# Properties

 Label 1.9.3t1.a.a Dimension $1$ Group $C_3$ Conductor $9$ Root number not computed Indicator $0$

# Related objects

## Basic invariants

 Dimension: $1$ Group: $C_3$ Conductor: $$9$$$$\medspace = 3^{2}$$ Artin number field: Galois closure of $$\Q(\zeta_{9})^+$$ Galois orbit size: $2$ Smallest permutation container: $C_3$ Parity: even Dirichlet character: $$\chi_{9}(7,\cdot)$$ Projective image: $C_1$ Projective field: $$\Q$$

## Defining polynomial

 $f(x)$ $=$ $x^{3} - 3 x - 1$.

The roots of $f$ are computed in $\Q_{ 17 }$ to precision 5.

Roots:
 $r_{ 1 }$ $=$ $3 + 12\cdot 17 + 14\cdot 17^{2} + 14\cdot 17^{3} + 4\cdot 17^{4} +O\left(17^{ 5 }\right)$ $r_{ 2 }$ $=$ $4 + 9\cdot 17 + 13\cdot 17^{2} + 10\cdot 17^{3} + 15\cdot 17^{4} +O\left(17^{ 5 }\right)$ $r_{ 3 }$ $=$ $10 + 12\cdot 17 + 5\cdot 17^{2} + 8\cdot 17^{3} + 13\cdot 17^{4} +O\left(17^{ 5 }\right)$

## Generators of the action on the roots $r_{ 1 }, r_{ 2 }, r_{ 3 }$

 Cycle notation $(1,2,3)$

## Character values on conjugacy classes

 Size Order Action on $r_{ 1 }, r_{ 2 }, r_{ 3 }$ Character value $1$ $1$ $()$ $1$ $1$ $3$ $(1,2,3)$ $\zeta_{3}$ $1$ $3$ $(1,3,2)$ $-\zeta_{3} - 1$

The blue line marks the conjugacy class containing complex conjugation.