Basic invariants
| Dimension: | $2$ |
| Group: | $D_{9}$ |
| Conductor: | \(1879\) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 9.1.12465425870881.1 |
| Galois orbit size: | $3$ |
| Smallest permutation container: | $D_{9}$ |
| Parity: | odd |
| Determinant: | 1.1879.2t1.a.a |
| Projective image: | $D_9$ |
| Projective stem field: | Galois closure of 9.1.12465425870881.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{9} - 4x^{8} + 10x^{7} - 23x^{6} + 37x^{5} - 97x^{4} + 165x^{3} - 278x^{2} + 239x - 143 \)
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The roots of $f$ are computed in an extension of $\Q_{ 61 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 61 }$:
\( x^{3} + 7x + 59 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 48 a^{2} + 34 a + 57 + \left(13 a^{2} + 2 a + 47\right)\cdot 61 + \left(4 a^{2} + 59 a + 37\right)\cdot 61^{2} + \left(33 a^{2} + 47 a + 42\right)\cdot 61^{3} + \left(56 a^{2} + 16 a + 30\right)\cdot 61^{4} +O(61^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 25 a^{2} + 44 a + 9 + \left(3 a^{2} + 2 a + 34\right)\cdot 61 + \left(22 a^{2} + 19 a + 23\right)\cdot 61^{2} + \left(42 a^{2} + 8 a + 51\right)\cdot 61^{3} + \left(34 a^{2} + 43 a + 10\right)\cdot 61^{4} +O(61^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 20 a^{2} + 52 a + 44 + \left(10 a^{2} + 24 a + 5\right)\cdot 61 + \left(12 a^{2} + 32 a + 57\right)\cdot 61^{2} + \left(35 a^{2} + 25 a + 35\right)\cdot 61^{3} + \left(15 a^{2} + 8 a + 9\right)\cdot 61^{4} +O(61^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 24 a^{2} + 24 a + 6 + \left(56 a^{2} + 42 a + 23\right)\cdot 61 + \left(18 a^{2} + 20 a + 45\right)\cdot 61^{2} + \left(24 a^{2} + 13 a + 1\right)\cdot 61^{3} + \left(38 a^{2} + 21 a + 7\right)\cdot 61^{4} +O(61^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 34 a^{2} + 59 a + 51 + \left(49 a^{2} + 22 a + 25\right)\cdot 61 + \left(11 a + 46\right)\cdot 61^{2} + \left(5 a^{2} + 21 a + 39\right)\cdot 61^{3} + \left(59 a^{2} + 32 a + 2\right)\cdot 61^{4} +O(61^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 44 a^{2} + 18 a + 34 + \left(59 a^{2} + 34 a + 12\right)\cdot 61 + \left(55 a^{2} + 16 a + 58\right)\cdot 61^{2} + \left(56 a^{2} + 12 a + 55\right)\cdot 61^{3} + \left(38 a^{2} + 6 a + 16\right)\cdot 61^{4} +O(61^{5})\)
|
| $r_{ 7 }$ | $=$ |
\( 2 a^{2} + 19 a + 44 + \left(8 a^{2} + 35 a + 55\right)\cdot 61 + \left(38 a^{2} + 30 a + 57\right)\cdot 61^{2} + \left(13 a^{2} + 31 a + 18\right)\cdot 61^{3} + \left(28 a^{2} + 46 a + 21\right)\cdot 61^{4} +O(61^{5})\)
|
| $r_{ 8 }$ | $=$ |
\( 50 a^{2} + 3 a + 46 + \left(51 a^{2} + 16 a + 1\right)\cdot 61 + \left(37 a^{2} + 42 a + 32\right)\cdot 61^{2} + \left(3 a^{2} + 60 a + 47\right)\cdot 61^{3} + \left(27 a^{2} + 22 a + 55\right)\cdot 61^{4} +O(61^{5})\)
|
| $r_{ 9 }$ | $=$ |
\( 58 a^{2} + 52 a + 18 + \left(51 a^{2} + a + 37\right)\cdot 61 + \left(53 a^{2} + 12 a + 7\right)\cdot 61^{2} + \left(29 a^{2} + 23 a + 11\right)\cdot 61^{3} + \left(6 a^{2} + 46 a + 28\right)\cdot 61^{4} +O(61^{5})\)
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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 |
| $1$ | $1$ | $()$ | $2$ |
| $9$ | $2$ | $(1,2)(4,5)(6,9)(7,8)$ | $0$ |
| $2$ | $3$ | $(1,8,4)(2,5,7)(3,9,6)$ | $-1$ |
| $2$ | $9$ | $(1,9,7,8,6,2,4,3,5)$ | $\zeta_{9}^{5} + \zeta_{9}^{4}$ |
| $2$ | $9$ | $(1,7,6,4,5,9,8,2,3)$ | $-\zeta_{9}^{5} - \zeta_{9}^{2} + \zeta_{9}$ |
| $2$ | $9$ | $(1,6,5,8,3,7,4,9,2)$ | $-\zeta_{9}^{4} + \zeta_{9}^{2} - \zeta_{9}$ |
The blue line marks the conjugacy class containing complex conjugation.