Basic invariants
| Dimension: | $2$ |
| Group: | $D_{9}$ |
| Conductor: | \(3671\) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 9.1.181609071490081.1 |
| Galois orbit size: | $3$ |
| Smallest permutation container: | $D_{9}$ |
| Parity: | odd |
| Determinant: | 1.3671.2t1.a.a |
| Projective image: | $D_9$ |
| Projective stem field: | Galois closure of 9.1.181609071490081.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{9} - 3x^{8} + 5x^{7} - 16x^{6} + 23x^{5} - 63x^{4} + 54x^{3} - 63x^{2} + 102x - 113 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 11 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 11 }$:
\( x^{3} + 2x + 9 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 7 a^{2} + 2 a + 6 + \left(10 a^{2} + 5 a + 10\right)\cdot 11 + \left(7 a^{2} + 6 a + 6\right)\cdot 11^{2} + \left(a^{2} + 10 a + 9\right)\cdot 11^{3} + \left(a + 3\right)\cdot 11^{4} +O(11^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 2 a^{2} + 3 a + 3 + \left(8 a^{2} + 3 a + 7\right)\cdot 11 + \left(9 a^{2} + 6 a + 5\right)\cdot 11^{2} + \left(2 a^{2} + 7\right)\cdot 11^{3} + \left(9 a^{2} + 9 a + 8\right)\cdot 11^{4} +O(11^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 5 a^{2} + 6 a + 7 + \left(6 a^{2} + 10 a + 8\right)\cdot 11 + 4 a\cdot 11^{2} + \left(5 a^{2} + 10 a + 3\right)\cdot 11^{3} + \left(7 a^{2} + 10 a + 6\right)\cdot 11^{4} +O(11^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 9 a^{2} + 7 a + 5 + \left(4 a^{2} + 3 a + 6\right)\cdot 11 + \left(3 a^{2} + 4 a + 4\right)\cdot 11^{2} + \left(3 a^{2} + 2 a + 4\right)\cdot 11^{3} + \left(5 a^{2} + 6 a + 3\right)\cdot 11^{4} +O(11^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 10 a^{2} + 10 + \left(8 a^{2} + 3 a\right)\cdot 11 + \left(8 a^{2} + 9 a + 8\right)\cdot 11^{2} + \left(10 a^{2} + 10\right)\cdot 11^{3} + \left(3 a^{2} + 7 a + 8\right)\cdot 11^{4} +O(11^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 2 a + 4 + \left(6 a^{2} + 6 a + 4\right)\cdot 11 + \left(4 a^{2} + 9 a + 2\right)\cdot 11^{2} + \left(7 a^{2} + 8 a + 6\right)\cdot 11^{3} + \left(5 a^{2} + a + 7\right)\cdot 11^{4} +O(11^{5})\)
|
| $r_{ 7 }$ | $=$ |
\( 10 a^{2} + 3 a + 10 + \left(4 a^{2} + 6 a + 2\right)\cdot 11 + \left(2 a^{2} + 10 a + 3\right)\cdot 11^{2} + \left(4 a^{2} + 9\right)\cdot 11^{3} + \left(3 a^{2} + 9 a\right)\cdot 11^{4} +O(11^{5})\)
|
| $r_{ 8 }$ | $=$ |
\( 2 a^{2} + 2 a + 3 + a\cdot 11 + \left(3 a^{2} + 8 a + 4\right)\cdot 11^{2} + 10 a\cdot 11^{3} + 2 a\cdot 11^{4} +O(11^{5})\)
|
| $r_{ 9 }$ | $=$ |
\( 10 a^{2} + 8 a + 10 + \left(4 a^{2} + 4 a + 2\right)\cdot 11 + \left(3 a^{2} + 6 a + 8\right)\cdot 11^{2} + \left(8 a^{2} + 9 a + 3\right)\cdot 11^{3} + \left(8 a^{2} + 5 a + 4\right)\cdot 11^{4} +O(11^{5})\)
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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 |
| $1$ | $1$ | $()$ | $2$ |
| $9$ | $2$ | $(1,3)(2,8)(4,5)(6,9)$ | $0$ |
| $2$ | $3$ | $(1,3,7)(2,5,9)(4,8,6)$ | $-1$ |
| $2$ | $9$ | $(1,2,8,3,5,6,7,9,4)$ | $-\zeta_{9}^{5} - \zeta_{9}^{2} + \zeta_{9}$ |
| $2$ | $9$ | $(1,8,5,7,4,2,3,6,9)$ | $-\zeta_{9}^{4} + \zeta_{9}^{2} - \zeta_{9}$ |
| $2$ | $9$ | $(1,5,4,3,9,8,7,2,6)$ | $\zeta_{9}^{5} + \zeta_{9}^{4}$ |
The blue line marks the conjugacy class containing complex conjugation.