Artin representation 2.128516.3t2.b.a

• Label: 2.128516.3t2.b.a
• Dimension: $2$
• Group: $S_3$
• Conductor: $128516$
• Root number: $1$
• Indicator: $1$

Basic invariants

Dimension:	$2$
Group:	$S_3$
Conductor:	$128516$$\medspace = 2^{2} \cdot 19^{2} \cdot 89$
Frobenius-Schur indicator:	$1$
Root number:	$1$
Artin stem field:	Galois closure of 3.1.128516.2
Galois orbit size:	$1$
Smallest permutation container:	$S_3$
Parity:	odd
Determinant:	1.356.2t1.a.a
Projective image:	$S_3$
Projective stem field:	Galois closure of 3.1.128516.2

Defining polynomial

$f(x)$

$=$

$x^{3} - 19x - 76$ .

The roots of $f$ are computed in $\Q_{ 23 }$ to precision 5.

Roots:

$r_{ 1 }$	$=$	$5 + 14\cdot 23 + 20\cdot 23^{2} + 4\cdot 23^{3} + 16\cdot 23^{4} +O(23^{5})$
$r_{ 2 }$	$=$	$20 + 5\cdot 23 + 15\cdot 23^{3} + 11\cdot 23^{4} +O(23^{5})$
$r_{ 3 }$	$=$	$21 + 2\cdot 23 + 2\cdot 23^{2} + 3\cdot 23^{3} + 18\cdot 23^{4} +O(23^{5})$

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
$1$	$1$	$()$	$2$
$3$	$2$	$(1,2)$	$0$	✓
$2$	$3$	$(1,2,3)$	$-1$