Basic invariants
| Dimension: | $1$ |
| Group: | $C_6$ |
| Conductor: | \(133\)\(\medspace = 7 \cdot 19 \) |
| Artin field: | Galois closure of 6.0.2190305047.1 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $C_6$ |
| Parity: | odd |
| Dirichlet character: | \(\chi_{133}(45,\cdot)\) |
| Projective image: | $C_1$ |
| Projective field: | Galois closure of \(\Q\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - x^{5} + x^{4} - 99x^{3} + 281x^{2} + 573x + 981 \)
|
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:
| $r_{ 1 }$ | $=$ |
\( 13 a + 19 + \left(37 a + 13\right)\cdot 47 + \left(36 a + 36\right)\cdot 47^{2} + \left(18 a + 32\right)\cdot 47^{3} + \left(6 a + 19\right)\cdot 47^{4} +O(47^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 17 a + 36 + \left(9 a + 17\right)\cdot 47 + \left(44 a + 38\right)\cdot 47^{2} + \left(27 a + 2\right)\cdot 47^{3} + \left(29 a + 35\right)\cdot 47^{4} +O(47^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 30 a + 23 + \left(37 a + 19\right)\cdot 47 + \left(2 a + 23\right)\cdot 47^{2} + \left(19 a + 14\right)\cdot 47^{3} + \left(17 a + 19\right)\cdot 47^{4} +O(47^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 35 a + 45 + \left(12 a + 35\right)\cdot 47 + \left(16 a + 45\right)\cdot 47^{2} + \left(17 a + 42\right)\cdot 47^{3} + \left(13 a + 21\right)\cdot 47^{4} +O(47^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 34 a + 45 + \left(9 a + 27\right)\cdot 47 + \left(10 a + 25\right)\cdot 47^{2} + \left(28 a + 33\right)\cdot 47^{3} + \left(40 a + 13\right)\cdot 47^{4} +O(47^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 12 a + 21 + \left(34 a + 26\right)\cdot 47 + \left(30 a + 18\right)\cdot 47^{2} + \left(29 a + 14\right)\cdot 47^{3} + \left(33 a + 31\right)\cdot 47^{4} +O(47^{5})\)
|
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 |
| $1$ | $1$ | $()$ | $1$ |
| $1$ | $2$ | $(1,5)(2,3)(4,6)$ | $-1$ |
| $1$ | $3$ | $(1,3,4)(2,6,5)$ | $\zeta_{3}$ |
| $1$ | $3$ | $(1,4,3)(2,5,6)$ | $-\zeta_{3} - 1$ |
| $1$ | $6$ | $(1,6,3,5,4,2)$ | $\zeta_{3} + 1$ |
| $1$ | $6$ | $(1,2,4,5,3,6)$ | $-\zeta_{3}$ |
The blue line marks the conjugacy class containing complex conjugation.