Basic invariants
| Dimension: | $2$ |
| Group: | $D_{9}$ |
| Conductor: | \(332\)\(\medspace = 2^{2} \cdot 83 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin number field: | Galois closure of 9.1.3037332544.1 |
| Galois orbit size: | $3$ |
| Smallest permutation container: | $D_{9}$ |
| Parity: | odd |
| Projective image: | $D_9$ |
| Projective field: | Galois closure of 9.1.3037332544.1 |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 23 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 23 }$:
\( x^{3} + 2x + 18 \)
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Roots:
| $r_{ 1 }$ | $=$ |
\( 20 a^{2} + 12 a + 7 + \left(6 a^{2} + 6 a + 4\right)\cdot 23 + \left(21 a^{2} + 20 a + 21\right)\cdot 23^{2} + \left(16 a^{2} + 4 a + 9\right)\cdot 23^{3} + \left(9 a^{2} + 7 a + 11\right)\cdot 23^{4} +O(23^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 4 a^{2} + 15 a + 12 + \left(17 a + 3\right)\cdot 23 + \left(13 a + 1\right)\cdot 23^{2} + \left(11 a^{2} + 6 a + 15\right)\cdot 23^{3} + \left(a^{2} + 10 a + 12\right)\cdot 23^{4} +O(23^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 17 a^{2} + 20 a + 13 + \left(12 a^{2} + 17 a + 18\right)\cdot 23 + \left(9 a^{2} + 12 a + 18\right)\cdot 23^{2} + \left(20 a^{2} + 15 a + 8\right)\cdot 23^{3} + \left(3 a^{2} + 17 a + 11\right)\cdot 23^{4} +O(23^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 21 a^{2} + 10 a + 4 + \left(12 a^{2} + a + 5\right)\cdot 23 + \left(18 a^{2} + 4 a + 18\right)\cdot 23^{2} + \left(10 a^{2} + 3 a + 14\right)\cdot 23^{3} + \left(a^{2} + 9 a + 12\right)\cdot 23^{4} +O(23^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 10 a^{2} + 7 a + 9 + \left(22 a^{2} + 10 a + 17\right)\cdot 23 + \left(22 a^{2} + 15 a + 15\right)\cdot 23^{2} + \left(18 a^{2} + 7 a + 12\right)\cdot 23^{3} + \left(14 a^{2} + 18 a + 10\right)\cdot 23^{4} +O(23^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 21 a^{2} + 21 a + 4 + \left(9 a^{2} + 3 a + 1\right)\cdot 23 + \left(4 a^{2} + 5 a + 7\right)\cdot 23^{2} + \left(a^{2} + 13 a + 17\right)\cdot 23^{3} + \left(20 a^{2} + 3 a + 6\right)\cdot 23^{4} +O(23^{5})\)
|
| $r_{ 7 }$ | $=$ |
\( 16 a^{2} + 4 a + 17 + \left(16 a^{2} + 6 a + 9\right)\cdot 23 + \left(a^{2} + 10 a + 10\right)\cdot 23^{2} + \left(10 a^{2} + 10 a + 8\right)\cdot 23^{3} + \left(21 a^{2} + 20 a + 19\right)\cdot 23^{4} +O(23^{5})\)
|
| $r_{ 8 }$ | $=$ |
\( 22 a^{2} + 12 a + 12 + \left(8 a^{2} + 19 a + 13\right)\cdot 23 + \left(13 a^{2} + 4 a + 8\right)\cdot 23^{2} + \left(8 a^{2} + 14 a + 8\right)\cdot 23^{3} + \left(13 a^{2} + 20 a + 16\right)\cdot 23^{4} +O(23^{5})\)
|
| $r_{ 9 }$ | $=$ |
\( 7 a^{2} + 14 a + 15 + \left(a^{2} + 8 a + 18\right)\cdot 23 + \left(5 a + 13\right)\cdot 23^{2} + \left(17 a^{2} + 16 a + 19\right)\cdot 23^{3} + \left(5 a^{2} + 7 a + 13\right)\cdot 23^{4} +O(23^{5})\)
|
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 | ||
| $c1$ | $c2$ | $c3$ | |||
| $1$ | $1$ | $()$ | $2$ | $2$ | $2$ |
| $9$ | $2$ | $(1,9)(2,4)(3,7)(5,8)$ | $0$ | $0$ | $0$ |
| $2$ | $3$ | $(1,7,5)(2,6,4)(3,9,8)$ | $-1$ | $-1$ | $-1$ |
| $2$ | $9$ | $(1,2,8,7,6,3,5,4,9)$ | $-\zeta_{9}^{4} + \zeta_{9}^{2} - \zeta_{9}$ | $\zeta_{9}^{5} + \zeta_{9}^{4}$ | $-\zeta_{9}^{5} - \zeta_{9}^{2} + \zeta_{9}$ |
| $2$ | $9$ | $(1,8,6,5,9,2,7,3,4)$ | $\zeta_{9}^{5} + \zeta_{9}^{4}$ | $-\zeta_{9}^{5} - \zeta_{9}^{2} + \zeta_{9}$ | $-\zeta_{9}^{4} + \zeta_{9}^{2} - \zeta_{9}$ |
| $2$ | $9$ | $(1,6,9,7,4,8,5,2,3)$ | $-\zeta_{9}^{5} - \zeta_{9}^{2} + \zeta_{9}$ | $-\zeta_{9}^{4} + \zeta_{9}^{2} - \zeta_{9}$ | $\zeta_{9}^{5} + \zeta_{9}^{4}$ |