211^{3} + 139\cdot 211^{4} +O\left(211^{ 5 }\right)$
$r_{ 2 }$	$=$	$ 107 + 131\cdot 211 + 30\cdot 211^{2} + 2\cdot 211^{3} + 32\cdot 211^{4} +O\left(211^{ 5 }\right)$
$r_{ 3 }$	$=$	$ 140 + 126\cdot 211 + 23\cdot 211^{2} + 119\cdot 211^{3} + 198\cdot 211^{4} +O\left(211^{ 5 }\right)$
$r_{ 4 }$	$=$	$ 170 + 15\cdot 211 + 177\cdot 211^{2} + 86\cdot 211^{3} + 52\cdot 211^{4} +O\left(211^{ 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.