Basic invariants
| Dimension: | $1$ |
| Group: | $C_6$ |
| Conductor: | \(1032\)\(\medspace = 2^{3} \cdot 3 \cdot 43 \) |
| Artin field: | Galois closure of 6.6.47261505024.1 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $C_6$ |
| Parity: | even |
| Dirichlet character: | \(\chi_{1032}(251,\cdot)\) |
| Projective image: | $C_1$ |
| Projective field: | Galois closure of \(\Q\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - 2x^{5} - 45x^{4} + 36x^{3} + 308x^{2} - 304x - 188 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 41 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 41 }$:
\( x^{2} + 38x + 6 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 33 a + 28 + \left(13 a + 1\right)\cdot 41 + \left(30 a + 29\right)\cdot 41^{2} + \left(15 a + 4\right)\cdot 41^{3} + \left(27 a + 22\right)\cdot 41^{4} +O(41^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 8 a + 6 + \left(27 a + 35\right)\cdot 41 + \left(10 a + 24\right)\cdot 41^{2} + \left(25 a + 32\right)\cdot 41^{3} + \left(13 a + 28\right)\cdot 41^{4} +O(41^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 8 a + 4 + \left(27 a + 10\right)\cdot 41 + \left(10 a + 24\right)\cdot 41^{2} + \left(25 a + 21\right)\cdot 41^{3} + \left(13 a + 6\right)\cdot 41^{4} +O(41^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 33 a + 20 + \left(13 a + 20\right)\cdot 41 + \left(30 a + 30\right)\cdot 41^{2} + \left(15 a + 15\right)\cdot 41^{3} + \left(27 a + 18\right)\cdot 41^{4} +O(41^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 8 a + 37 + \left(27 a + 28\right)\cdot 41 + \left(10 a + 25\right)\cdot 41^{2} + \left(25 a + 32\right)\cdot 41^{3} + \left(13 a + 2\right)\cdot 41^{4} +O(41^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 33 a + 30 + \left(13 a + 26\right)\cdot 41 + \left(30 a + 29\right)\cdot 41^{2} + \left(15 a + 15\right)\cdot 41^{3} + \left(27 a + 3\right)\cdot 41^{4} +O(41^{5})\)
|
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 |
| $1$ | $1$ | $()$ | $1$ |
| $1$ | $2$ | $(1,3)(2,6)(4,5)$ | $-1$ |
| $1$ | $3$ | $(1,4,6)(2,3,5)$ | $\zeta_{3}$ |
| $1$ | $3$ | $(1,6,4)(2,5,3)$ | $-\zeta_{3} - 1$ |
| $1$ | $6$ | $(1,2,4,3,6,5)$ | $\zeta_{3} + 1$ |
| $1$ | $6$ | $(1,5,6,3,4,2)$ | $-\zeta_{3}$ |
The blue line marks the conjugacy class containing complex conjugation.