Galois orbit of Artin representations 1.19109.2t1.a

• Label: 1.19109.2t1.a
• Dimension: $1$
• Group: $C_2$
• Conductor: $19109$
• Indicator: $1$

Basic invariants

Dimension:	$1$
Group:	$C_2$
Conductor:	$19109$$\medspace = 97 \cdot 197$
Frobenius-Schur indicator:	$1$
Root number:	$1$
Artin number field:	Galois closure of $\Q(\sqrt{19109})$
Galois orbit size:	$1$
Smallest permutation container:	$C_2$
Parity:	even
Projective image:	$C_1$
Projective field:	Galois closure of $\Q$

Galois action

Roots of defining polynomial

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

Roots:

$r_{ 1 }$	$=$	$2 + 2\cdot 5^{2} + 4\cdot 5^{3} + 2\cdot 5^{4} +O(5^{5})$
$r_{ 2 }$	$=$	$4 + 4\cdot 5 + 2\cdot 5^{2} + 2\cdot 5^{4} +O(5^{5})$

Generators of the action on the roots $r_{ 1 }, r_{ 2 }$

Cycle notation

$(1,2)$

Character values on conjugacy classes

Size	Order	Action on $r_{ 1 }, r_{ 2 }$	Character values
			$c1$
$1$	$1$	$()$	$1$
$1$	$2$	$(1,2)$	$-1$

The blue line marks the conjugacy class containing complex conjugation.