Basic invariants
| Dimension: | $1$ |
| Group: | $C_6$ |
| Conductor: | \(209\)\(\medspace = 11 \cdot 19 \) |
| Artin field: | Galois closure of 6.0.173457251.1 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $C_6$ |
| Parity: | odd |
| Dirichlet character: | \(\chi_{209}(197,\cdot)\) |
| Projective image: | $C_1$ |
| Projective field: | Galois closure of \(\Q\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - x^{5} - 4x^{4} - 7x^{3} + 67x^{2} + 122x + 229 \)
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The roots of $f$ are computed in an extension of $\Q_{ 83 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 83 }$:
\( x^{2} + 82x + 2 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 76 a + 68 + \left(39 a + 13\right)\cdot 83 + \left(57 a + 69\right)\cdot 83^{2} + \left(68 a + 71\right)\cdot 83^{3} + \left(17 a + 45\right)\cdot 83^{4} +O(83^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 76 a + 47 + \left(39 a + 12\right)\cdot 83 + \left(57 a + 38\right)\cdot 83^{2} + \left(68 a + 2\right)\cdot 83^{3} + \left(17 a + 45\right)\cdot 83^{4} +O(83^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 7 a + 40 + \left(43 a + 59\right)\cdot 83 + \left(25 a + 55\right)\cdot 83^{2} + \left(14 a + 13\right)\cdot 83^{3} + \left(65 a + 77\right)\cdot 83^{4} +O(83^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 7 a + 55 + \left(43 a + 74\right)\cdot 83 + \left(25 a + 49\right)\cdot 83^{2} + \left(14 a + 44\right)\cdot 83^{3} + \left(65 a + 17\right)\cdot 83^{4} +O(83^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 76 a + 62 + \left(39 a + 27\right)\cdot 83 + \left(57 a + 32\right)\cdot 83^{2} + \left(68 a + 33\right)\cdot 83^{3} + \left(17 a + 68\right)\cdot 83^{4} +O(83^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 7 a + 61 + \left(43 a + 60\right)\cdot 83 + \left(25 a + 3\right)\cdot 83^{2} + 14 a\cdot 83^{3} + \left(65 a + 78\right)\cdot 83^{4} +O(83^{5})\)
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Generators of the action on the roots $r_1, \ldots, r_{ 6 }$
| Cycle notation |
Character values on conjugacy classes
| Size | Order | Action on $r_1, \ldots, r_{ 6 }$ | Character value |
| $1$ | $1$ | $()$ | $1$ |
| $1$ | $2$ | $(1,6)(2,3)(4,5)$ | $-1$ |
| $1$ | $3$ | $(1,2,5)(3,4,6)$ | $-\zeta_{3} - 1$ |
| $1$ | $3$ | $(1,5,2)(3,6,4)$ | $\zeta_{3}$ |
| $1$ | $6$ | $(1,3,5,6,2,4)$ | $\zeta_{3} + 1$ |
| $1$ | $6$ | $(1,4,2,6,5,3)$ | $-\zeta_{3}$ |
The blue line marks the conjugacy class containing complex conjugation.