Basic invariants
| Dimension: | $1$ |
| Group: | $C_9$ |
| Conductor: | \(189\)\(\medspace = 3^{3} \cdot 7 \) |
| Artin number field: | Galois closure of 9.9.3691950281939241.1 |
| Galois orbit size: | $6$ |
| Smallest permutation container: | $C_9$ |
| 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 an extension of $\Q_{ 17 }$ to precision 9.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 17 }$:
\( x^{3} + x + 14 \)
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Roots:
| $r_{ 1 }$ | $=$ |
\( 4 a^{2} + 6 a + 14 + \left(9 a^{2} + 11\right)\cdot 17 + \left(6 a^{2} + 5 a + 15\right)\cdot 17^{2} + \left(14 a^{2} + 5 a + 3\right)\cdot 17^{3} + \left(12 a^{2} + 3 a + 14\right)\cdot 17^{4} + \left(7 a^{2} + 13 a + 10\right)\cdot 17^{5} + \left(a^{2} + 7 a + 6\right)\cdot 17^{6} + \left(5 a^{2} + 13 a + 3\right)\cdot 17^{7} + \left(7 a^{2} + 5 a + 16\right)\cdot 17^{8} +O(17^{9})\)
|
| $r_{ 2 }$ | $=$ |
\( 13 a^{2} + 4 a + 3 + \left(9 a^{2} + 15 a + 12\right)\cdot 17 + \left(8 a^{2} + 4 a + 5\right)\cdot 17^{2} + \left(6 a^{2} + 12 a + 4\right)\cdot 17^{3} + \left(2 a^{2} + 4 a + 7\right)\cdot 17^{4} + \left(16 a^{2} + 16 a + 16\right)\cdot 17^{5} + \left(16 a^{2} + 2 a + 16\right)\cdot 17^{6} + \left(8 a^{2} + 12 a + 5\right)\cdot 17^{7} + \left(10 a^{2} + 9 a + 1\right)\cdot 17^{8} +O(17^{9})\)
|
| $r_{ 3 }$ | $=$ |
\( a^{2} + 14 a + 12 + \left(11 a^{2} + 10 a + 1\right)\cdot 17 + \left(15 a^{2} + 16 a + 16\right)\cdot 17^{2} + \left(14 a^{2} + 13 a + 9\right)\cdot 17^{3} + \left(6 a^{2} + 5 a + 4\right)\cdot 17^{4} + \left(16 a^{2} + 8 a + 5\right)\cdot 17^{5} + \left(12 a^{2} + 8 a + 14\right)\cdot 17^{6} + \left(2 a^{2} + 3 a + 1\right)\cdot 17^{7} + \left(10 a^{2} + 1\right)\cdot 17^{8} +O(17^{9})\)
|
| $r_{ 4 }$ | $=$ |
\( 5 a^{2} + 8 a + 9 + \left(4 a^{2} + 9 a + 8\right)\cdot 17 + \left(14 a^{2} + 9 a + 9\right)\cdot 17^{2} + \left(2 a^{2} + 10 a + 7\right)\cdot 17^{3} + \left(11 a^{2} + 5 a + 7\right)\cdot 17^{4} + \left(5 a^{2} + 9\right)\cdot 17^{5} + \left(12 a^{2} + a + 2\right)\cdot 17^{6} + \left(14 a^{2} + 5 a + 4\right)\cdot 17^{7} + \left(15 a^{2} + 2 a + 16\right)\cdot 17^{8} +O(17^{9})\)
|
| $r_{ 5 }$ | $=$ |
\( 16 a^{2} + 5 a + 5 + \left(2 a^{2} + 9 a + 13\right)\cdot 17 + \left(11 a^{2} + 2 a + 1\right)\cdot 17^{2} + \left(7 a^{2} + 11 a + 5\right)\cdot 17^{3} + \left(3 a^{2} + 6 a + 2\right)\cdot 17^{4} + \left(12 a^{2} + 8\right)\cdot 17^{5} + \left(4 a^{2} + 13 a + 14\right)\cdot 17^{6} + \left(10 a^{2} + 16 a + 6\right)\cdot 17^{7} + \left(7 a^{2} + 4 a + 16\right)\cdot 17^{8} +O(17^{9})\)
|
| $r_{ 6 }$ | $=$ |
\( 8 a + \left(5 a^{2} + 2 a + 9\right)\cdot 17 + \left(3 a^{2} + 6 a + 13\right)\cdot 17^{2} + \left(13 a^{2} + 11 a + 8\right)\cdot 17^{3} + \left(16 a^{2} + 8 a + 5\right)\cdot 17^{4} + \left(16 a^{2} + 8 a + 11\right)\cdot 17^{5} + \left(10 a^{2} + 10 a + 1\right)\cdot 17^{6} + \left(13 a^{2} + 9 a + 9\right)\cdot 17^{7} + \left(7 a^{2} + 4 a + 16\right)\cdot 17^{8} +O(17^{9})\)
|
| $r_{ 7 }$ | $=$ |
\( 2 a^{2} + 5 a + 7 + \left(9 a^{2} + 3 a\right)\cdot 17 + \left(14 a^{2} + 15 a + 4\right)\cdot 17^{2} + \left(4 a^{2} + 10 a + 3\right)\cdot 17^{3} + \left(3 a^{2} + 11 a + 2\right)\cdot 17^{4} + \left(16 a^{2} + 12 a + 5\right)\cdot 17^{5} + \left(4 a^{2} + 5 a + 3\right)\cdot 17^{6} + \left(6 a^{2} + 11 a + 4\right)\cdot 17^{7} + \left(a^{2} + 16 a + 12\right)\cdot 17^{8} +O(17^{9})\)
|
| $r_{ 8 }$ | $=$ |
\( 12 a^{2} + 14 a + 8 + \left(13 a^{2} + 5 a + 3\right)\cdot 17 + \left(11 a^{2} + 12 a + 2\right)\cdot 17^{2} + \left(4 a^{2} + 14 a + 3\right)\cdot 17^{3} + \left(14 a^{2} + 7 a + 15\right)\cdot 17^{4} + \left(9 a^{2} + 12 a\right)\cdot 17^{5} + \left(2 a^{2} + 13\right)\cdot 17^{6} + \left(9 a^{2} + 11\right)\cdot 17^{7} + \left(16 a^{2} + 11 a + 16\right)\cdot 17^{8} +O(17^{9})\)
|
| $r_{ 9 }$ | $=$ |
\( 15 a^{2} + 4 a + 10 + \left(2 a^{2} + 11 a + 7\right)\cdot 17 + \left(16 a^{2} + 12 a + 16\right)\cdot 17^{2} + \left(15 a^{2} + 11 a + 4\right)\cdot 17^{3} + \left(13 a^{2} + 13 a + 9\right)\cdot 17^{4} + 12 a\cdot 17^{5} + \left(a^{2} + 12\right)\cdot 17^{6} + \left(14 a^{2} + 13 a + 3\right)\cdot 17^{7} + \left(7 a^{2} + 12 a + 5\right)\cdot 17^{8} +O(17^{9})\)
|
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 values | |||||
| $c1$ | $c2$ | $c3$ | $c4$ | $c5$ | $c6$ | |||
| $1$ | $1$ | $()$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$ |
| $1$ | $3$ | $(1,3,8)(2,4,5)(6,9,7)$ | $\zeta_{9}^{3}$ | $-\zeta_{9}^{3} - 1$ | $\zeta_{9}^{3}$ | $-\zeta_{9}^{3} - 1$ | $\zeta_{9}^{3}$ | $-\zeta_{9}^{3} - 1$ |
| $1$ | $3$ | $(1,8,3)(2,5,4)(6,7,9)$ | $-\zeta_{9}^{3} - 1$ | $\zeta_{9}^{3}$ | $-\zeta_{9}^{3} - 1$ | $\zeta_{9}^{3}$ | $-\zeta_{9}^{3} - 1$ | $\zeta_{9}^{3}$ |
| $1$ | $9$ | $(1,5,9,3,2,7,8,4,6)$ | $\zeta_{9}$ | $\zeta_{9}^{2}$ | $\zeta_{9}^{4}$ | $\zeta_{9}^{5}$ | $-\zeta_{9}^{4} - \zeta_{9}$ | $-\zeta_{9}^{5} - \zeta_{9}^{2}$ |
| $1$ | $9$ | $(1,9,2,8,6,5,3,7,4)$ | $\zeta_{9}^{2}$ | $\zeta_{9}^{4}$ | $-\zeta_{9}^{5} - \zeta_{9}^{2}$ | $\zeta_{9}$ | $\zeta_{9}^{5}$ | $-\zeta_{9}^{4} - \zeta_{9}$ |
| $1$ | $9$ | $(1,2,6,3,4,9,8,5,7)$ | $\zeta_{9}^{4}$ | $-\zeta_{9}^{5} - \zeta_{9}^{2}$ | $-\zeta_{9}^{4} - \zeta_{9}$ | $\zeta_{9}^{2}$ | $\zeta_{9}$ | $\zeta_{9}^{5}$ |
| $1$ | $9$ | $(1,7,5,8,9,4,3,6,2)$ | $\zeta_{9}^{5}$ | $\zeta_{9}$ | $\zeta_{9}^{2}$ | $-\zeta_{9}^{4} - \zeta_{9}$ | $-\zeta_{9}^{5} - \zeta_{9}^{2}$ | $\zeta_{9}^{4}$ |
| $1$ | $9$ | $(1,4,7,3,5,6,8,2,9)$ | $-\zeta_{9}^{4} - \zeta_{9}$ | $\zeta_{9}^{5}$ | $\zeta_{9}$ | $-\zeta_{9}^{5} - \zeta_{9}^{2}$ | $\zeta_{9}^{4}$ | $\zeta_{9}^{2}$ |
| $1$ | $9$ | $(1,6,4,8,7,2,3,9,5)$ | $-\zeta_{9}^{5} - \zeta_{9}^{2}$ | $-\zeta_{9}^{4} - \zeta_{9}$ | $\zeta_{9}^{5}$ | $\zeta_{9}^{4}$ | $\zeta_{9}^{2}$ | $\zeta_{9}$ |