Basic invariants
| Dimension: | $1$ |
| Group: | $C_6$ |
| Conductor: | \(105\)\(\medspace = 3 \cdot 5 \cdot 7 \) |
| Artin field: | Galois closure of 6.6.56723625.1 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $C_6$ |
| Parity: | even |
| Dirichlet character: | \(\chi_{105}(59,\cdot)\) |
| Projective image: | $C_1$ |
| Projective field: | Galois closure of \(\Q\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - x^{5} - 27x^{4} + 27x^{3} + 197x^{2} - 197x - 251 \)
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The roots of $f$ are computed in an extension of $\Q_{ 29 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 29 }$:
\( x^{2} + 24x + 2 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 17 a + 14 + \left(25 a + 24\right)\cdot 29 + \left(6 a + 17\right)\cdot 29^{2} + \left(8 a + 18\right)\cdot 29^{3} + \left(28 a + 19\right)\cdot 29^{4} +O(29^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 9 a + 12 + \left(20 a + 17\right)\cdot 29 + \left(10 a + 12\right)\cdot 29^{2} + \left(26 a + 19\right)\cdot 29^{3} + \left(27 a + 17\right)\cdot 29^{4} +O(29^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 12 a + 12 + \left(3 a + 19\right)\cdot 29 + \left(22 a + 26\right)\cdot 29^{2} + \left(20 a + 23\right)\cdot 29^{3} + 7\cdot 29^{4} +O(29^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 7 a + 8 + \left(4 a + 23\right)\cdot 29 + \left(9 a + 14\right)\cdot 29^{2} + \left(16 a + 7\right)\cdot 29^{3} + \left(5 a + 22\right)\cdot 29^{4} +O(29^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 20 a + 28 + \left(8 a + 22\right)\cdot 29 + \left(18 a + 16\right)\cdot 29^{2} + \left(2 a + 24\right)\cdot 29^{3} + \left(a + 14\right)\cdot 29^{4} +O(29^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 22 a + 14 + \left(24 a + 8\right)\cdot 29 + \left(19 a + 27\right)\cdot 29^{2} + \left(12 a + 21\right)\cdot 29^{3} + \left(23 a + 4\right)\cdot 29^{4} +O(29^{5})\)
|
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 |
| $1$ | $1$ | $()$ | $1$ |
| $1$ | $2$ | $(1,3)(2,5)(4,6)$ | $-1$ |
| $1$ | $3$ | $(1,6,2)(3,4,5)$ | $\zeta_{3}$ |
| $1$ | $3$ | $(1,2,6)(3,5,4)$ | $-\zeta_{3} - 1$ |
| $1$ | $6$ | $(1,4,2,3,6,5)$ | $-\zeta_{3}$ |
| $1$ | $6$ | $(1,5,6,3,2,4)$ | $\zeta_{3} + 1$ |
The blue line marks the conjugacy class containing complex conjugation.