Galois orbit of Artin representations 2.76.3t2.a

• Label: 2.76.3t2.a
• Dimension: $2$
• Group: $S_3$
• Conductor: $76$
• Indicator: $1$

Basic invariants

Dimension:	$2$
Group:	$S_3$
Conductor:	$76$$\medspace = 2^{2} \cdot 19$
Frobenius-Schur indicator:	$1$
Root number:	$1$
Artin number field:	Galois closure of 3.1.76.1
Galois orbit size:	$1$
Smallest permutation container:	$S_3$
Parity:	odd
Projective image:	$S_3$
Projective field:	Galois closure of 3.1.76.1

Galois action

Roots of defining polynomial

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

Roots:

$r_{ 1 }$	$=$	$12 + 4\cdot 23 + 8\cdot 23^{2} + 22\cdot 23^{3} + 10\cdot 23^{4} +O(23^{5})$
$r_{ 2 }$	$=$	$14 + 6\cdot 23 + 5\cdot 23^{2} + 7\cdot 23^{3} + 15\cdot 23^{4} +O(23^{5})$
$r_{ 3 }$	$=$	$20 + 11\cdot 23 + 9\cdot 23^{2} + 16\cdot 23^{3} + 19\cdot 23^{4} +O(23^{5})$

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 values
			$c1$
$1$	$1$	$()$	$2$
$3$	$2$	$(1,2)$	$0$
$2$	$3$	$(1,2,3)$	$-1$

The blue line marks the conjugacy class containing complex conjugation.