Galois orbit of Artin representations 1.2e3_3_5_7.2t1.2

• Label: 1.2e3_3_5_7.2t1.2
• Dimension: 1
• Group: $C_2$
• Conductor: $ 2^{3} \cdot 3 \cdot 5 \cdot 7 $
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$1$
Group:	$C_2$
Conductor:	$840= 2^{3} \cdot 3 \cdot 5 \cdot 7 $
Artin number field:	Splitting field of $f= x^{2} + 210 $ 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 $\Q_{ 29 }$ to precision 5.

Roots:

$r_{ 1 }$	$=$	$ 14 + 14\cdot 29 + 7\cdot 29^{2} + 7\cdot 29^{3} + 27\cdot 29^{4} +O\left(29^{ 5 }\right)$
$r_{ 2 }$	$=$	$ 15 + 14\cdot 29 + 21\cdot 29^{2} + 21\cdot 29^{3} + 29^{4} +O\left(29^{ 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.