Basic invariants
| Dimension: | $2$ |
| Group: | $D_{9}$ |
| Conductor: | \(2759\)\(\medspace = 31 \cdot 89 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 9.1.57943777150561.1 |
| Galois orbit size: | $3$ |
| Smallest permutation container: | $D_{9}$ |
| Parity: | odd |
| Determinant: | 1.2759.2t1.a.a |
| Projective image: | $D_9$ |
| Projective stem field: | Galois closure of 9.1.57943777150561.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{9} - 3x^{8} + 8x^{7} - 32x^{6} + 59x^{5} - 86x^{4} + 158x^{3} - 102x^{2} - 21x + 67 \)
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The roots of $f$ are computed in an extension of $\Q_{ 97 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 97 }$:
\( x^{3} + 9x + 92 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 61 a^{2} + 84 a + 43 + \left(45 a^{2} + 87 a + 47\right)\cdot 97 + \left(14 a^{2} + 80 a + 54\right)\cdot 97^{2} + \left(7 a^{2} + 85 a + 10\right)\cdot 97^{3} + \left(78 a^{2} + 64 a + 48\right)\cdot 97^{4} +O(97^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 29 a^{2} + 73 a + 45 + \left(25 a^{2} + 47 a + 22\right)\cdot 97 + \left(24 a^{2} + 74 a + 16\right)\cdot 97^{2} + \left(33 a^{2} + 42 a + 70\right)\cdot 97^{3} + \left(23 a^{2} + 27 a + 10\right)\cdot 97^{4} +O(97^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 25 a^{2} + 33 a + 21 + \left(78 a^{2} + 51 a + 49\right)\cdot 97 + \left(78 a^{2} + 12 a + 52\right)\cdot 97^{2} + \left(23 a^{2} + 41 a + 13\right)\cdot 97^{3} + \left(25 a^{2} + 84 a + 22\right)\cdot 97^{4} +O(97^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 6 a^{2} + 10 a + 4 + \left(5 a^{2} + 34 a + 95\right)\cdot 97 + \left(50 a^{2} + 91 a + 73\right)\cdot 97^{2} + \left(82 a^{2} + 58 a + 74\right)\cdot 97^{3} + \left(56 a^{2} + 41 a + 17\right)\cdot 97^{4} +O(97^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 45 a^{2} + 46 a + 44 + \left(44 a^{2} + 32 a + 40\right)\cdot 97 + \left(10 a^{2} + 9 a + 30\right)\cdot 97^{2} + \left(39 a^{2} + 23 a + 8\right)\cdot 97^{3} + \left(37 a^{2} + 5 a + 95\right)\cdot 97^{4} +O(97^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 88 a^{2} + 9 a + 11 + \left(24 a^{2} + 17 a + 20\right)\cdot 97 + \left(25 a^{2} + 50 a + 22\right)\cdot 97^{2} + \left(35 a^{2} + 89 a + 82\right)\cdot 97^{3} + \left(48 a^{2} + 68 a + 63\right)\cdot 97^{4} +O(97^{5})\)
|
| $r_{ 7 }$ | $=$ |
\( 62 a^{2} + 14 a + 49 + \left(66 a^{2} + 15 a + 76\right)\cdot 97 + \left(22 a^{2} + 28 a + 6\right)\cdot 97^{2} + \left(78 a^{2} + 92 a + 49\right)\cdot 97^{3} + \left(16 a^{2} + 27 a + 68\right)\cdot 97^{4} +O(97^{5})\)
|
| $r_{ 8 }$ | $=$ |
\( 81 a^{2} + 55 a + 66 + \left(90 a^{2} + 28 a + 27\right)\cdot 97 + \left(89 a^{2} + 34 a + 22\right)\cdot 97^{2} + \left(37 a^{2} + 63 a + 1\right)\cdot 97^{3} + \left(23 a^{2} + 40 a + 11\right)\cdot 97^{4} +O(97^{5})\)
|
| $r_{ 9 }$ | $=$ |
\( 88 a^{2} + 64 a + 11 + \left(6 a^{2} + 73 a + 9\right)\cdot 97 + \left(72 a^{2} + 6 a + 12\right)\cdot 97^{2} + \left(50 a^{2} + 85 a + 78\right)\cdot 97^{3} + \left(78 a^{2} + 26 a + 50\right)\cdot 97^{4} +O(97^{5})\)
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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 |
| $1$ | $1$ | $()$ | $2$ |
| $9$ | $2$ | $(1,6)(2,4)(3,9)(5,8)$ | $0$ |
| $2$ | $3$ | $(1,5,9)(2,4,7)(3,8,6)$ | $-1$ |
| $2$ | $9$ | $(1,8,7,5,6,2,9,3,4)$ | $-\zeta_{9}^{4} + \zeta_{9}^{2} - \zeta_{9}$ |
| $2$ | $9$ | $(1,7,6,9,4,8,5,2,3)$ | $\zeta_{9}^{5} + \zeta_{9}^{4}$ |
| $2$ | $9$ | $(1,6,4,5,3,7,9,8,2)$ | $-\zeta_{9}^{5} - \zeta_{9}^{2} + \zeta_{9}$ |
The blue line marks the conjugacy class containing complex conjugation.