Basic invariants
Dimension: | $1$ |
Group: | $C_6$ |
Conductor: | \(99\)\(\medspace = 3^{2} \cdot 11 \) |
Artin field: | Galois closure of 6.0.8732691.1 |
Galois orbit size: | $2$ |
Smallest permutation container: | $C_6$ |
Parity: | odd |
Dirichlet character: | \(\chi_{99}(43,\cdot)\) |
Projective image: | $C_1$ |
Projective field: | Galois closure of \(\Q\) |
Defining polynomial
$f(x)$ | $=$ | \( x^{6} - 3x^{5} + 6x^{4} - 5x^{3} + 33x^{2} - 54x + 111 \) . |
The roots of $f$ are computed in an extension of $\Q_{ 17 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 17 }$: \( x^{2} + 16x + 3 \)
Roots:
$r_{ 1 }$ | $=$ |
\( a + 14 + \left(3 a + 3\right)\cdot 17 + \left(a + 3\right)\cdot 17^{2} + \left(16 a + 3\right)\cdot 17^{3} + \left(3 a + 1\right)\cdot 17^{4} +O(17^{5})\)
$r_{ 2 }$ |
$=$ |
\( 16 a + 14 + \left(13 a + 8\right)\cdot 17 + \left(15 a + 2\right)\cdot 17^{2} + 5\cdot 17^{3} + \left(13 a + 12\right)\cdot 17^{4} +O(17^{5})\)
| $r_{ 3 }$ |
$=$ |
\( 16 a + 15 + \left(13 a + 5\right)\cdot 17 + \left(15 a + 1\right)\cdot 17^{2} + 17^{3} + \left(13 a + 6\right)\cdot 17^{4} +O(17^{5})\)
| $r_{ 4 }$ |
$=$ |
\( a + 13 + \left(3 a + 6\right)\cdot 17 + \left(a + 4\right)\cdot 17^{2} + \left(16 a + 7\right)\cdot 17^{3} + \left(3 a + 7\right)\cdot 17^{4} +O(17^{5})\)
| $r_{ 5 }$ |
$=$ |
\( 16 a + 8 + \left(13 a + 5\right)\cdot 17 + \left(15 a + 10\right)\cdot 17^{2} + 7\cdot 17^{3} + \left(13 a + 14\right)\cdot 17^{4} +O(17^{5})\)
| $r_{ 6 }$ |
$=$ |
\( a + 7 + \left(3 a + 3\right)\cdot 17 + \left(a + 12\right)\cdot 17^{2} + \left(16 a + 9\right)\cdot 17^{3} + \left(3 a + 9\right)\cdot 17^{4} +O(17^{5})\)
| |
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 |
$1$ | $1$ | $()$ | $1$ |
$1$ | $2$ | $(1,3)(2,4)(5,6)$ | $-1$ |
$1$ | $3$ | $(1,6,4)(2,3,5)$ | $-\zeta_{3} - 1$ |
$1$ | $3$ | $(1,4,6)(2,5,3)$ | $\zeta_{3}$ |
$1$ | $6$ | $(1,2,6,3,4,5)$ | $-\zeta_{3}$ |
$1$ | $6$ | $(1,5,4,3,6,2)$ | $\zeta_{3} + 1$ |
The blue line marks the conjugacy class containing complex conjugation.