Artin representation 1.2e2_37.2t1.1c1

• Label: 1.2e2_37.2t1.1c1
• Dimension: 1
• Group: $C_2$
• Conductor: $ 2^{2} \cdot 37 $
• Root number: 1
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$1$
Group:	$C_2$
Conductor:	$148= 2^{2} \cdot 37 $
Artin number field:	Splitting field of $f= x^{2} + 37 $ over $\Q$
Size of Galois orbit:	1
Smallest containing permutation representation:	$C_2$
Parity:	Odd
Corresponding Dirichlet character:	\(\displaystyle\left(\frac{-148}{\bullet}\right)\)

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 19 }$ to precision 5.

Roots:

$r_{ 1 }$	$=$	$ 1 + 18\cdot 19 + 8\cdot 19^{2} + 18\cdot 19^{3} + 15\cdot 19^{4} +O\left(19^{ 5 }\right)$
$r_{ 2 }$	$=$	$ 18 + 10\cdot 19^{2} + 3\cdot 19^{4} +O\left(19^{ 5 }\right)$

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 value
$1$	$1$	$()$	$1$
$1$	$2$	$(1,2)$	$-1$

The blue line marks the conjugacy class containing complex conjugation.