Basic invariants
| Dimension: | $2$ |
| Group: | $D_{9}$ |
| Conductor: | \(1759\) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 9.1.9573337234561.1 |
| Galois orbit size: | $3$ |
| Smallest permutation container: | $D_{9}$ |
| Parity: | odd |
| Determinant: | 1.1759.2t1.a.a |
| Projective image: | $D_9$ |
| Projective stem field: | Galois closure of 9.1.9573337234561.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{9} - 2x^{8} + 2x^{7} + x^{6} + 33x^{5} + 33x^{4} + 77x^{3} + 72x^{2} + 99x + 27 \)
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The roots of $f$ are computed in an extension of $\Q_{ 31 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 31 }$:
\( x^{3} + x + 28 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 19 a^{2} + 6 a + 21 + \left(15 a^{2} + 2 a + 7\right)\cdot 31 + \left(28 a^{2} + a + 5\right)\cdot 31^{2} + \left(13 a^{2} + 9 a + 5\right)\cdot 31^{3} + \left(12 a^{2} + 11 a + 10\right)\cdot 31^{4} +O(31^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 11 a^{2} + 4 a + 14 + \left(3 a^{2} + 27 a + 15\right)\cdot 31 + \left(15 a^{2} + 10 a + 6\right)\cdot 31^{2} + \left(6 a^{2} + 23 a + 24\right)\cdot 31^{3} + \left(2 a^{2} + 14 a + 29\right)\cdot 31^{4} +O(31^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 7 a^{2} + 26 a + 1 + \left(4 a^{2} + 20 a + 16\right)\cdot 31 + \left(29 a^{2} + 27 a + 5\right)\cdot 31^{2} + \left(3 a^{2} + 12 a + 12\right)\cdot 31^{3} + 20 a^{2} 31^{4} +O(31^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 13 a^{2} + a + 5 + \left(23 a^{2} + 14 a + 8\right)\cdot 31 + \left(17 a^{2} + 23 a + 8\right)\cdot 31^{2} + \left(20 a^{2} + 25 a + 23\right)\cdot 31^{3} + \left(8 a^{2} + 15 a + 23\right)\cdot 31^{4} +O(31^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 4 a + 29 + \left(10 a^{2} + 5 a + 3\right)\cdot 31 + \left(27 a^{2} + a + 25\right)\cdot 31^{2} + \left(25 a^{2} + 5 a + 2\right)\cdot 31^{3} + \left(15 a^{2} + 14 a + 2\right)\cdot 31^{4} +O(31^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 12 a^{2} + 21 a + 6 + \left(5 a^{2} + 23 a + 11\right)\cdot 31 + \left(6 a^{2} + 28 a + 21\right)\cdot 31^{2} + \left(22 a^{2} + 16 a + 10\right)\cdot 31^{3} + \left(2 a^{2} + 5 a + 24\right)\cdot 31^{4} +O(31^{5})\)
|
| $r_{ 7 }$ | $=$ |
\( 10 a^{2} + 24 a + 13 + \left(7 a^{2} + 15 a + 25\right)\cdot 31 + \left(a^{2} + 23 a + 7\right)\cdot 31^{2} + \left(25 a^{2} + 9 a + 11\right)\cdot 31^{3} + \left(13 a^{2} + 27 a + 20\right)\cdot 31^{4} +O(31^{5})\)
|
| $r_{ 8 }$ | $=$ |
\( 25 a^{2} + 12 a + 23 + \left(4 a^{2} + 23 a + 23\right)\cdot 31 + \left(21 a^{2} + 14 a + 10\right)\cdot 31^{2} + \left(4 a^{2} + 23 a + 18\right)\cdot 31^{3} + \left(23 a^{2} + 10 a + 26\right)\cdot 31^{4} +O(31^{5})\)
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| $r_{ 9 }$ | $=$ |
\( 27 a^{2} + 26 a + 14 + \left(18 a^{2} + 22 a + 12\right)\cdot 31 + \left(8 a^{2} + 23 a + 2\right)\cdot 31^{2} + \left(a^{2} + 28 a + 16\right)\cdot 31^{3} + \left(25 a^{2} + 23 a + 17\right)\cdot 31^{4} +O(31^{5})\)
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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 |
| $1$ | $1$ | $()$ | $2$ |
| $9$ | $2$ | $(1,3)(2,6)(4,5)(7,8)$ | $0$ |
| $2$ | $3$ | $(1,6,5)(2,3,4)(7,9,8)$ | $-1$ |
| $2$ | $9$ | $(1,2,9,6,3,8,5,4,7)$ | $-\zeta_{9}^{4} + \zeta_{9}^{2} - \zeta_{9}$ |
| $2$ | $9$ | $(1,9,3,5,7,2,6,8,4)$ | $\zeta_{9}^{5} + \zeta_{9}^{4}$ |
| $2$ | $9$ | $(1,3,7,6,4,9,5,2,8)$ | $-\zeta_{9}^{5} - \zeta_{9}^{2} + \zeta_{9}$ |
The blue line marks the conjugacy class containing complex conjugation.