# 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 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: Galois closure of $$\Q$$

## Defining polynomial

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

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

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

## 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.