Galois orbit of Artin representations 1.2e2_31_367.2t1.1

• Label: 1.2e2_31_367.2t1.1
• Dimension: 1
• Group: $C_2$
• Conductor: $ 2^{2} \cdot 31 \cdot 367 $
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$1$
Group:	$C_2$
Conductor:	$45508= 2^{2} \cdot 31 \cdot 367 $
Artin number field:	Splitting field of $f= x^{2} + 11377 $ 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 $\Q_{ 17 }$ to precision 5.

Roots:

$r_{ 1 }$	$=$	$ 8 + 10\cdot 17 + 13\cdot 17^{2} + 9\cdot 17^{3} +O\left(17^{ 5 }\right)$
$r_{ 2 }$	$=$	$ 9 + 6\cdot 17 + 3\cdot 17^{2} + 7\cdot 17^{3} + 16\cdot 17^{4} +O\left(17^{ 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.