Basic invariants
| Dimension: | $2$ |
| Group: | $D_{9}$ |
| Conductor: | \(14580\)\(\medspace = 2^{2} \cdot 3^{6} \cdot 5 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 9.1.5020969537440000.1 |
| Galois orbit size: | $3$ |
| Smallest permutation container: | $D_{9}$ |
| Parity: | odd |
| Determinant: | 1.20.2t1.a.a |
| Projective image: | $D_9$ |
| Projective stem field: | Galois closure of 9.1.5020969537440000.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{9} - 9x^{7} - 18x^{6} + 27x^{5} + 108x^{4} + 609x^{3} - 162x^{2} - 1908x - 5048 \)
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The roots of $f$ are computed in an extension of $\Q_{ 43 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 43 }$:
\( x^{3} + x + 40 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 9 a^{2} + 3 a + 6 + \left(16 a^{2} + 6 a + 25\right)\cdot 43 + \left(9 a^{2} + 32 a + 20\right)\cdot 43^{2} + \left(27 a^{2} + 5 a + 32\right)\cdot 43^{3} + \left(38 a^{2} + a + 25\right)\cdot 43^{4} +O(43^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 18 a^{2} + 39 a + 12 + \left(10 a^{2} + 2 a + 21\right)\cdot 43 + \left(7 a^{2} + 19 a + 33\right)\cdot 43^{2} + \left(30 a + 28\right)\cdot 43^{3} + \left(13 a^{2} + 36 a + 8\right)\cdot 43^{4} +O(43^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 13 a + \left(18 a^{2} + 18 a + 12\right)\cdot 43 + \left(15 a^{2} + 35 a + 10\right)\cdot 43^{2} + \left(19 a^{2} + 22 a + 27\right)\cdot 43^{3} + \left(32 a^{2} + 4 a + 21\right)\cdot 43^{4} +O(43^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 35 a^{2} + 22 a + 9 + \left(7 a^{2} + 22 a + 5\right)\cdot 43 + \left(13 a^{2} + 12 a + 23\right)\cdot 43^{2} + \left(25 a^{2} + 42 a + 2\right)\cdot 43^{3} + \left(18 a^{2} + 30 a + 41\right)\cdot 43^{4} +O(43^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 8 a^{2} + 8 a + 34 + \left(17 a^{2} + 2 a + 25\right)\cdot 43 + \left(14 a^{2} + 38 a + 9\right)\cdot 43^{2} + \left(41 a^{2} + 20 a + 13\right)\cdot 43^{3} + \left(34 a^{2} + 7 a + 23\right)\cdot 43^{4} +O(43^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 40 a^{2} + 16 a + 41 + \left(38 a^{2} + 37 a + 25\right)\cdot 43 + \left(18 a^{2} + 39 a + 12\right)\cdot 43^{2} + \left(4 a^{2} + 35 a + 17\right)\cdot 43^{3} + \left(31 a^{2} + 42 a + 6\right)\cdot 43^{4} +O(43^{5})\)
|
| $r_{ 7 }$ | $=$ |
\( 27 a^{2} + 7 a + 18 + \left(3 a^{2} + 20 a + 2\right)\cdot 43 + \left(3 a^{2} + 21 a + 2\right)\cdot 43^{2} + \left(35 a^{2} + 3 a + 9\right)\cdot 43^{3} + \left(11 a^{2} + a + 22\right)\cdot 43^{4} +O(43^{5})\)
|
| $r_{ 8 }$ | $=$ |
\( 7 a^{2} + 33 a + 19 + \left(23 a^{2} + 16 a + 15\right)\cdot 43 + \left(30 a^{2} + 32 a + 20\right)\cdot 43^{2} + \left(23 a^{2} + 33 a + 1\right)\cdot 43^{3} + \left(35 a^{2} + 40 a + 38\right)\cdot 43^{4} +O(43^{5})\)
|
| $r_{ 9 }$ | $=$ |
\( 28 a^{2} + 31 a + 33 + \left(36 a^{2} + 2 a + 38\right)\cdot 43 + \left(16 a^{2} + 27 a + 39\right)\cdot 43^{2} + \left(38 a^{2} + 19 a + 39\right)\cdot 43^{3} + \left(41 a^{2} + 6 a + 27\right)\cdot 43^{4} +O(43^{5})\)
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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 |
| $1$ | $1$ | $()$ | $2$ |
| $9$ | $2$ | $(1,7)(2,4)(3,6)(5,9)$ | $0$ |
| $2$ | $3$ | $(1,8,7)(2,6,9)(3,4,5)$ | $-1$ |
| $2$ | $9$ | $(1,3,2,8,4,6,7,5,9)$ | $-\zeta_{9}^{5} - \zeta_{9}^{2} + \zeta_{9}$ |
| $2$ | $9$ | $(1,2,4,7,9,3,8,6,5)$ | $-\zeta_{9}^{4} + \zeta_{9}^{2} - \zeta_{9}$ |
| $2$ | $9$ | $(1,4,9,8,5,2,7,3,6)$ | $\zeta_{9}^{5} + \zeta_{9}^{4}$ |
The blue line marks the conjugacy class containing complex conjugation.