# Properties

 Label 1.171.3t1.b Dimension $1$ Group $C_3$ Conductor $171$ Indicator $0$

# Related objects

## Basic invariants

 Dimension: $1$ Group: $C_3$ Conductor: $$171$$$$\medspace = 3^{2} \cdot 19$$ Artin number field: Galois closure of 3.3.29241.2 Galois orbit size: $2$ Smallest permutation container: $C_3$ Parity: even Projective image: $C_1$ Projective field: Galois closure of $$\Q$$

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 29 }$ to precision 5.
Roots:
 $r_{ 1 }$ $=$ $$6 + 20\cdot 29 + 2\cdot 29^{2} + 29^{3} + 11\cdot 29^{4} +O(29^{5})$$ 6 + 20*29 + 2*29^2 + 29^3 + 11*29^4+O(29^5) $r_{ 2 }$ $=$ $$25 + 6\cdot 29 + 15\cdot 29^{2} + 26\cdot 29^{3} + 11\cdot 29^{4} +O(29^{5})$$ 25 + 6*29 + 15*29^2 + 26*29^3 + 11*29^4+O(29^5) $r_{ 3 }$ $=$ $$27 + 29 + 11\cdot 29^{2} + 29^{3} + 6\cdot 29^{4} +O(29^{5})$$ 27 + 29 + 11*29^2 + 29^3 + 6*29^4+O(29^5)

### 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 values $c1$ $c2$ $1$ $1$ $()$ $1$ $1$ $1$ $3$ $(1,2,3)$ $\zeta_{3}$ $-\zeta_{3} - 1$ $1$ $3$ $(1,3,2)$ $-\zeta_{3} - 1$ $\zeta_{3}$
The blue line marks the conjugacy class containing complex conjugation.