Artin representation 3.2e6_43e2.4t4.1c1

• Label: 3.2e6_43e2.4t4.1c1
• Dimension: 3
• Group: $A_4$
• Conductor: $ 2^{6} \cdot 43^{2}$
• Root number: 1
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$3$
Group:	$A_4$
Conductor:	$118336= 2^{6} \cdot 43^{2} $
Artin number field:	Splitting field of $f= x^{4} - 2 x^{3} + 9 x^{2} - 6 x + 7 $ 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 $\Q_{ 97 }$ to precision 5.

Roots:

$r_{ 1 }$	$=$	$ 3 + 58\cdot 97 + 34\cdot 97^{2} + 4\cdot 97^{3} + 21\cdot 97^{4} +O\left(97^{ 5 }\right)$
$r_{ 2 }$	$=$	$ 18 + 12\cdot 97 + 14\cdot 97^{2} + 14\cdot 97^{3} + 52\cdot 97^{4} +O\left(97^{ 5 }\right)$
$r_{ 3 }$	$=$	$ 29 + 72\cdot 97 + 54\cdot 97^{2} + 26\cdot 97^{3} + 36\cdot 97^{4} +O\left(97^{ 5 }\right)$
$r_{ 4 }$	$=$	$ 49 + 51\cdot 97 + 90\cdot 97^{2} + 51\cdot 97^{3} + 84\cdot 97^{4} +O\left(97^{ 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.