Basic invariants
| Dimension: | $2$ |
| Group: | $D_{9}$ |
| Conductor: | \(1944\)\(\medspace = 2^{3} \cdot 3^{5} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 9.1.1586874322944.5 |
| Galois orbit size: | $3$ |
| Smallest permutation container: | $D_{9}$ |
| Parity: | odd |
| Determinant: | 1.24.2t1.b.a |
| Projective image: | $D_9$ |
| Projective stem field: | Galois closure of 9.1.1586874322944.5 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{9} + 9x^{7} - 6x^{6} + 27x^{5} - 36x^{4} + 27x^{3} - 54x^{2} - 32 \)
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The roots of $f$ are computed in an extension of $\Q_{ 29 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 29 }$:
\( x^{3} + 2x + 27 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 13 a^{2} + 22 a + 27 + \left(12 a^{2} + 18 a + 6\right)\cdot 29 + \left(21 a^{2} + 28 a + 9\right)\cdot 29^{2} + \left(7 a^{2} + 21 a + 10\right)\cdot 29^{3} + \left(10 a^{2} + 10 a + 23\right)\cdot 29^{4} +O(29^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 6 a^{2} + 27 a + 8 + \left(17 a^{2} + 25 a + 13\right)\cdot 29 + \left(3 a^{2} + 16 a + 14\right)\cdot 29^{2} + \left(11 a^{2} + 9 a + 24\right)\cdot 29^{3} + \left(26 a^{2} + 6 a + 15\right)\cdot 29^{4} +O(29^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 3 a^{2} + 11 a + 4 + \left(14 a^{2} + 7 a + 9\right)\cdot 29 + \left(9 a^{2} + 22\right)\cdot 29^{2} + \left(a^{2} + 14 a + 1\right)\cdot 29^{3} + \left(13 a^{2} + 8 a + 27\right)\cdot 29^{4} +O(29^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 26 a^{2} + 2 a + 25 + \left(18 a^{2} + 27 a + 5\right)\cdot 29 + \left(10 a^{2} + a + 14\right)\cdot 29^{2} + \left(7 a^{2} + 13 a + 19\right)\cdot 29^{3} + \left(21 a^{2} + 17 a + 18\right)\cdot 29^{4} +O(29^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 23 a^{2} + 9 a + 21 + \left(2 a^{2} + 3\right)\cdot 29 + \left(3 a^{2} + 8 a + 4\right)\cdot 29^{2} + \left(2 a^{2} + 2 a + 22\right)\cdot 29^{3} + \left(9 a^{2} + 22 a + 21\right)\cdot 29^{4} +O(29^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 4 a^{2} + 19 a + 15 + \left(11 a^{2} + 9 a + 24\right)\cdot 29 + \left(16 a^{2} + 16 a + 21\right)\cdot 29^{2} + \left(25 a^{2} + 10 a + 14\right)\cdot 29^{3} + \left(6 a^{2} + 5 a + 28\right)\cdot 29^{4} +O(29^{5})\)
|
| $r_{ 7 }$ | $=$ |
\( 12 a^{2} + 17 a + 16 + \left(5 a^{2} + 26\right)\cdot 29 + \left(20 a^{2} + 13 a + 26\right)\cdot 29^{2} + \left(24 a^{2} + 25 a + 3\right)\cdot 29^{3} + \left(11 a^{2} + 12 a + 6\right)\cdot 29^{4} +O(29^{5})\)
|
| $r_{ 8 }$ | $=$ |
\( 9 a^{2} + 18 a + 12 + \left(7 a^{2} + a + 19\right)\cdot 29 + \left(15 a^{2} + 19 a + 10\right)\cdot 29^{2} + \left(19 a^{2} + 13 a + 16\right)\cdot 29^{3} + \left(27 a^{2} + 18 a + 17\right)\cdot 29^{4} +O(29^{5})\)
|
| $r_{ 9 }$ | $=$ |
\( 20 a^{2} + 20 a + 17 + \left(26 a^{2} + 24 a + 6\right)\cdot 29 + \left(15 a^{2} + 11 a + 21\right)\cdot 29^{2} + \left(16 a^{2} + 5 a + 2\right)\cdot 29^{3} + \left(18 a^{2} + 14 a + 15\right)\cdot 29^{4} +O(29^{5})\)
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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 |
| $1$ | $1$ | $()$ | $2$ |
| $9$ | $2$ | $(1,5)(2,3)(4,6)(7,8)$ | $0$ |
| $2$ | $3$ | $(1,6,7)(2,3,9)(4,5,8)$ | $-1$ |
| $2$ | $9$ | $(1,9,5,6,2,8,7,3,4)$ | $-\zeta_{9}^{4} + \zeta_{9}^{2} - \zeta_{9}$ |
| $2$ | $9$ | $(1,5,2,7,4,9,6,8,3)$ | $\zeta_{9}^{5} + \zeta_{9}^{4}$ |
| $2$ | $9$ | $(1,2,4,6,3,5,7,9,8)$ | $-\zeta_{9}^{5} - \zeta_{9}^{2} + \zeta_{9}$ |
The blue line marks the conjugacy class containing complex conjugation.