# Properties

 Label 1.9.3t1.a.b Dimension 1 Group $C_3$ Conductor $3^{2}$ Root number not computed Frobenius-Schur indicator 0

# Related objects

## Basic invariants

 Dimension: $1$ Group: $C_3$ Conductor: $9= 3^{2}$ Artin number field: Splitting field of $$\Q(\zeta_{9})^+$$ defined by $f= x^{3} - 3 x - 1$ over $\Q$ Size of Galois orbit: 2 Smallest containing permutation representation: $C_3$ Parity: Even Corresponding Dirichlet character: $$\chi_{9}(4,\cdot)$$ Projective image: $C_1$ Projective field: $$\Q$$

## Galois action

### Roots of defining polynomial

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 $1$ $1$ $()$ $1$ $1$ $3$ $(1,2,3)$ $-\zeta_{3} - 1$ $1$ $3$ $(1,3,2)$ $\zeta_{3}$
The blue line marks the conjugacy class containing complex conjugation.