Basic invariants
| Dimension: | $1$ |
| Group: | $C_3$ |
| Conductor: | \(117\)\(\medspace = 3^{2} \cdot 13 \) |
| Artin number field: | Galois closure of 3.3.13689.2 |
| 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 $\Q_{ 7 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 2\cdot 7 + 2\cdot 7^{2} + 4\cdot 7^{3} + 2\cdot 7^{4} +O(7^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 2 + 2\cdot 7 + 7^{2} + 5\cdot 7^{3} + 3\cdot 7^{4} +O(7^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 5 + 2\cdot 7 + 3\cdot 7^{2} + 4\cdot 7^{3} +O(7^{5})\)
|
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 values | |
| $c1$ | $c2$ | |||
| $1$ | $1$ | $()$ | $1$ | $1$ |
| $1$ | $3$ | $(1,2,3)$ | $\zeta_{3}$ | $-\zeta_{3} - 1$ |
| $1$ | $3$ | $(1,3,2)$ | $-\zeta_{3} - 1$ | $\zeta_{3}$ |