Basic invariants
| Dimension: | $1$ |
| Group: | $C_6$ |
| Conductor: | \(1035\)\(\medspace = 3^{2} \cdot 5 \cdot 23 \) |
| Artin field: | Galois closure of 6.0.9978460875.4 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $C_6$ |
| Parity: | odd |
| Dirichlet character: | \(\chi_{1035}(229,\cdot)\) |
| Projective image: | $C_1$ |
| Projective field: | Galois closure of \(\Q\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - 3x^{5} + 84x^{4} - 161x^{3} + 2607x^{2} - 2706x + 29699 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 19 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 19 }$:
\( x^{2} + 18x + 2 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 7 a + 6 + \left(14 a + 14\right)\cdot 19 + \left(4 a + 15\right)\cdot 19^{2} + \left(3 a + 12\right)\cdot 19^{3} + \left(13 a + 12\right)\cdot 19^{4} +O(19^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 7 a + \left(14 a + 2\right)\cdot 19 + \left(4 a + 9\right)\cdot 19^{2} + \left(3 a + 9\right)\cdot 19^{3} + \left(13 a + 12\right)\cdot 19^{4} +O(19^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 12 a + 7 + \left(4 a + 9\right)\cdot 19 + \left(14 a + 18\right)\cdot 19^{2} + \left(15 a + 7\right)\cdot 19^{3} + \left(5 a + 3\right)\cdot 19^{4} +O(19^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 12 a + 11 + \left(4 a + 8\right)\cdot 19 + \left(14 a + 8\right)\cdot 19^{2} + \left(15 a + 16\right)\cdot 19^{3} + \left(5 a + 7\right)\cdot 19^{4} +O(19^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 12 a + 13 + \left(4 a + 2\right)\cdot 19 + \left(14 a + 6\right)\cdot 19^{2} + \left(15 a + 11\right)\cdot 19^{3} + \left(5 a + 3\right)\cdot 19^{4} +O(19^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 7 a + 4 + \left(14 a + 1\right)\cdot 19 + \left(4 a + 18\right)\cdot 19^{2} + \left(3 a + 17\right)\cdot 19^{3} + \left(13 a + 16\right)\cdot 19^{4} +O(19^{5})\)
|
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 |
| $1$ | $1$ | $()$ | $1$ |
| $1$ | $2$ | $(1,5)(2,3)(4,6)$ | $-1$ |
| $1$ | $3$ | $(1,6,2)(3,5,4)$ | $-\zeta_{3} - 1$ |
| $1$ | $3$ | $(1,2,6)(3,4,5)$ | $\zeta_{3}$ |
| $1$ | $6$ | $(1,4,2,5,6,3)$ | $\zeta_{3} + 1$ |
| $1$ | $6$ | $(1,3,6,5,2,4)$ | $-\zeta_{3}$ |
The blue line marks the conjugacy class containing complex conjugation.