Basic invariants
| Dimension: | $1$ |
| Group: | $C_9$ |
| Conductor: | \(109\) |
| Artin field: | Galois closure of 9.9.19925626416901921.1 |
| Galois orbit size: | $6$ |
| Smallest permutation container: | $C_9$ |
| Parity: | even |
| Dirichlet character: | \(\chi_{109}(105,\cdot)\) |
| Projective image: | $C_1$ |
| Projective field: | Galois closure of \(\Q\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{9} - x^{8} - 48x^{7} + 73x^{6} + 660x^{5} - 1454x^{4} - 2149x^{3} + 8350x^{2} - 7432x + 2008 \)
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The roots of $f$ are computed in an extension of $\Q_{ 19 }$ to precision 6.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 19 }$:
\( x^{3} + 4x + 17 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 15 a^{2} + 11 a + 8 + \left(2 a^{2} + 2 a + 6\right)\cdot 19 + \left(17 a^{2} + 5 a + 7\right)\cdot 19^{2} + \left(4 a^{2} + 5 a + 16\right)\cdot 19^{3} + \left(a^{2} + 9\right)\cdot 19^{4} + \left(2 a^{2} + 18 a + 10\right)\cdot 19^{5} +O(19^{6})\)
|
| $r_{ 2 }$ | $=$ |
\( 17 a^{2} + 7 + \left(7 a^{2} + 17 a + 7\right)\cdot 19 + \left(5 a^{2} + 12 a + 1\right)\cdot 19^{2} + \left(6 a^{2} + a + 1\right)\cdot 19^{3} + \left(7 a^{2} + 14 a + 7\right)\cdot 19^{4} + \left(7 a^{2} + 5 a + 18\right)\cdot 19^{5} +O(19^{6})\)
|
| $r_{ 3 }$ | $=$ |
\( 5 a^{2} + 15 a + 15 + \left(16 a^{2} + 2 a + 1\right)\cdot 19 + \left(8 a^{2} + 6 a + 1\right)\cdot 19^{2} + \left(13 a^{2} + a + 18\right)\cdot 19^{3} + \left(10 a^{2} + 12 a + 13\right)\cdot 19^{4} + \left(13 a^{2} + 18 a + 13\right)\cdot 19^{5} +O(19^{6})\)
|
| $r_{ 4 }$ | $=$ |
\( a + 18 + \left(15 a^{2} + 13 a + 12\right)\cdot 19 + \left(9 a^{2} + 14 a + 4\right)\cdot 19^{2} + \left(9 a^{2} + 18 a + 8\right)\cdot 19^{3} + \left(9 a^{2} + 13 a + 14\right)\cdot 19^{4} + \left(12 a^{2} + 7 a + 12\right)\cdot 19^{5} +O(19^{6})\)
|
| $r_{ 5 }$ | $=$ |
\( 17 a^{2} + 13 a + 9 + \left(5 a^{2} + 15 a + 18\right)\cdot 19 + \left(15 a^{2} + a + 11\right)\cdot 19^{2} + \left(2 a^{2} + 11 a + 8\right)\cdot 19^{3} + \left(8 a^{2} + 12 a + 13\right)\cdot 19^{4} + \left(8 a^{2} + 3 a + 12\right)\cdot 19^{5} +O(19^{6})\)
|
| $r_{ 6 }$ | $=$ |
\( 6 a^{2} + 8 a + 3 + \left(8 a^{2} + 18 a + 2\right)\cdot 19 + \left(15 a^{2} + 9\right)\cdot 19^{2} + \left(7 a^{2} + 12 a + 11\right)\cdot 19^{3} + \left(10 a^{2} + 4 a + 2\right)\cdot 19^{4} + \left(9 a^{2} + 14 a + 5\right)\cdot 19^{5} +O(19^{6})\)
|
| $r_{ 7 }$ | $=$ |
\( 16 a^{2} + 10 a + \left(15 a^{2} + 7\right)\cdot 19 + \left(13 a^{2} + 11 a + 14\right)\cdot 19^{2} + \left(2 a^{2} + 6 a + 14\right)\cdot 19^{3} + \left(13 a + 4\right)\cdot 19^{4} + \left(16 a^{2} + 15 a + 1\right)\cdot 19^{5} +O(19^{6})\)
|
| $r_{ 8 }$ | $=$ |
\( 16 a^{2} + 2 a + 10 + \left(4 a^{2} + 17 a + 17\right)\cdot 19 + \left(15 a^{2} + 15 a + 12\right)\cdot 19^{2} + \left(2 a^{2} + 4 a + 15\right)\cdot 19^{3} + \left(10 a^{2} + 2 a + 9\right)\cdot 19^{4} + \left(3 a^{2} + 18 a + 1\right)\cdot 19^{5} +O(19^{6})\)
|
| $r_{ 9 }$ | $=$ |
\( 3 a^{2} + 16 a + 7 + \left(18 a^{2} + 7 a + 2\right)\cdot 19 + \left(12 a^{2} + 7 a + 13\right)\cdot 19^{2} + \left(6 a^{2} + 14 a\right)\cdot 19^{3} + \left(18 a^{2} + 2 a\right)\cdot 19^{4} + \left(2 a^{2} + 12 a\right)\cdot 19^{5} +O(19^{6})\)
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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 |
| $1$ | $1$ | $()$ | $1$ |
| $1$ | $3$ | $(1,2,6)(3,7,5)(4,8,9)$ | $\zeta_{9}^{3}$ |
| $1$ | $3$ | $(1,6,2)(3,5,7)(4,9,8)$ | $-\zeta_{9}^{3} - 1$ |
| $1$ | $9$ | $(1,4,7,2,8,5,6,9,3)$ | $\zeta_{9}$ |
| $1$ | $9$ | $(1,7,8,6,3,4,2,5,9)$ | $\zeta_{9}^{2}$ |
| $1$ | $9$ | $(1,8,3,2,9,7,6,4,5)$ | $\zeta_{9}^{4}$ |
| $1$ | $9$ | $(1,5,4,6,7,9,2,3,8)$ | $\zeta_{9}^{5}$ |
| $1$ | $9$ | $(1,9,5,2,4,3,6,8,7)$ | $-\zeta_{9}^{4} - \zeta_{9}$ |
| $1$ | $9$ | $(1,3,9,6,5,8,2,7,4)$ | $-\zeta_{9}^{5} - \zeta_{9}^{2}$ |
The blue line marks the conjugacy class containing complex conjugation.