Basic invariants
| Dimension: | $1$ |
| Group: | $C_3$ |
| Conductor: | \(63\)\(\medspace = 3^{2} \cdot 7 \) |
| Artin field: | Galois closure of 3.3.3969.1 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $C_3$ |
| Parity: | even |
| Dirichlet character: | \(\chi_{63}(25,\cdot)\) |
| Projective image: | $C_1$ |
| Projective field: | Galois closure of \(\Q\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{3} - 21x - 28 \)
|
The roots of $f$ are computed in $\Q_{ 31 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 2 + 17\cdot 31 + 13\cdot 31^{2} + 30\cdot 31^{3} + 15\cdot 31^{4} +O(31^{5})\)
| $r_{ 2 }$ |
$=$ |
\( 6 + 21\cdot 31 + 5\cdot 31^{2} + 23\cdot 31^{3} + 9\cdot 31^{4} +O(31^{5})\)
| $r_{ 3 }$ |
$=$ |
\( 23 + 23\cdot 31 + 11\cdot 31^{2} + 8\cdot 31^{3} + 5\cdot 31^{4} +O(31^{5})\)
| |
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.