Basic invariants
| Dimension: | $1$ |
| Group: | $C_6$ |
| Conductor: | \(185\)\(\medspace = 5 \cdot 37 \) |
| Artin field: | Galois closure of 6.6.8667994625.1 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $C_6$ |
| Parity: | even |
| Dirichlet character: | \(\chi_{185}(64,\cdot)\) |
| Projective image: | $C_1$ |
| Projective field: | Galois closure of \(\Q\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - x^{5} - 52x^{4} + 139x^{3} + 126x^{2} - 408x + 184 \)
|
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:
| $r_{ 1 }$ | $=$ |
\( 12 a + 10 + \left(21 a + 15\right)\cdot 29 + \left(4 a + 17\right)\cdot 29^{2} + \left(6 a + 12\right)\cdot 29^{3} + \left(27 a + 21\right)\cdot 29^{4} +O(29^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 19 a + 13 + \left(9 a + 17\right)\cdot 29 + \left(26 a + 19\right)\cdot 29^{2} + \left(22 a + 7\right)\cdot 29^{3} + \left(25 a + 9\right)\cdot 29^{4} +O(29^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 2 a + 11 + \left(16 a + 25\right)\cdot 29 + \left(3 a + 15\right)\cdot 29^{2} + \left(12 a + 24\right)\cdot 29^{3} + \left(24 a + 13\right)\cdot 29^{4} +O(29^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 27 a + 21 + \left(12 a + 16\right)\cdot 29 + \left(25 a + 17\right)\cdot 29^{2} + \left(16 a + 23\right)\cdot 29^{3} + \left(4 a + 7\right)\cdot 29^{4} +O(29^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 17 a + 12 + \left(7 a + 23\right)\cdot 29 + \left(24 a + 19\right)\cdot 29^{2} + \left(22 a + 9\right)\cdot 29^{3} + \left(a + 6\right)\cdot 29^{4} +O(29^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 10 a + 21 + \left(19 a + 17\right)\cdot 29 + \left(2 a + 25\right)\cdot 29^{2} + \left(6 a + 8\right)\cdot 29^{3} + \left(3 a + 28\right)\cdot 29^{4} +O(29^{5})\)
|
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 |
| $1$ | $1$ | $()$ | $1$ |
| $1$ | $2$ | $(1,5)(2,6)(3,4)$ | $-1$ |
| $1$ | $3$ | $(1,2,3)(4,5,6)$ | $\zeta_{3}$ |
| $1$ | $3$ | $(1,3,2)(4,6,5)$ | $-\zeta_{3} - 1$ |
| $1$ | $6$ | $(1,6,3,5,2,4)$ | $-\zeta_{3}$ |
| $1$ | $6$ | $(1,4,2,5,3,6)$ | $\zeta_{3} + 1$ |
The blue line marks the conjugacy class containing complex conjugation.