Basic invariants
| Dimension: | $1$ |
| Group: | $C_6$ |
| Conductor: | \(1027\)\(\medspace = 13 \cdot 79 \) |
| Artin field: | Galois closure of 6.6.85573327957.2 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $C_6$ |
| Parity: | even |
| Dirichlet character: | \(\chi_{1027}(181,\cdot)\) |
| Projective image: | $C_1$ |
| Projective field: | Galois closure of \(\Q\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - x^{5} - 62x^{4} + 141x^{3} + 665x^{2} - 1882x + 1037 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 67 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 67 }$:
\( x^{2} + 63x + 2 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 55 a + 39 + \left(6 a + 5\right)\cdot 67 + \left(26 a + 10\right)\cdot 67^{2} + \left(23 a + 63\right)\cdot 67^{3} + \left(62 a + 50\right)\cdot 67^{4} +O(67^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 12 a + 29 + \left(60 a + 2\right)\cdot 67 + \left(40 a + 62\right)\cdot 67^{2} + \left(43 a + 54\right)\cdot 67^{3} + \left(4 a + 57\right)\cdot 67^{4} +O(67^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 12 a + 9 + \left(60 a + 45\right)\cdot 67 + \left(40 a + 43\right)\cdot 67^{2} + \left(43 a + 49\right)\cdot 67^{3} + \left(4 a + 37\right)\cdot 67^{4} +O(67^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 55 a + 10 + \left(6 a + 30\right)\cdot 67 + \left(26 a + 31\right)\cdot 67^{2} + \left(23 a + 54\right)\cdot 67^{3} + \left(62 a + 32\right)\cdot 67^{4} +O(67^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 55 a + 57 + \left(6 a + 5\right)\cdot 67 + \left(26 a + 13\right)\cdot 67^{2} + \left(23 a + 49\right)\cdot 67^{3} + \left(62 a + 12\right)\cdot 67^{4} +O(67^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 12 a + 58 + \left(60 a + 44\right)\cdot 67 + \left(40 a + 40\right)\cdot 67^{2} + \left(43 a + 63\right)\cdot 67^{3} + \left(4 a + 8\right)\cdot 67^{4} +O(67^{5})\)
|
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 |
| $1$ | $1$ | $()$ | $1$ |
| $1$ | $2$ | $(1,6)(2,4)(3,5)$ | $-1$ |
| $1$ | $3$ | $(1,5,4)(2,6,3)$ | $\zeta_{3}$ |
| $1$ | $3$ | $(1,4,5)(2,3,6)$ | $-\zeta_{3} - 1$ |
| $1$ | $6$ | $(1,2,5,6,4,3)$ | $\zeta_{3} + 1$ |
| $1$ | $6$ | $(1,3,4,6,5,2)$ | $-\zeta_{3}$ |
The blue line marks the conjugacy class containing complex conjugation.