Basic invariants
| Dimension: | $2$ |
| Group: | $D_{9}$ |
| Conductor: | \(11191\)\(\medspace = 19^{2} \cdot 31 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 9.1.43447859067001.1 |
| Galois orbit size: | $3$ |
| Smallest permutation container: | $D_{9}$ |
| Parity: | odd |
| Determinant: | 1.31.2t1.a.a |
| Projective image: | $D_9$ |
| Projective stem field: | Galois closure of 9.1.43447859067001.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{9} - 2x^{8} + 20x^{7} - 34x^{6} + 162x^{5} - 269x^{4} + 606x^{3} - 1419x^{2} - 527x + 2139 \)
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The roots of $f$ are computed in an extension of $\Q_{ 47 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 47 }$:
\( x^{3} + 3x + 42 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 46 a^{2} + 27 a + 40 + \left(16 a^{2} + 34 a + 8\right)\cdot 47 + \left(10 a^{2} + 44 a + 27\right)\cdot 47^{2} + \left(4 a^{2} + 35 a + 6\right)\cdot 47^{3} + \left(38 a^{2} + 21 a + 46\right)\cdot 47^{4} +O(47^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 34 a^{2} + 19 a + 16 + \left(4 a^{2} + 7 a + 31\right)\cdot 47 + \left(38 a^{2} + 40 a + 35\right)\cdot 47^{2} + \left(3 a^{2} + 30 a + 5\right)\cdot 47^{3} + \left(31 a^{2} + 16 a + 32\right)\cdot 47^{4} +O(47^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 9 a^{2} + 33 a + 20 + \left(44 a^{2} + 3 a + 24\right)\cdot 47 + \left(12 a^{2} + 26 a + 40\right)\cdot 47^{2} + \left(9 a^{2} + 36 a + 6\right)\cdot 47^{3} + \left(3 a + 23\right)\cdot 47^{4} +O(47^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 36 a^{2} + 10 a + 13 + \left(12 a^{2} + 36\right)\cdot 47 + \left(21 a^{2} + 39 a + 5\right)\cdot 47^{2} + \left(27 a^{2} + 41 a + 37\right)\cdot 47^{3} + \left(17 a^{2} + 2 a + 26\right)\cdot 47^{4} +O(47^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 25 a^{2} + 19 a + 5 + \left(20 a^{2} + 33 a + 24\right)\cdot 47 + \left(3 a^{2} + 23 a + 21\right)\cdot 47^{2} + \left(10 a^{2} + 21 a + 8\right)\cdot 47^{3} + \left(44 a^{2} + 22 a + 17\right)\cdot 47^{4} +O(47^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 14 a^{2} + a + 23 + \left(25 a^{2} + 5 a + 25\right)\cdot 47 + \left(45 a^{2} + 9 a + 3\right)\cdot 47^{2} + \left(38 a^{2} + 27 a + 29\right)\cdot 47^{3} + \left(24 a^{2} + 8 a + 19\right)\cdot 47^{4} +O(47^{5})\)
|
| $r_{ 7 }$ | $=$ |
\( 13 a^{2} + 42 a + 28 + \left(29 a^{2} + 9 a + 41\right)\cdot 47 + \left(30 a^{2} + 44 a + 28\right)\cdot 47^{2} + \left(27 a^{2} + 35 a + 43\right)\cdot 47^{3} + \left(2 a^{2} + 20 a + 27\right)\cdot 47^{4} +O(47^{5})\)
|
| $r_{ 8 }$ | $=$ |
\( 21 a^{2} + 34 a + 30 + \left(5 a^{2} + 32 a + 21\right)\cdot 47 + \left(22 a^{2} + 7 a + 7\right)\cdot 47^{2} + \left(11 a^{2} + 37 a + 5\right)\cdot 47^{3} + \left(29 a^{2} + 9 a + 3\right)\cdot 47^{4} +O(47^{5})\)
|
| $r_{ 9 }$ | $=$ |
\( 37 a^{2} + 3 a + 15 + \left(28 a^{2} + 14 a + 21\right)\cdot 47 + \left(3 a^{2} + 17\right)\cdot 47^{2} + \left(8 a^{2} + 15 a + 45\right)\cdot 47^{3} + \left(34 a + 38\right)\cdot 47^{4} +O(47^{5})\)
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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 |
| $1$ | $1$ | $()$ | $2$ |
| $9$ | $2$ | $(1,3)(2,5)(4,8)(6,7)$ | $0$ |
| $2$ | $3$ | $(1,6,2)(3,5,7)(4,9,8)$ | $-1$ |
| $2$ | $9$ | $(1,5,8,6,7,4,2,3,9)$ | $-\zeta_{9}^{5} - \zeta_{9}^{2} + \zeta_{9}$ |
| $2$ | $9$ | $(1,8,7,2,9,5,6,4,3)$ | $-\zeta_{9}^{4} + \zeta_{9}^{2} - \zeta_{9}$ |
| $2$ | $9$ | $(1,7,9,6,3,8,2,5,4)$ | $\zeta_{9}^{5} + \zeta_{9}^{4}$ |
The blue line marks the conjugacy class containing complex conjugation.