# Properties

 Label 1.3e3.9t1.1c1 Dimension 1 Group $C_9$ Conductor $3^{3}$ Root number not computed Frobenius-Schur indicator 0

# Related objects

## Basic invariants

 Dimension: $1$ Group: $C_9$ Conductor: $27= 3^{3}$ Artin number field: Splitting field of $f= x^{9} - 9 x^{7} + 27 x^{5} - 30 x^{3} + 9 x - 1$ over $\Q$ Size of Galois orbit: 6 Smallest containing permutation representation: $C_9$ Parity: Even Corresponding Dirichlet character: $$\chi_{27}(16,\cdot)$$

## Galois action

### Roots of defining polynomial

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

### 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 $1$ $1$ $()$ $1$ $1$ $3$ $(1,8,6)(2,5,7)(3,4,9)$ $\zeta_{9}^{3}$ $1$ $3$ $(1,6,8)(2,7,5)(3,9,4)$ $-\zeta_{9}^{3} - 1$ $1$ $9$ $(1,7,9,8,2,3,6,5,4)$ $\zeta_{9}$ $1$ $9$ $(1,9,2,6,4,7,8,3,5)$ $\zeta_{9}^{2}$ $1$ $9$ $(1,2,4,8,5,9,6,7,3)$ $\zeta_{9}^{4}$ $1$ $9$ $(1,3,7,6,9,5,8,4,2)$ $\zeta_{9}^{5}$ $1$ $9$ $(1,5,3,8,7,4,6,2,9)$ $-\zeta_{9}^{4} - \zeta_{9}$ $1$ $9$ $(1,4,5,6,3,2,8,9,7)$ $-\zeta_{9}^{5} - \zeta_{9}^{2}$
The blue line marks the conjugacy class containing complex conjugation.