Artin representation 3.2e6_3e4_37e2.4t4.3c1

• Label: 3.2e6_3e4_37e2.4t4.3c1
• Dimension: 3
• Group: $A_4$
• Conductor: $ 2^{6} \cdot 3^{4} \cdot 37^{2}$
• Root number: 1
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$3$
Group:	$A_4$
Conductor:	$7096896= 2^{6} \cdot 3^{4} \cdot 37^{2} $
Artin number field:	Splitting field of $f= x^{4} - 2 x^{3} - 42 x^{2} + 6 x + 231 $ 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 $\Q_{ 109 }$ to precision 5.

Roots:

$r_{ 1 }$	$=$	$ 11 + 78\cdot 109 + 20\cdot 109^{2} + 81\cdot 109^{3} + 24\cdot 109^{4} +O\left(109^{ 5 }\right)$
$r_{ 2 }$	$=$	$ 39 + 47\cdot 109 + 102\cdot 109^{2} + 99\cdot 109^{3} + 100\cdot 109^{4} +O\left(109^{ 5 }\right)$
$r_{ 3 }$	$=$	$ 77 + 75\cdot 109 + 38\cdot 109^{2} + 11\cdot 109^{3} + 66\cdot 109^{4} +O\left(109^{ 5 }\right)$
$r_{ 4 }$	$=$	$ 93 + 16\cdot 109 + 56\cdot 109^{2} + 25\cdot 109^{3} + 26\cdot 109^{4} +O\left(109^{ 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.