Basic invariants
| Dimension: | $1$ |
| Group: | $C_6$ |
| Conductor: | \(93\)\(\medspace = 3 \cdot 31 \) |
| Artin field: | Galois closure of 6.6.772987077.1 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $C_6$ |
| Parity: | even |
| Dirichlet character: | \(\chi_{93}(68,\cdot)\) |
| Projective image: | $C_1$ |
| Projective field: | Galois closure of \(\Q\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - x^{5} - 28x^{4} + 51x^{3} + 75x^{2} - 98x - 92 \)
|
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:
| $r_{ 1 }$ | $=$ |
\( 37 a + 46 + \left(7 a + 37\right)\cdot 47 + \left(26 a + 11\right)\cdot 47^{2} + \left(8 a + 10\right)\cdot 47^{3} + \left(27 a + 20\right)\cdot 47^{4} +O(47^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 40 a + 11 + \left(4 a + 28\right)\cdot 47 + \left(3 a + 34\right)\cdot 47^{2} + \left(15 a + 1\right)\cdot 47^{3} + \left(43 a + 42\right)\cdot 47^{4} +O(47^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 18 a + 13 + \left(14 a + 1\right)\cdot 47 + \left(24 a + 31\right)\cdot 47^{2} + \left(26 a + 11\right)\cdot 47^{3} + \left(39 a + 17\right)\cdot 47^{4} +O(47^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 7 a + 44 + \left(42 a + 44\right)\cdot 47 + \left(43 a + 35\right)\cdot 47^{2} + \left(31 a + 28\right)\cdot 47^{3} + \left(3 a + 19\right)\cdot 47^{4} +O(47^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 10 a + 26 + \left(39 a + 16\right)\cdot 47 + \left(20 a + 9\right)\cdot 47^{2} + \left(38 a + 1\right)\cdot 47^{3} + \left(19 a + 19\right)\cdot 47^{4} +O(47^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 29 a + 2 + \left(32 a + 12\right)\cdot 47 + \left(22 a + 18\right)\cdot 47^{2} + \left(20 a + 40\right)\cdot 47^{3} + \left(7 a + 22\right)\cdot 47^{4} +O(47^{5})\)
|
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 |
| $1$ | $1$ | $()$ | $1$ |
| $1$ | $2$ | $(1,5)(2,4)(3,6)$ | $-1$ |
| $1$ | $3$ | $(1,2,6)(3,5,4)$ | $\zeta_{3}$ |
| $1$ | $3$ | $(1,6,2)(3,4,5)$ | $-\zeta_{3} - 1$ |
| $1$ | $6$ | $(1,4,6,5,2,3)$ | $-\zeta_{3}$ |
| $1$ | $6$ | $(1,3,2,5,6,4)$ | $\zeta_{3} + 1$ |
The blue line marks the conjugacy class containing complex conjugation.