Artin representation 2.2e2_7_61.3t2.1c1

• Label: 2.2e2_7_61.3t2.1c1
• Dimension: 2
• Group: $S_3$
• Conductor: $ 2^{2} \cdot 7 \cdot 61 $
• Root number: 1
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$2$
Group:	$S_3$
Conductor:	$1708= 2^{2} \cdot 7 \cdot 61 $
Artin number field:	Splitting field of $f= x^{3} - x^{2} - 8 x - 2 $ over $\Q$
Size of Galois orbit:	1
Smallest containing permutation representation:	$S_3$
Parity:	Even
Determinant:	1.2e2_7_61.2t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 101 }$ to precision 5.

Roots:

$r_{ 1 }$	$=$	$ 53 + 61\cdot 101 + 69\cdot 101^{2} + 80\cdot 101^{3} + 45\cdot 101^{4} +O\left(101^{ 5 }\right)$
$r_{ 2 }$	$=$	$ 68 + 4\cdot 101 + 99\cdot 101^{2} + 101^{3} + 96\cdot 101^{4} +O\left(101^{ 5 }\right)$
$r_{ 3 }$	$=$	$ 82 + 34\cdot 101 + 33\cdot 101^{2} + 18\cdot 101^{3} + 60\cdot 101^{4} +O\left(101^{ 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.