# Properties

 Label 1.9.6t1.a.a Dimension $1$ Group $C_6$ Conductor $9$ Root number not computed Indicator $0$

# Related objects

## Basic invariants

 Dimension: $1$ Group: $C_6$ Conductor: $$9$$$$\medspace = 3^{2}$$ Artin field: Galois closure of $$\Q(\zeta_{9})$$ Galois orbit size: $2$ Smallest permutation container: $C_6$ Parity: odd Dirichlet character: $$\chi_{9}(5,\cdot)$$ Projective image: $C_1$ Projective field: Galois closure of $$\Q$$

## Defining polynomial

 $f(x)$ $=$ $$x^{6} - x^{3} + 1$$ x^6 - x^3 + 1 .

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

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 17 }$: $$x^{2} + 16x + 3$$

Roots:
 $r_{ 1 }$ $=$ $$11 a + 5 + \left(8 a + 14\right)\cdot 17 + \left(13 a + 12\right)\cdot 17^{2} + \left(8 a + 7\right)\cdot 17^{3} + \left(a + 11\right)\cdot 17^{4} +O(17^{5})$$ 11*a + 5 + (8*a + 14)*17 + (13*a + 12)*17^2 + (8*a + 7)*17^3 + (a + 11)*17^4+O(17^5) $r_{ 2 }$ $=$ $$4 a + 3 + \left(6 a + 5\right)\cdot 17 + \left(16 a + 6\right)\cdot 17^{2} + \left(13 a + 5\right)\cdot 17^{3} + \left(3 a + 3\right)\cdot 17^{4} +O(17^{5})$$ 4*a + 3 + (6*a + 5)*17 + (16*a + 6)*17^2 + (13*a + 5)*17^3 + (3*a + 3)*17^4+O(17^5) $r_{ 3 }$ $=$ $$10 a + 5 + \left(14 a + 12\right)\cdot 17 + \left(2 a + 4\right)\cdot 17^{2} + \left(5 a + 6\right)\cdot 17^{3} + \left(2 a + 12\right)\cdot 17^{4} +O(17^{5})$$ 10*a + 5 + (14*a + 12)*17 + (2*a + 4)*17^2 + (5*a + 6)*17^3 + (2*a + 12)*17^4+O(17^5) $r_{ 4 }$ $=$ $$7 a + 15 + \left(2 a + 16\right)\cdot 17 + \left(14 a + 9\right)\cdot 17^{2} + \left(11 a + 8\right)\cdot 17^{3} + \left(14 a + 9\right)\cdot 17^{4} +O(17^{5})$$ 7*a + 15 + (2*a + 16)*17 + (14*a + 9)*17^2 + (11*a + 8)*17^3 + (14*a + 9)*17^4+O(17^5) $r_{ 5 }$ $=$ $$13 a + 7 + \left(10 a + 7\right)\cdot 17 + 16\cdot 17^{2} + \left(3 a + 2\right)\cdot 17^{3} + \left(13 a + 10\right)\cdot 17^{4} +O(17^{5})$$ 13*a + 7 + (10*a + 7)*17 + 16*17^2 + (3*a + 2)*17^3 + (13*a + 10)*17^4+O(17^5) $r_{ 6 }$ $=$ $$6 a + 16 + \left(8 a + 11\right)\cdot 17 + 3 a\cdot 17^{2} + \left(8 a + 3\right)\cdot 17^{3} + \left(15 a + 4\right)\cdot 17^{4} +O(17^{5})$$ 6*a + 16 + (8*a + 11)*17 + 3*a*17^2 + (8*a + 3)*17^3 + (15*a + 4)*17^4+O(17^5)

## Generators of the action on the roots $r_1, \ldots, r_{ 6 }$

 Cycle notation $(1,2,3,6,5,4)$ $(1,6)(2,5)(3,4)$

## Character values on conjugacy classes

 Size Order Action on $r_1, \ldots, r_{ 6 }$ Character value $1$ $1$ $()$ $1$ $1$ $2$ $(1,6)(2,5)(3,4)$ $-1$ $1$ $3$ $(1,3,5)(2,6,4)$ $\zeta_{3}$ $1$ $3$ $(1,5,3)(2,4,6)$ $-\zeta_{3} - 1$ $1$ $6$ $(1,2,3,6,5,4)$ $\zeta_{3} + 1$ $1$ $6$ $(1,4,5,6,3,2)$ $-\zeta_{3}$

The blue line marks the conjugacy class containing complex conjugation.