Basic invariants
| Dimension: | $2$ |
| Group: | $D_{9}$ |
| Conductor: | \(419\) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 9.1.30821664721.1 |
| Galois orbit size: | $3$ |
| Smallest permutation container: | $D_{9}$ |
| Parity: | odd |
| Determinant: | 1.419.2t1.a.a |
| Projective image: | $D_9$ |
| Projective stem field: | Galois closure of 9.1.30821664721.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{9} - 2x^{7} - 4x^{6} + 4x^{5} + 10x^{4} - x^{3} - 8x^{2} + 8x + 8 \)
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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:
| $r_{ 1 }$ | $=$ |
\( 29 a^{2} + 4 a + 38 + \left(3 a^{2} + 38 a + 25\right)\cdot 43 + \left(3 a^{2} + 25 a + 10\right)\cdot 43^{2} + \left(19 a^{2} + 37 a\right)\cdot 43^{3} + \left(31 a^{2} + 22 a + 29\right)\cdot 43^{4} +O(43^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 29 a^{2} + 24 a + 18 + \left(7 a^{2} + 40 a + 5\right)\cdot 43 + \left(19 a^{2} + 33 a + 2\right)\cdot 43^{2} + \left(27 a^{2} + 10 a + 2\right)\cdot 43^{3} + \left(42 a^{2} + 32 a + 13\right)\cdot 43^{4} +O(43^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 19 a^{2} + 8 a + 24 + \left(11 a^{2} + 12\right)\cdot 43 + \left(10 a^{2} + 40 a + 23\right)\cdot 43^{2} + \left(24 a^{2} + 5 a + 30\right)\cdot 43^{3} + \left(22 a^{2} + a + 36\right)\cdot 43^{4} +O(43^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 4 a^{2} + 14 + \left(8 a^{2} + 41 a + 10\right)\cdot 43 + \left(26 a^{2} + 21 a + 5\right)\cdot 43^{2} + \left(30 a^{2} + 8 a + 6\right)\cdot 43^{3} + \left(34 a^{2} + 34 a + 16\right)\cdot 43^{4} +O(43^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( a^{2} + 13 a + 28 + \left(26 a^{2} + 23 a + 17\right)\cdot 43 + \left(19 a^{2} + 5 a + 2\right)\cdot 43^{2} + \left(23 a^{2} + 22 a + 28\right)\cdot 43^{3} + \left(26 a^{2} + 16\right)\cdot 43^{4} +O(43^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 22 a^{2} + 35 a + 19 + \left(22 a^{2} + 17 a + 38\right)\cdot 43 + \left(31 a^{2} + 31 a\right)\cdot 43^{2} + \left(6 a^{2} + 22 a + 35\right)\cdot 43^{3} + \left(27 a^{2} + 17 a + 11\right)\cdot 43^{4} +O(43^{5})\)
|
| $r_{ 7 }$ | $=$ |
\( 35 a^{2} + 4 a + 42 + \left(16 a^{2} + 30 a + 5\right)\cdot 43 + \left(8 a^{2} + 28 a + 14\right)\cdot 43^{2} + \left(17 a^{2} + 25 a + 13\right)\cdot 43^{3} + \left(27 a^{2} + 2 a + 26\right)\cdot 43^{4} +O(43^{5})\)
|
| $r_{ 8 }$ | $=$ |
\( 13 a^{2} + 6 a + 36 + \left(9 a^{2} + 22 a + 20\right)\cdot 43 + \left(4 a^{2} + 3 a + 6\right)\cdot 43^{2} + \left(35 a^{2} + 10 a + 7\right)\cdot 43^{3} + \left(16 a^{2} + 10 a + 10\right)\cdot 43^{4} +O(43^{5})\)
|
| $r_{ 9 }$ | $=$ |
\( 20 a^{2} + 35 a + 39 + \left(23 a^{2} + a + 34\right)\cdot 43 + \left(6 a^{2} + 24 a + 20\right)\cdot 43^{2} + \left(31 a^{2} + 28 a + 6\right)\cdot 43^{3} + \left(28 a^{2} + 7 a + 12\right)\cdot 43^{4} +O(43^{5})\)
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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 |
| $1$ | $1$ | $()$ | $2$ |
| $9$ | $2$ | $(1,2)(3,4)(5,6)(7,8)$ | $0$ |
| $2$ | $3$ | $(1,6,7)(2,8,5)(3,9,4)$ | $-1$ |
| $2$ | $9$ | $(1,9,2,6,4,8,7,3,5)$ | $-\zeta_{9}^{4} + \zeta_{9}^{2} - \zeta_{9}$ |
| $2$ | $9$ | $(1,2,4,7,5,9,6,8,3)$ | $\zeta_{9}^{5} + \zeta_{9}^{4}$ |
| $2$ | $9$ | $(1,4,5,6,3,2,7,9,8)$ | $-\zeta_{9}^{5} - \zeta_{9}^{2} + \zeta_{9}$ |
The blue line marks the conjugacy class containing complex conjugation.