Basic invariants
| Dimension: | $1$ |
| Group: | $C_9$ |
| Conductor: | \(27\)\(\medspace = 3^{3} \) |
| Artin field: | Galois closure of \(\Q(\zeta_{27})^+\) |
| Galois orbit size: | $6$ |
| Smallest permutation container: | $C_9$ |
| Parity: | even |
| Dirichlet character: | \(\chi_{27}(16,\cdot)\) |
| Projective image: | $C_1$ |
| Projective field: | Galois closure of \(\Q\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{9} - 9x^{7} + 27x^{5} - 30x^{3} + 9x - 1 \)
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The roots of $f$ are computed in an extension of $\Q_{ 17 }$ to precision 6.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 17 }$:
\( x^{3} + x + 14 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 16 a^{2} + 15 a + 5 + \left(12 a^{2} + 14 a + 14\right)\cdot 17 + \left(2 a^{2} + 11 a + 1\right)\cdot 17^{2} + \left(5 a^{2} + 5 a + 9\right)\cdot 17^{3} + \left(16 a^{2} + 8 a + 16\right)\cdot 17^{4} + \left(a^{2} + 14 a + 6\right)\cdot 17^{5} +O(17^{6})\)
|
| $r_{ 2 }$ | $=$ |
\( 13 a^{2} + 2 a + 3 + \left(6 a^{2} + 7 a + 10\right)\cdot 17 + \left(14 a^{2} + 13 a + 9\right)\cdot 17^{2} + \left(2 a^{2} + 3 a + 7\right)\cdot 17^{3} + \left(10 a + 11\right)\cdot 17^{4} + \left(13 a^{2} + 7 a + 8\right)\cdot 17^{5} +O(17^{6})\)
|
| $r_{ 3 }$ | $=$ |
\( 13 a^{2} + 13 a + 3 + \left(14 a^{2} + 7 a + 4\right)\cdot 17 + \left(13 a^{2} + 4 a + 9\right)\cdot 17^{2} + \left(12 a^{2} + 8\right)\cdot 17^{3} + \left(8 a^{2} + 11 a + 11\right)\cdot 17^{4} + \left(8 a^{2} + 5 a + 5\right)\cdot 17^{5} +O(17^{6})\)
|
| $r_{ 4 }$ | $=$ |
\( 14 a^{2} + 15 + \left(a^{2} + 3 a + 6\right)\cdot 17 + \left(14 a^{2} + 8 a + 9\right)\cdot 17^{2} + \left(a^{2} + 3 a + 12\right)\cdot 17^{3} + \left(12 a^{2} + 13\right)\cdot 17^{4} + \left(16 a^{2} + 2 a + 16\right)\cdot 17^{5} +O(17^{6})\)
|
| $r_{ 5 }$ | $=$ |
\( 7 a^{2} + 8 a + 16 + \left(6 a^{2} + 6 a + 9\right)\cdot 17 + \left(7 a^{2} + 6 a + 10\right)\cdot 17^{2} + \left(15 a^{2} + 16 a + 4\right)\cdot 17^{3} + \left(14 a^{2} + 15 a + 4\right)\cdot 17^{4} + \left(a^{2} + 15 a + 1\right)\cdot 17^{5} +O(17^{6})\)
|
| $r_{ 6 }$ | $=$ |
\( 2 a^{2} + 9 a + 7 + \left(7 a^{2} + a + 10\right)\cdot 17 + \left(4 a^{2} + a + 8\right)\cdot 17^{2} + \left(13 a^{2} + 3 a + 14\right)\cdot 17^{3} + \left(4 a^{2} + 6 a + 8\right)\cdot 17^{4} + \left(16 a^{2} + 4 a + 16\right)\cdot 17^{5} +O(17^{6})\)
|
| $r_{ 7 }$ | $=$ |
\( 14 a^{2} + 7 a + 15 + \left(3 a^{2} + 3 a + 13\right)\cdot 17 + \left(12 a^{2} + 14 a + 13\right)\cdot 17^{2} + \left(15 a^{2} + 13 a + 4\right)\cdot 17^{3} + \left(a^{2} + 7 a + 1\right)\cdot 17^{4} + \left(2 a^{2} + 10 a + 7\right)\cdot 17^{5} +O(17^{6})\)
|
| $r_{ 8 }$ | $=$ |
\( 16 a^{2} + 10 a + 5 + \left(13 a^{2} + 9\right)\cdot 17 + \left(9 a^{2} + 4 a + 6\right)\cdot 17^{2} + \left(15 a^{2} + 8 a + 10\right)\cdot 17^{3} + \left(12 a^{2} + 2 a + 8\right)\cdot 17^{4} + \left(15 a^{2} + 15 a + 10\right)\cdot 17^{5} +O(17^{6})\)
|
| $r_{ 9 }$ | $=$ |
\( 7 a^{2} + 4 a + 16 + \left(6 a + 5\right)\cdot 17 + \left(6 a^{2} + 4 a + 15\right)\cdot 17^{2} + \left(2 a^{2} + 13 a + 12\right)\cdot 17^{3} + \left(13 a^{2} + 5 a + 8\right)\cdot 17^{4} + \left(8 a^{2} + 9 a + 11\right)\cdot 17^{5} +O(17^{6})\)
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Generators of the action on the roots $r_1, \ldots, r_{ 9 }$
| Cycle notation |
Character values on conjugacy classes
| Size | Order | Action on $r_1, \ldots, r_{ 9 }$ | Character value | Complex conjugation |
| $1$ | $1$ | $()$ | $1$ | ✓ |
| $1$ | $3$ | $(1,8,6)(2,5,7)(3,4,9)$ | $\zeta_{9}^{3}$ | |
| $1$ | $3$ | $(1,6,8)(2,7,5)(3,9,4)$ | $-\zeta_{9}^{3} - 1$ | |
| $1$ | $9$ | $(1,7,9,8,2,3,6,5,4)$ | $\zeta_{9}$ | |
| $1$ | $9$ | $(1,9,2,6,4,7,8,3,5)$ | $\zeta_{9}^{2}$ | |
| $1$ | $9$ | $(1,2,4,8,5,9,6,7,3)$ | $\zeta_{9}^{4}$ | |
| $1$ | $9$ | $(1,3,7,6,9,5,8,4,2)$ | $\zeta_{9}^{5}$ | |
| $1$ | $9$ | $(1,5,3,8,7,4,6,2,9)$ | $-\zeta_{9}^{4} - \zeta_{9}$ | |
| $1$ | $9$ | $(1,4,5,6,3,2,8,9,7)$ | $-\zeta_{9}^{5} - \zeta_{9}^{2}$ |