Basic invariants
Dimension: | $1$ |
Group: | $C_6$ |
Conductor: | \(1027\)\(\medspace = 13 \cdot 79 \) |
Artin field: | Galois closure of 6.0.14081686879.2 |
Galois orbit size: | $2$ |
Smallest permutation container: | $C_6$ |
Parity: | odd |
Dirichlet character: | \(\chi_{1027}(315,\cdot)\) |
Projective image: | $C_1$ |
Projective field: | Galois closure of \(\Q\) |
Defining polynomial
$f(x)$ | $=$ | \( x^{6} - x^{5} + 51x^{4} - 29x^{3} + 1195x^{2} - 703x + 11545 \) . |
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:
$r_{ 1 }$ | $=$ |
\( 40 a + 6 + \left(14 a + 29\right)\cdot 47 + \left(16 a + 34\right)\cdot 47^{2} + \left(2 a + 31\right)\cdot 47^{3} + \left(24 a + 19\right)\cdot 47^{4} +O(47^{5})\)
$r_{ 2 }$ |
$=$ |
\( 7 a + 8 + \left(32 a + 7\right)\cdot 47 + \left(30 a + 5\right)\cdot 47^{2} + 44 a\cdot 47^{3} + \left(22 a + 39\right)\cdot 47^{4} +O(47^{5})\)
| $r_{ 3 }$ |
$=$ |
\( 40 a + 17 + \left(14 a + 39\right)\cdot 47 + \left(16 a + 21\right)\cdot 47^{2} + \left(2 a + 44\right)\cdot 47^{3} + \left(24 a + 35\right)\cdot 47^{4} +O(47^{5})\)
| $r_{ 4 }$ |
$=$ |
\( 40 a + 22 + \left(14 a + 17\right)\cdot 47 + \left(16 a + 34\right)\cdot 47^{2} + \left(2 a + 11\right)\cdot 47^{3} + \left(24 a + 40\right)\cdot 47^{4} +O(47^{5})\)
| $r_{ 5 }$ |
$=$ |
\( 7 a + 39 + \left(32 a + 18\right)\cdot 47 + \left(30 a + 5\right)\cdot 47^{2} + \left(44 a + 20\right)\cdot 47^{3} + \left(22 a + 18\right)\cdot 47^{4} +O(47^{5})\)
| $r_{ 6 }$ |
$=$ |
\( 7 a + 3 + \left(32 a + 29\right)\cdot 47 + \left(30 a + 39\right)\cdot 47^{2} + \left(44 a + 32\right)\cdot 47^{3} + \left(22 a + 34\right)\cdot 47^{4} +O(47^{5})\)
| |
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 |
$1$ | $1$ | $()$ | $1$ |
$1$ | $2$ | $(1,5)(2,4)(3,6)$ | $-1$ |
$1$ | $3$ | $(1,3,4)(2,5,6)$ | $-\zeta_{3} - 1$ |
$1$ | $3$ | $(1,4,3)(2,6,5)$ | $\zeta_{3}$ |
$1$ | $6$ | $(1,2,3,5,4,6)$ | $-\zeta_{3}$ |
$1$ | $6$ | $(1,6,4,5,3,2)$ | $\zeta_{3} + 1$ |
The blue line marks the conjugacy class containing complex conjugation.