Basic invariants
| Dimension: | $1$ |
| Group: | $C_6$ |
| Conductor: | \(2639\)\(\medspace = 7 \cdot 13 \cdot 29 \) |
| Artin field: | Galois closure of 6.6.128651901833.1 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $C_6$ |
| Parity: | even |
| Dirichlet character: | \(\chi_{2639}(1507,\cdot)\) |
| Projective image: | $C_1$ |
| Projective field: | Galois closure of \(\Q\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - x^{5} - 287x^{4} + 191x^{3} + 26607x^{2} - 9025x - 795523 \)
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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:
| $r_{ 1 }$ | $=$ |
\( 13 a + 13 + \left(26 a + 20\right)\cdot 43 + \left(22 a + 30\right)\cdot 43^{2} + \left(9 a + 31\right)\cdot 43^{3} + \left(20 a + 15\right)\cdot 43^{4} +O(43^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 13 a + 9 + \left(26 a + 41\right)\cdot 43 + \left(22 a + 17\right)\cdot 43^{2} + \left(9 a + 1\right)\cdot 43^{3} + 20 a\cdot 43^{4} +O(43^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 30 a + 15 + \left(16 a + 39\right)\cdot 43 + \left(20 a + 17\right)\cdot 43^{2} + \left(33 a + 16\right)\cdot 43^{3} + 22 a\cdot 43^{4} +O(43^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 13 a + 2 + \left(26 a + 26\right)\cdot 43 + \left(22 a + 21\right)\cdot 43^{2} + \left(9 a + 29\right)\cdot 43^{3} + \left(20 a + 32\right)\cdot 43^{4} +O(43^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 30 a + 22 + \left(16 a + 11\right)\cdot 43 + \left(20 a + 14\right)\cdot 43^{2} + \left(33 a + 31\right)\cdot 43^{3} + \left(22 a + 10\right)\cdot 43^{4} +O(43^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 30 a + 26 + \left(16 a + 33\right)\cdot 43 + \left(20 a + 26\right)\cdot 43^{2} + \left(33 a + 18\right)\cdot 43^{3} + \left(22 a + 26\right)\cdot 43^{4} +O(43^{5})\)
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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 |
| $1$ | $1$ | $()$ | $1$ |
| $1$ | $2$ | $(1,6)(2,5)(3,4)$ | $-1$ |
| $1$ | $3$ | $(1,4,2)(3,5,6)$ | $\zeta_{3}$ |
| $1$ | $3$ | $(1,2,4)(3,6,5)$ | $-\zeta_{3} - 1$ |
| $1$ | $6$ | $(1,5,4,6,2,3)$ | $\zeta_{3} + 1$ |
| $1$ | $6$ | $(1,3,2,6,4,5)$ | $-\zeta_{3}$ |
The blue line marks the conjugacy class containing complex conjugation.