Artin representation 2.2e2_19_23_409.3t2.1c1

• Label: 2.2e2_19_23_409.3t2.1c1
• Dimension: 2
• Group: $S_3$
• Conductor: $ 2^{2} \cdot 19 \cdot 23 \cdot 409 $
• Root number: 1
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$2$
Group:	$S_3$
Conductor:	$714932= 2^{2} \cdot 19 \cdot 23 \cdot 409 $
Artin number field:	Splitting field of $f= x^{3} - x^{2} - 116 x + 1128 $ over $\Q$
Size of Galois orbit:	1
Smallest containing permutation representation:	$S_3$
Parity:	Odd
Determinant:	1.2e2_19_23_409.2t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 47 }$ to precision 5.

Roots:

$r_{ 1 }$	$=$	$ 31\cdot 47 + 2\cdot 47^{2} + 27\cdot 47^{3} + 32\cdot 47^{4} +O\left(47^{ 5 }\right)$
$r_{ 2 }$	$=$	$ 15 + 19\cdot 47 + 41\cdot 47^{2} + 29\cdot 47^{3} + 37\cdot 47^{4} +O\left(47^{ 5 }\right)$
$r_{ 3 }$	$=$	$ 33 + 43\cdot 47 + 2\cdot 47^{2} + 37\cdot 47^{3} + 23\cdot 47^{4} +O\left(47^{ 5 }\right)$

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

The blue line marks the conjugacy class containing complex conjugation.