Basic invariants
| Dimension: | $1$ |
| Group: | $C_6$ |
| Conductor: | \(39\)\(\medspace = 3 \cdot 13 \) |
| Artin field: | Galois closure of 6.0.771147.1 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $C_6$ |
| Parity: | odd |
| Dirichlet character: | \(\chi_{39}(35,\cdot)\) |
| Projective image: | $C_1$ |
| Projective field: | Galois closure of \(\Q\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - x^{5} + 5x^{4} + 6x^{3} + 15x^{2} + 4x + 1 \)
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The roots of $f$ are computed in an extension of $\Q_{ 47 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 47 }$:
\( x^{2} + 45x + 5 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 20 a + 8 + \left(24 a + 38\right)\cdot 47 + 47^{2} + \left(11 a + 45\right)\cdot 47^{3} + \left(37 a + 18\right)\cdot 47^{4} +O(47^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 19 a + 17 + \left(42 a + 37\right)\cdot 47 + \left(43 a + 37\right)\cdot 47^{2} + \left(10 a + 9\right)\cdot 47^{3} + \left(43 a + 23\right)\cdot 47^{4} +O(47^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 5 a + 2 + \left(23 a + 21\right)\cdot 47 + \left(37 a + 17\right)\cdot 47^{2} + \left(32 a + 25\right)\cdot 47^{3} + \left(12 a + 9\right)\cdot 47^{4} +O(47^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 27 a + 1 + \left(22 a + 20\right)\cdot 47 + \left(46 a + 25\right)\cdot 47^{2} + \left(35 a + 19\right)\cdot 47^{3} + \left(9 a + 35\right)\cdot 47^{4} +O(47^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 28 a + 8 + \left(4 a + 9\right)\cdot 47 + \left(3 a + 36\right)\cdot 47^{2} + \left(36 a + 34\right)\cdot 47^{3} + \left(3 a + 4\right)\cdot 47^{4} +O(47^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 42 a + 12 + \left(23 a + 15\right)\cdot 47 + \left(9 a + 22\right)\cdot 47^{2} + \left(14 a + 6\right)\cdot 47^{3} + \left(34 a + 2\right)\cdot 47^{4} +O(47^{5})\)
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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 |
| $1$ | $1$ | $()$ | $1$ |
| $1$ | $2$ | $(1,4)(2,5)(3,6)$ | $-1$ |
| $1$ | $3$ | $(1,2,3)(4,5,6)$ | $\zeta_{3}$ |
| $1$ | $3$ | $(1,3,2)(4,6,5)$ | $-\zeta_{3} - 1$ |
| $1$ | $6$ | $(1,5,3,4,2,6)$ | $-\zeta_{3}$ |
| $1$ | $6$ | $(1,6,2,4,3,5)$ | $\zeta_{3} + 1$ |
The blue line marks the conjugacy class containing complex conjugation.