Basic invariants
| Dimension: | $2$ |
| Group: | $D_{9}$ |
| Conductor: | \(3800\)\(\medspace = 2^{3} \cdot 5^{2} \cdot 19 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 9.1.8340544000000.1 |
| Galois orbit size: | $3$ |
| Smallest permutation container: | $D_{9}$ |
| Parity: | odd |
| Determinant: | 1.152.2t1.b.a |
| Projective image: | $D_9$ |
| Projective stem field: | Galois closure of 9.1.8340544000000.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{9} - 5x^{7} - 18x^{6} + 60x^{5} + 40x^{4} - 252x^{3} + 280x^{2} - 160x + 64 \)
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The roots of $f$ are computed in an extension of $\Q_{ 37 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 37 }$:
\( x^{3} + 6x + 35 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 21 a^{2} + 21 a + 24 + \left(23 a^{2} + 13 a + 3\right)\cdot 37 + \left(12 a^{2} + 18 a + 20\right)\cdot 37^{2} + \left(31 a^{2} + 26 a\right)\cdot 37^{3} + \left(a^{2} + 7 a + 25\right)\cdot 37^{4} +O(37^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 26 a^{2} + 24 a + 7 + \left(9 a^{2} + 28 a + 22\right)\cdot 37 + \left(14 a^{2} + 34 a + 26\right)\cdot 37^{2} + \left(25 a^{2} + 29 a + 13\right)\cdot 37^{3} + \left(34 a^{2} + 6 a + 8\right)\cdot 37^{4} +O(37^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 15 a^{2} + 34 a + 29 + \left(22 a^{2} + 6 a + 32\right)\cdot 37 + \left(31 a^{2} + a + 17\right)\cdot 37^{2} + \left(19 a^{2} + 7 a + 4\right)\cdot 37^{3} + \left(25 a^{2} + 13 a + 9\right)\cdot 37^{4} +O(37^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 26 a^{2} + 5 a + 10 + \left(36 a^{2} + 12 a + 35\right)\cdot 37 + \left(13 a^{2} + 16 a + 9\right)\cdot 37^{2} + \left(21 a^{2} + 30 a + 26\right)\cdot 37^{3} + \left(a^{2} + 2 a + 7\right)\cdot 37^{4} +O(37^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 27 a^{2} + 29 a + 11 + \left(3 a^{2} + 31 a + 35\right)\cdot 37 + \left(10 a^{2} + 20 a + 9\right)\cdot 37^{2} + \left(17 a^{2} + 17 a + 18\right)\cdot 37^{3} + \left(22 a + 19\right)\cdot 37^{4} +O(37^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 30 a^{2} + 16 a + 15 + \left(12 a^{2} + 34 a + 31\right)\cdot 37 + \left(35 a^{2} + 32\right)\cdot 37^{2} + \left(16 a^{2} + 35 a + 29\right)\cdot 37^{3} + \left(13 a^{2} + 27 a + 34\right)\cdot 37^{4} +O(37^{5})\)
|
| $r_{ 7 }$ | $=$ |
\( 6 a^{2} + 23 a + 4 + \left(35 a^{2} + 10 a + 29\right)\cdot 37 + \left(23 a^{2} + 18 a + 12\right)\cdot 37^{2} + \left(28 a^{2} + 6 a + 18\right)\cdot 37^{3} + \left(20 a^{2} + 16 a + 10\right)\cdot 37^{4} +O(37^{5})\)
|
| $r_{ 8 }$ | $=$ |
\( 5 a^{2} + 9 a + \left(2 a^{2} + 14 a + 8\right)\cdot 37 + \left(36 a^{2} + 2 a + 24\right)\cdot 37^{2} + \left(23 a^{2} + 36\right)\cdot 37^{3} + \left(14 a^{2} + 18 a + 22\right)\cdot 37^{4} +O(37^{5})\)
|
| $r_{ 9 }$ | $=$ |
\( 29 a^{2} + 24 a + 11 + \left(a^{2} + 32 a + 24\right)\cdot 37 + \left(7 a^{2} + 34 a + 30\right)\cdot 37^{2} + \left(31 a + 36\right)\cdot 37^{3} + \left(35 a^{2} + 32 a + 9\right)\cdot 37^{4} +O(37^{5})\)
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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 |
| $1$ | $1$ | $()$ | $2$ |
| $9$ | $2$ | $(1,2)(3,8)(4,6)(7,9)$ | $0$ |
| $2$ | $3$ | $(1,5,2)(3,9,6)(4,7,8)$ | $-1$ |
| $2$ | $9$ | $(1,4,9,5,7,6,2,8,3)$ | $-\zeta_{9}^{4} + \zeta_{9}^{2} - \zeta_{9}$ |
| $2$ | $9$ | $(1,9,7,2,3,4,5,6,8)$ | $\zeta_{9}^{5} + \zeta_{9}^{4}$ |
| $2$ | $9$ | $(1,7,3,5,8,9,2,4,6)$ | $-\zeta_{9}^{5} - \zeta_{9}^{2} + \zeta_{9}$ |
The blue line marks the conjugacy class containing complex conjugation.