Artin representation 2.2e3_3_37.3t2.1c1

• Label: 2.2e3_3_37.3t2.1c1
• Dimension: 2
• Group: $S_3$
• Conductor: $ 2^{3} \cdot 3 \cdot 37 $
• Root number: 1
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$2$
Group:	$S_3$
Conductor:	$888= 2^{3} \cdot 3 \cdot 37 $
Artin number field:	Splitting field of $f= x^{3} - x^{2} + 9 x + 3 $ over $\Q$
Size of Galois orbit:	1
Smallest containing permutation representation:	$S_3$
Parity:	Odd
Determinant:	1.2e3_3_37.2t1.2c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 43 }$ to precision 5.

Roots:

$r_{ 1 }$	$=$	$ 20 + 15\cdot 43 + 36\cdot 43^{2} + 8\cdot 43^{3} + 7\cdot 43^{4} +O\left(43^{ 5 }\right)$
$r_{ 2 }$	$=$	$ 32 + 41\cdot 43 + 36\cdot 43^{2} + 4\cdot 43^{3} + 43^{4} +O\left(43^{ 5 }\right)$
$r_{ 3 }$	$=$	$ 35 + 28\cdot 43 + 12\cdot 43^{2} + 29\cdot 43^{3} + 34\cdot 43^{4} +O\left(43^{ 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.