# Properties

 Label 1.27.9t1.a.c Dimension $1$ Group $C_9$ Conductor $27$ Root number not computed Indicator $0$

# Related objects

## Basic invariants

 Dimension: $1$ Group: $C_9$ Conductor: $$27$$$$\medspace = 3^{3}$$ Artin field: Galois closure of $$\Q(\zeta_{27})^+$$ Galois orbit size: $6$ Smallest permutation container: $C_9$ Parity: even Dirichlet character: $$\chi_{27}(7,\cdot)$$ Projective image: $C_1$ Projective field: Galois closure of $$\Q$$

## Defining polynomial

 $f(x)$ $=$ $$x^{9} - 9x^{7} + 27x^{5} - 30x^{3} + 9x - 1$$ x^9 - 9*x^7 + 27*x^5 - 30*x^3 + 9*x - 1 .

The roots of $f$ are computed in an extension of $\Q_{ 17 }$ to precision 6.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 17 }$: $$x^{3} + x + 14$$

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

## Generators of the action on the roots $r_1, \ldots, r_{ 9 }$

 Cycle notation $(1,7,9,8,2,3,6,5,4)$ $(1,8,6)(2,5,7)(3,4,9)$

## Character values on conjugacy classes

 Size Order Action on $r_1, \ldots, r_{ 9 }$ Character value $1$ $1$ $()$ $1$ $1$ $3$ $(1,8,6)(2,5,7)(3,4,9)$ $\zeta_{9}^{3}$ $1$ $3$ $(1,6,8)(2,7,5)(3,9,4)$ $-\zeta_{9}^{3} - 1$ $1$ $9$ $(1,7,9,8,2,3,6,5,4)$ $\zeta_{9}^{4}$ $1$ $9$ $(1,9,2,6,4,7,8,3,5)$ $-\zeta_{9}^{5} - \zeta_{9}^{2}$ $1$ $9$ $(1,2,4,8,5,9,6,7,3)$ $-\zeta_{9}^{4} - \zeta_{9}$ $1$ $9$ $(1,3,7,6,9,5,8,4,2)$ $\zeta_{9}^{2}$ $1$ $9$ $(1,5,3,8,7,4,6,2,9)$ $\zeta_{9}$ $1$ $9$ $(1,4,5,6,3,2,8,9,7)$ $\zeta_{9}^{5}$

The blue line marks the conjugacy class containing complex conjugation.