Artin representation 2.8619.3t2.a.a

• Label: 2.8619.3t2.a.a
• Dimension: $2$
• Group: $S_3$
• Conductor: $8619$
• Root number: $1$
• Indicator: $1$

Basic invariants

Dimension:	$2$
Group:	$S_3$
Conductor:	$8619$$\medspace = 3 \cdot 13^{2} \cdot 17$
Frobenius-Schur indicator:	$1$
Root number:	$1$
Artin stem field:	Galois closure of 3.1.8619.1
Galois orbit size:	$1$
Smallest permutation container:	$S_3$
Parity:	odd
Determinant:	1.51.2t1.a.a
Projective image:	$S_3$
Projective stem field:	Galois closure of 3.1.8619.1

Defining polynomial

$f(x)$

$=$

$x^{3} - x^{2} + 9x + 12$ .

The roots of $f$ are computed in $\Q_{ 19 }$ to precision 5.

Roots:

$r_{ 1 }$	$=$	$3 + 17\cdot 19 + 5\cdot 19^{2} + 12\cdot 19^{3} + 10\cdot 19^{4} +O(19^{5})$
$r_{ 2 }$	$=$	$8 + 17\cdot 19 + 10\cdot 19^{2} + 12\cdot 19^{3} + 15\cdot 19^{4} +O(19^{5})$
$r_{ 3 }$	$=$	$9 + 3\cdot 19 + 2\cdot 19^{2} + 13\cdot 19^{3} + 11\cdot 19^{4} +O(19^{5})$

Generators of the action on the roots $r_{ 1 }, r_{ 2 }, r_{ 3 }$

Cycle notation

$(1,2,3)$

$(1,2)$

Character values on conjugacy classes

Size	Order	Action on $r_{ 1 }, r_{ 2 }, r_{ 3 }$	Character value	Complex conjugation
$1$	$1$	$()$	$2$
$3$	$2$	$(1,2)$	$0$	✓
$2$	$3$	$(1,2,3)$	$-1$