Artin representation 3.7e2_13e2_29e2.4t4.3c1

• Label: 3.7e2_13e2_29e2.4t4.3c1
• Dimension: 3
• Group: $A_4$
• Conductor: $ 7^{2} \cdot 13^{2} \cdot 29^{2}$
• Root number: 1
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$3$
Group:	$A_4$
Conductor:	$6964321= 7^{2} \cdot 13^{2} \cdot 29^{2} $
Artin number field:	Splitting field of $f= x^{4} - x^{3} + 26 x^{2} - 60 x + 584 $ 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 $\Q_{ 181 }$ to precision 5.

Roots:

$r_{ 1 }$	$=$	$ 10 + 142\cdot 181 + 85\cdot 181^{2} + 175\cdot 181^{3} + 6\cdot 181^{4} +O\left(181^{ 5 }\right)$
$r_{ 2 }$	$=$	$ 69 + 115\cdot 181 + 115\cdot 181^{2} + 40\cdot 181^{3} + 97\cdot 181^{4} +O\left(181^{ 5 }\right)$
$r_{ 3 }$	$=$	$ 115 + 79\cdot 181 + 9\cdot 181^{2} + 180\cdot 181^{3} + 178\cdot 181^{4} +O\left(181^{ 5 }\right)$
$r_{ 4 }$	$=$	$ 169 + 24\cdot 181 + 151\cdot 181^{2} + 146\cdot 181^{3} + 78\cdot 181^{4} +O\left(181^{ 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.