Artin representation 2.2e2_3_5_13.3t2.1c1

• Label: 2.2e2_3_5_13.3t2.1c1
• Dimension: 2
• Group: $S_3$
• Conductor: $ 2^{2} \cdot 3 \cdot 5 \cdot 13 $
• Root number: 1
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$2$
Group:	$S_3$
Conductor:	$780= 2^{2} \cdot 3 \cdot 5 \cdot 13 $
Artin number field:	Splitting field of $f= x^{3} - x^{2} - x - 5 $ over $\Q$
Size of Galois orbit:	1
Smallest containing permutation representation:	$S_3$
Parity:	Odd
Determinant:	1.3_5_13.2t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 59 }$ to precision 5.

Roots:

$r_{ 1 }$	$=$	$ 10 + 12\cdot 59 + 47\cdot 59^{2} + 10\cdot 59^{3} + 43\cdot 59^{4} +O\left(59^{ 5 }\right)$
$r_{ 2 }$	$=$	$ 17 + 46\cdot 59 + 37\cdot 59^{2} + 40\cdot 59^{3} + 31\cdot 59^{4} +O\left(59^{ 5 }\right)$
$r_{ 3 }$	$=$	$ 33 + 33\cdot 59^{2} + 7\cdot 59^{3} + 43\cdot 59^{4} +O\left(59^{ 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.