Artin representation 3.7e2_13e2_41e2.4t4.1c1

• Label: 3.7e2_13e2_41e2.4t4.1c1
• Dimension: 3
• Group: $A_4$
• Conductor: $ 7^{2} \cdot 13^{2} \cdot 41^{2}$
• Root number: 1
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$3$
Group:	$A_4$
Conductor:	$13920361= 7^{2} \cdot 13^{2} \cdot 41^{2} $
Artin number field:	Splitting field of $f= x^{4} - x^{3} - 65 x^{2} - 34 x + 623 $ 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 $\Q_{ 127 }$ to precision 5.

Roots:

$r_{ 1 }$	$=$	$ 21 + 76\cdot 127 + 50\cdot 127^{2} + 23\cdot 127^{3} + 22\cdot 127^{4} +O\left(127^{ 5 }\right)$
$r_{ 2 }$	$=$	$ 31 + 71\cdot 127 + 28\cdot 127^{2} + 124\cdot 127^{3} + 66\cdot 127^{4} +O\left(127^{ 5 }\right)$
$r_{ 3 }$	$=$	$ 34 + 41\cdot 127 + 79\cdot 127^{2} + 104\cdot 127^{3} + 10\cdot 127^{4} +O\left(127^{ 5 }\right)$
$r_{ 4 }$	$=$	$ 42 + 65\cdot 127 + 95\cdot 127^{2} + 127^{3} + 27\cdot 127^{4} +O\left(127^{ 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.