Basic invariants
| Dimension: | $1$ |
| Group: | $C_3$ |
| Conductor: | \(171\)\(\medspace = 3^{2} \cdot 19 \) |
| Artin field: | Galois closure of 3.3.29241.2 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $C_3$ |
| Parity: | even |
| Dirichlet character: | \(\chi_{171}(121,\cdot)\) |
| Projective image: | $C_1$ |
| Projective field: | Galois closure of \(\Q\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{3} - 57x - 19 \)
|
The roots of $f$ are computed in $\Q_{ 29 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 6 + 20\cdot 29 + 2\cdot 29^{2} + 29^{3} + 11\cdot 29^{4} +O(29^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 25 + 6\cdot 29 + 15\cdot 29^{2} + 26\cdot 29^{3} + 11\cdot 29^{4} +O(29^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 27 + 29 + 11\cdot 29^{2} + 29^{3} + 6\cdot 29^{4} +O(29^{5})\)
|
Generators of the action on the roots $ r_{ 1 }, r_{ 2 }, r_{ 3 } $
| Cycle notation |
Character values on conjugacy classes
| Size | Order | Action on $ r_{ 1 }, r_{ 2 }, r_{ 3 } $ | Character value |
| $1$ | $1$ | $()$ | $1$ |
| $1$ | $3$ | $(1,2,3)$ | $\zeta_{3}$ |
| $1$ | $3$ | $(1,3,2)$ | $-\zeta_{3} - 1$ |
The blue line marks the conjugacy class containing complex conjugation.