Basic invariants
| Dimension: | $1$ |
| Group: | $C_6$ |
| Conductor: | \(260\)\(\medspace = 2^{2} \cdot 5 \cdot 13 \) |
| Artin field: | Galois closure of 6.0.2970344000.1 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $C_6$ |
| Parity: | odd |
| Dirichlet character: | \(\chi_{260}(199,\cdot)\) |
| Projective image: | $C_1$ |
| Projective field: | Galois closure of \(\Q\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} + 65x^{4} + 650x^{2} + 1625 \)
|
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:
| $r_{ 1 }$ | $=$ |
\( 14 a + 33 + \left(5 a + 1\right)\cdot 47 + \left(17 a + 9\right)\cdot 47^{2} + \left(20 a + 35\right)\cdot 47^{3} + \left(23 a + 33\right)\cdot 47^{4} +O(47^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 8 a + 39 + \left(14 a + 36\right)\cdot 47 + \left(36 a + 17\right)\cdot 47^{2} + \left(37 a + 27\right)\cdot 47^{3} + \left(25 a + 16\right)\cdot 47^{4} +O(47^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 12 a + 35 + \left(17 a + 35\right)\cdot 47 + \left(28 a + 3\right)\cdot 47^{2} + \left(44 a + 40\right)\cdot 47^{3} + \left(29 a + 15\right)\cdot 47^{4} +O(47^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 33 a + 14 + \left(41 a + 45\right)\cdot 47 + \left(29 a + 37\right)\cdot 47^{2} + \left(26 a + 11\right)\cdot 47^{3} + \left(23 a + 13\right)\cdot 47^{4} +O(47^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 39 a + 8 + \left(32 a + 10\right)\cdot 47 + \left(10 a + 29\right)\cdot 47^{2} + \left(9 a + 19\right)\cdot 47^{3} + \left(21 a + 30\right)\cdot 47^{4} +O(47^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 35 a + 12 + \left(29 a + 11\right)\cdot 47 + \left(18 a + 43\right)\cdot 47^{2} + \left(2 a + 6\right)\cdot 47^{3} + \left(17 a + 31\right)\cdot 47^{4} +O(47^{5})\)
|
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 | Complex conjugation |
| $1$ | $1$ | $()$ | $1$ | |
| $1$ | $2$ | $(1,4)(2,5)(3,6)$ | $-1$ | ✓ |
| $1$ | $3$ | $(1,2,3)(4,5,6)$ | $-\zeta_{3} - 1$ | |
| $1$ | $3$ | $(1,3,2)(4,6,5)$ | $\zeta_{3}$ | |
| $1$ | $6$ | $(1,5,3,4,2,6)$ | $\zeta_{3} + 1$ | |
| $1$ | $6$ | $(1,6,2,4,3,5)$ | $-\zeta_{3}$ |