Basic invariants
| Dimension: | $2$ |
| Group: | $D_{9}$ |
| Conductor: | \(2003\) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 9.1.16096216216081.1 |
| Galois orbit size: | $3$ |
| Smallest permutation container: | $D_{9}$ |
| Parity: | odd |
| Determinant: | 1.2003.2t1.a.a |
| Projective image: | $D_9$ |
| Projective stem field: | Galois closure of 9.1.16096216216081.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{9} - 3x^{8} - 2x^{7} + 24x^{6} - 40x^{5} - 88x^{4} + 259x^{3} + 75x^{2} - 318x + 148 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 19 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 19 }$:
\( x^{3} + 4x + 17 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 13 a^{2} + 12 a + 16 + \left(3 a^{2} + 9\right)\cdot 19 + \left(3 a^{2} + 4 a + 8\right)\cdot 19^{2} + \left(15 a^{2} + 11 a + 2\right)\cdot 19^{3} + \left(6 a^{2} + 4 a + 18\right)\cdot 19^{4} +O(19^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 16 a^{2} + 14 a + 5 + \left(6 a^{2} + 10 a + 18\right)\cdot 19 + \left(11 a^{2} + 12 a + 4\right)\cdot 19^{2} + \left(5 a + 14\right)\cdot 19^{3} + \left(13 a^{2} + 15 a + 15\right)\cdot 19^{4} +O(19^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 10 a^{2} + 10 a + 8 + \left(3 a^{2} + 12 a + 9\right)\cdot 19 + \left(15 a^{2} + 2 a + 2\right)\cdot 19^{2} + \left(5 a^{2} + 18 a + 9\right)\cdot 19^{3} + \left(3 a^{2} + 9 a + 2\right)\cdot 19^{4} +O(19^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 5 a^{2} + 16 a + 1 + \left(3 a^{2} + 17 a + 15\right)\cdot 19 + \left(12 a^{2} + 13 a\right)\cdot 19^{2} + \left(11 a^{2} + 17 a + 12\right)\cdot 19^{3} + \left(6 a^{2} + 17\right)\cdot 19^{4} +O(19^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( a^{2} + 10 a + 3 + \left(12 a^{2} + 13\right)\cdot 19 + \left(3 a^{2} + a + 9\right)\cdot 19^{2} + \left(11 a^{2} + 9 a + 4\right)\cdot 19^{3} + \left(5 a^{2} + 13 a + 2\right)\cdot 19^{4} +O(19^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 17 a^{2} + 3 a + 14 + \left(14 a^{2} + 17 a + 1\right)\cdot 19 + \left(5 a^{2} + 15 a + 9\right)\cdot 19^{2} + \left(13 a^{2} + 16\right)\cdot 19^{3} + \left(2 a^{2} + 14 a\right)\cdot 19^{4} +O(19^{5})\)
|
| $r_{ 7 }$ | $=$ |
\( 12 a^{2} + 10 a + 7 + \left(6 a^{2} + 2 a + 11\right)\cdot 19 + \left(4 a^{2} + 18 a + 11\right)\cdot 19^{2} + \left(6 a^{2} + 13 a + 16\right)\cdot 19^{3} +O(19^{5})\)
|
| $r_{ 8 }$ | $=$ |
\( 16 a^{2} + 18 a + 5 + \left(8 a^{2} + 3 a + 17\right)\cdot 19 + \left(18 a^{2} + 17 a + 4\right)\cdot 19^{2} + \left(6 a^{2} + 5 a + 12\right)\cdot 19^{3} + \left(15 a^{2} + 8 a + 15\right)\cdot 19^{4} +O(19^{5})\)
|
| $r_{ 9 }$ | $=$ |
\( 5 a^{2} + 2 a + 1 + \left(16 a^{2} + 10 a + 18\right)\cdot 19 + \left(a^{2} + 9 a + 4\right)\cdot 19^{2} + \left(5 a^{2} + 12 a + 7\right)\cdot 19^{3} + \left(3 a^{2} + 8 a + 2\right)\cdot 19^{4} +O(19^{5})\)
|
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 |
| $1$ | $1$ | $()$ | $2$ |
| $9$ | $2$ | $(1,5)(2,7)(3,6)(8,9)$ | $0$ |
| $2$ | $3$ | $(1,5,4)(2,9,6)(3,8,7)$ | $-1$ |
| $2$ | $9$ | $(1,8,9,5,7,6,4,3,2)$ | $-\zeta_{9}^{4} + \zeta_{9}^{2} - \zeta_{9}$ |
| $2$ | $9$ | $(1,9,7,4,2,8,5,6,3)$ | $\zeta_{9}^{5} + \zeta_{9}^{4}$ |
| $2$ | $9$ | $(1,7,2,5,3,9,4,8,6)$ | $-\zeta_{9}^{5} - \zeta_{9}^{2} + \zeta_{9}$ |
The blue line marks the conjugacy class containing complex conjugation.