Basic invariants
| Dimension: | $1$ |
| Group: | $C_6$ |
| Conductor: | \(973\)\(\medspace = 7 \cdot 139 \) |
| Artin field: | Galois closure of 6.0.6448171219.1 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $C_6$ |
| Parity: | odd |
| Dirichlet character: | \(\chi_{973}(277,\cdot)\) |
| Projective image: | $C_1$ |
| Projective field: | Galois closure of \(\Q\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - x^{5} + 100x^{4} - 67x^{3} + 3645x^{2} - 1156x + 47879 \)
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The roots of $f$ are computed in an extension of $\Q_{ 43 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 43 }$:
\( x^{2} + 42x + 3 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 13 a + 13 + \left(8 a + 29\right)\cdot 43 + \left(36 a + 14\right)\cdot 43^{2} + \left(9 a + 38\right)\cdot 43^{3} + \left(25 a + 34\right)\cdot 43^{4} +O(43^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 13 a + 9 + \left(8 a + 7\right)\cdot 43 + \left(36 a + 2\right)\cdot 43^{2} + \left(9 a + 8\right)\cdot 43^{3} + \left(25 a + 19\right)\cdot 43^{4} +O(43^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 30 a + 15 + \left(34 a + 30\right)\cdot 43 + \left(6 a + 33\right)\cdot 43^{2} + \left(33 a + 9\right)\cdot 43^{3} + \left(17 a + 24\right)\cdot 43^{4} +O(43^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 13 a + 2 + \left(8 a + 35\right)\cdot 43 + \left(36 a + 5\right)\cdot 43^{2} + \left(9 a + 36\right)\cdot 43^{3} + \left(25 a + 8\right)\cdot 43^{4} +O(43^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 30 a + 22 + \left(34 a + 2\right)\cdot 43 + \left(6 a + 30\right)\cdot 43^{2} + \left(33 a + 24\right)\cdot 43^{3} + \left(17 a + 34\right)\cdot 43^{4} +O(43^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 30 a + 26 + \left(34 a + 24\right)\cdot 43 + \left(6 a + 42\right)\cdot 43^{2} + \left(33 a + 11\right)\cdot 43^{3} + \left(17 a + 7\right)\cdot 43^{4} +O(43^{5})\)
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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 |
| $1$ | $1$ | $()$ | $1$ |
| $1$ | $2$ | $(1,6)(2,5)(3,4)$ | $-1$ |
| $1$ | $3$ | $(1,4,2)(3,5,6)$ | $-\zeta_{3} - 1$ |
| $1$ | $3$ | $(1,2,4)(3,6,5)$ | $\zeta_{3}$ |
| $1$ | $6$ | $(1,5,4,6,2,3)$ | $-\zeta_{3}$ |
| $1$ | $6$ | $(1,3,2,6,4,5)$ | $\zeta_{3} + 1$ |
The blue line marks the conjugacy class containing complex conjugation.