Basic invariants
| Dimension: | $2$ |
| Group: | $D_{9}$ |
| Conductor: | \(1607\) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin number field: | Galois closure of 9.1.6669042837601.1 |
| Galois orbit size: | $3$ |
| Smallest permutation container: | $D_{9}$ |
| Parity: | odd |
| Projective image: | $D_9$ |
| Projective field: | Galois closure of 9.1.6669042837601.1 |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 101 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 101 }$:
\( x^{3} + 3x + 99 \)
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Roots:
| $r_{ 1 }$ | $=$ |
\( 98 a^{2} + 6 a + 46 + \left(82 a^{2} + 63 a + 64\right)\cdot 101 + \left(16 a^{2} + 97 a\right)\cdot 101^{2} + \left(53 a^{2} + 66 a + 75\right)\cdot 101^{3} + \left(27 a^{2} + 59 a + 67\right)\cdot 101^{4} +O(101^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 71 a^{2} + 37 a + 93 + \left(7 a^{2} + 6 a + 14\right)\cdot 101 + \left(23 a^{2} + 4 a + 13\right)\cdot 101^{2} + \left(49 a^{2} + 62 a + 67\right)\cdot 101^{3} + \left(51 a^{2} + 89 a + 14\right)\cdot 101^{4} +O(101^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 99 a^{2} + 10 a + 14 + \left(13 a^{2} + 69 a + 34\right)\cdot 101 + \left(27 a^{2} + 9 a + 84\right)\cdot 101^{2} + \left(36 a^{2} + 74 a + 40\right)\cdot 101^{3} + \left(62 a^{2} + 39 a + 80\right)\cdot 101^{4} +O(101^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 98 a^{2} + 49 a + 60 + \left(27 a^{2} + 12 a + 16\right)\cdot 101 + \left(80 a^{2} + 53 a + 96\right)\cdot 101^{2} + \left(68 a^{2} + 30 a + 65\right)\cdot 101^{3} + \left(69 a^{2} + 33 a + 2\right)\cdot 101^{4} +O(101^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 43 a^{2} + 23 a + 3 + \left(48 a^{2} + 65 a + 2\right)\cdot 101 + \left(52 a^{2} + 48 a + 34\right)\cdot 101^{2} + \left(34 a^{2} + 96 a + 37\right)\cdot 101^{3} + \left(5 a^{2} + 80 a + 67\right)\cdot 101^{4} +O(101^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 32 a^{2} + 86 a + 29 + \left(61 a^{2} + 99 a + 83\right)\cdot 101 + \left(86 a^{2} + 71 a + 7\right)\cdot 101^{2} + \left(51 a^{2} + 56 a + 32\right)\cdot 101^{3} + \left(13 a^{2} + 40 a + 92\right)\cdot 101^{4} +O(101^{5})\)
|
| $r_{ 7 }$ | $=$ |
\( 72 a^{2} + 67 a + 8 + \left(11 a^{2} + 89 a + 85\right)\cdot 101 + \left(35 a^{2} + 76 a + 5\right)\cdot 101^{2} + \left(81 a^{2} + 13 a + 91\right)\cdot 101^{3} + \left(17 a^{2} + 27 a + 100\right)\cdot 101^{4} +O(101^{5})\)
|
| $r_{ 8 }$ | $=$ |
\( 33 a^{2} + 58 a + 17 + \left(10 a^{2} + 31 a + 20\right)\cdot 101 + \left(61 a^{2} + 100 a + 89\right)\cdot 101^{2} + \left(99 a^{2} + 72 a + 66\right)\cdot 101^{3} + \left(21 a^{2} + 52 a + 56\right)\cdot 101^{4} +O(101^{5})\)
|
| $r_{ 9 }$ | $=$ |
\( 60 a^{2} + 68 a + 37 + \left(38 a^{2} + 67 a + 83\right)\cdot 101 + \left(21 a^{2} + 42 a + 72\right)\cdot 101^{2} + \left(30 a^{2} + 31 a + 28\right)\cdot 101^{3} + \left(33 a^{2} + 81 a + 22\right)\cdot 101^{4} +O(101^{5})\)
|
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 | ||
| $c1$ | $c2$ | $c3$ | |||
| $1$ | $1$ | $()$ | $2$ | $2$ | $2$ |
| $9$ | $2$ | $(1,6)(2,7)(3,5)(4,8)$ | $0$ | $0$ | $0$ |
| $2$ | $3$ | $(1,8,2)(3,5,9)(4,6,7)$ | $-1$ | $-1$ | $-1$ |
| $2$ | $9$ | $(1,3,4,8,5,6,2,9,7)$ | $-\zeta_{9}^{4} + \zeta_{9}^{2} - \zeta_{9}$ | $\zeta_{9}^{5} + \zeta_{9}^{4}$ | $-\zeta_{9}^{5} - \zeta_{9}^{2} + \zeta_{9}$ |
| $2$ | $9$ | $(1,4,5,2,7,3,8,6,9)$ | $\zeta_{9}^{5} + \zeta_{9}^{4}$ | $-\zeta_{9}^{5} - \zeta_{9}^{2} + \zeta_{9}$ | $-\zeta_{9}^{4} + \zeta_{9}^{2} - \zeta_{9}$ |
| $2$ | $9$ | $(1,5,7,8,9,4,2,3,6)$ | $-\zeta_{9}^{5} - \zeta_{9}^{2} + \zeta_{9}$ | $-\zeta_{9}^{4} + \zeta_{9}^{2} - \zeta_{9}$ | $\zeta_{9}^{5} + \zeta_{9}^{4}$ |