Basic invariants
| Dimension: | $2$ |
| Group: | $D_{9}$ |
| Conductor: | \(1999\) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin number field: | Galois closure of 9.1.15968023992001.1 |
| Galois orbit size: | $3$ |
| Smallest permutation container: | $D_{9}$ |
| Parity: | odd |
| Projective image: | $D_9$ |
| Projective field: | Galois closure of 9.1.15968023992001.1 |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 41 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 41 }$:
\( x^{3} + x + 35 \)
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Roots:
| $r_{ 1 }$ | $=$ |
\( 3 a^{2} + 14 a + 22 + \left(24 a^{2} + 31 a + 37\right)\cdot 41 + \left(3 a^{2} + 36 a + 28\right)\cdot 41^{2} + \left(23 a + 18\right)\cdot 41^{3} + \left(9 a^{2} + 12 a + 2\right)\cdot 41^{4} +O(41^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 26 a^{2} + 32 a + 37 + \left(3 a^{2} + 40 a + 15\right)\cdot 41 + \left(11 a^{2} + 14 a + 8\right)\cdot 41^{2} + \left(26 a^{2} + 32 a\right)\cdot 41^{3} + \left(20 a^{2} + 22 a + 15\right)\cdot 41^{4} +O(41^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 15 a^{2} + 15 a + 16 + \left(24 a^{2} + 34 a + 2\right)\cdot 41 + \left(12 a^{2} + 30 a + 23\right)\cdot 41^{2} + \left(2 a^{2} + 37 a + 11\right)\cdot 41^{3} + \left(15 a^{2} + 13 a + 11\right)\cdot 41^{4} +O(41^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 35 a + 6 + \left(13 a^{2} + 6 a + 22\right)\cdot 41 + \left(17 a^{2} + 36 a + 12\right)\cdot 41^{2} + \left(12 a^{2} + 11 a + 18\right)\cdot 41^{3} + \left(5 a^{2} + 4 a + 18\right)\cdot 41^{4} +O(41^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 22 a^{2} + 2 a + 17 + \left(34 a^{2} + 26 a + 15\right)\cdot 41 + \left(2 a^{2} + 4 a + 15\right)\cdot 41^{2} + \left(19 a^{2} + 24 a + 11\right)\cdot 41^{3} + \left(31 a^{2} + 12 a + 23\right)\cdot 41^{4} +O(41^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 32 a^{2} + 35 a + 10 + \left(21 a^{2} + 19 a + 34\right)\cdot 41 + \left(32 a^{2} + 9 a + 7\right)\cdot 41^{2} + \left(10 a^{2} + 16 a + 33\right)\cdot 41^{3} + \left(39 a^{2} + 20 a + 14\right)\cdot 41^{4} +O(41^{5})\)
|
| $r_{ 7 }$ | $=$ |
\( 28 a^{2} + 4 a + 21 + \left(25 a^{2} + 36 a + 9\right)\cdot 41 + \left(5 a^{2} + 26 a + 17\right)\cdot 41^{2} + \left(11 a^{2} + 33\right)\cdot 41^{3} + \left(11 a^{2} + 8 a + 9\right)\cdot 41^{4} +O(41^{5})\)
|
| $r_{ 8 }$ | $=$ |
\( 7 a^{2} + 3 a + 11 + \left(4 a^{2} + 2 a + 24\right)\cdot 41 + \left(23 a^{2} + 30 a + 14\right)\cdot 41^{2} + \left(20 a^{2} + 34 a + 32\right)\cdot 41^{3} + \left(37 a^{2} + 18 a + 7\right)\cdot 41^{4} +O(41^{5})\)
|
| $r_{ 9 }$ | $=$ |
\( 31 a^{2} + 24 a + 27 + \left(12 a^{2} + 7 a + 2\right)\cdot 41 + \left(14 a^{2} + 15 a + 36\right)\cdot 41^{2} + \left(20 a^{2} + 23 a + 4\right)\cdot 41^{3} + \left(35 a^{2} + 9 a + 20\right)\cdot 41^{4} +O(41^{5})\)
|
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 values | ||
| $c1$ | $c2$ | $c3$ | |||
| $1$ | $1$ | $()$ | $2$ | $2$ | $2$ |
| $9$ | $2$ | $(1,6)(2,3)(5,8)(7,9)$ | $0$ | $0$ | $0$ |
| $2$ | $3$ | $(1,9,8)(2,4,3)(5,7,6)$ | $-1$ | $-1$ | $-1$ |
| $2$ | $9$ | $(1,5,3,9,7,2,8,6,4)$ | $-\zeta_{9}^{4} + \zeta_{9}^{2} - \zeta_{9}$ | $\zeta_{9}^{5} + \zeta_{9}^{4}$ | $-\zeta_{9}^{5} - \zeta_{9}^{2} + \zeta_{9}$ |
| $2$ | $9$ | $(1,3,7,8,4,5,9,2,6)$ | $\zeta_{9}^{5} + \zeta_{9}^{4}$ | $-\zeta_{9}^{5} - \zeta_{9}^{2} + \zeta_{9}$ | $-\zeta_{9}^{4} + \zeta_{9}^{2} - \zeta_{9}$ |
| $2$ | $9$ | $(1,7,4,9,6,3,8,5,2)$ | $-\zeta_{9}^{5} - \zeta_{9}^{2} + \zeta_{9}$ | $-\zeta_{9}^{4} + \zeta_{9}^{2} - \zeta_{9}$ | $\zeta_{9}^{5} + \zeta_{9}^{4}$ |