Basic invariants
| Dimension: | $1$ |
| Group: | $C_3$ |
| Conductor: | \(7\) |
| Artin field: | Galois closure of \(\Q(\zeta_{7})^+\) |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $C_3$ |
| Parity: | even |
| Dirichlet character: | \(\chi_{7}(2,\cdot)\) |
| Projective image: | $C_1$ |
| Projective field: | Galois closure of \(\Q\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{3} - x^{2} - 2x + 1 \)
|
The roots of $f$ are computed in $\Q_{ 13 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 3 + 2\cdot 13 + 5\cdot 13^{2} + 5\cdot 13^{3} + 11\cdot 13^{4} +O(13^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 5 + 10\cdot 13 + 3\cdot 13^{2} + 8\cdot 13^{3} + 12\cdot 13^{4} +O(13^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 6 + 4\cdot 13^{2} + 12\cdot 13^{3} + 13^{4} +O(13^{5})\)
|
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 | Complex conjugation |
| $1$ | $1$ | $()$ | $1$ | ✓ |
| $1$ | $3$ | $(1,2,3)$ | $\zeta_{3}$ | |
| $1$ | $3$ | $(1,3,2)$ | $-\zeta_{3} - 1$ |