# Properties

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

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## Basic invariants

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

## Defining polynomial

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

The roots of $f$ are computed in $\Q_{ 7 }$ to precision 5.

Roots:
 $r_{ 1 }$ $=$ $3 + 7 + 2\cdot 7^{2} + 6\cdot 7^{3} + 7^{4} +O\left(7^{ 5 }\right)$ $r_{ 2 }$ $=$ $4 + 5\cdot 7 + 4\cdot 7^{2} + 5\cdot 7^{4} +O\left(7^{ 5 }\right)$

## 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.