# Properties

 Label 1.171.3t1.a.b Dimension $1$ Group $C_3$ Conductor $171$ Root number not computed Indicator $0$

# Related objects

## Basic invariants

 Dimension: $1$ Group: $C_3$ Conductor: $$171$$$$\medspace = 3^{2} \cdot 19$$ Artin field: 3.3.29241.1 Galois orbit size: $2$ Smallest permutation container: $C_3$ Parity: even Dirichlet character: $$\chi_{171}(49,\cdot)$$ Projective image: $C_1$ Projective field: $$\Q$$

## Defining polynomial

 $f(x)$ $=$ $$x^{3} - 57 x - 152$$  .

The roots of $f$ are computed in $\Q_{ 13 }$ to precision 5.

Roots:
 $r_{ 1 }$ $=$ $$1 + 5\cdot 13 + 7\cdot 13^{2} + 12\cdot 13^{3} + 9\cdot 13^{4} +O(13^{5})$$ $r_{ 2 }$ $=$ $$5 + 10\cdot 13 + 7\cdot 13^{2} + 10\cdot 13^{3} +O(13^{5})$$ $r_{ 3 }$ $=$ $$7 + 10\cdot 13 + 10\cdot 13^{2} + 2\cdot 13^{3} + 2\cdot 13^{4} +O(13^{5})$$

## 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 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.