Galois orbit of Artin representations 1.2e2_3_59.2t1.1

• Label: 1.2e2_3_59.2t1.1
• Dimension: 1
• Group: $C_2$
• Conductor: $ 2^{2} \cdot 3 \cdot 59 $
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$1$
Group:	$C_2$
Conductor:	$708= 2^{2} \cdot 3 \cdot 59 $
Artin number field:	Splitting field of $f= x^{2} + 177 $ over $\Q$
Size of Galois orbit:	1
Smallest containing permutation representation:	$C_2$
Parity:	Odd

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 31 }$ to precision 5.

Roots:

$r_{ 1 }$	$=$	$ 3 + 30\cdot 31 + 4\cdot 31^{2} + 17\cdot 31^{3} + 31^{4} +O\left(31^{ 5 }\right)$
$r_{ 2 }$	$=$	$ 28 + 26\cdot 31^{2} + 13\cdot 31^{3} + 29\cdot 31^{4} +O\left(31^{ 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 values
			$c1$
$1$	$1$	$()$	$1$
$1$	$2$	$(1,2)$	$-1$

The blue line marks the conjugacy class containing complex conjugation.