# Properties

 Label 1.8.2t1.b.a Dimension $1$ Group $C_2$ Conductor $8$ Root number $1$ Indicator $1$

# Related objects

## Basic invariants

 Dimension: $1$ Group: $C_2$ Conductor: $$8$$$$\medspace = 2^{3}$$ Frobenius-Schur indicator: $1$ Root number: $1$ Artin field: Galois closure of $$\Q(\sqrt{-2})$$ Galois orbit size: $1$ Smallest permutation container: $C_2$ Parity: odd Dirichlet character: $$\displaystyle\left(\frac{-8}{\bullet}\right)$$ Projective image: $C_1$ Projective field: Galois closure of $$\Q$$

## Defining polynomial

 $f(x)$ $=$ $$x^{2} + 2$$ x^2 + 2 .

The roots of $f$ are computed in $\Q_{ 3 }$ to precision 5.

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

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

The blue line marks the conjugacy class containing complex conjugation.