Basic invariants
| Dimension: | $1$ |
| Group: | $C_6$ |
| Conductor: | \(693\)\(\medspace = 3^{2} \cdot 7 \cdot 11 \) |
| Artin field: | Galois closure of 6.0.20967191091.5 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $C_6$ |
| Parity: | odd |
| Dirichlet character: | \(\chi_{693}(142,\cdot)\) |
| Projective image: | $C_1$ |
| Projective field: | Galois closure of \(\Q\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - 3x^{5} - 30x^{4} - 5x^{3} + 519x^{2} + 1548x + 1875 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 101 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 101 }$:
\( x^{2} + 97x + 2 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 84 a + 36 + \left(37 a + 20\right)\cdot 101 + \left(38 a + 30\right)\cdot 101^{2} + \left(91 a + 78\right)\cdot 101^{3} + \left(86 a + 81\right)\cdot 101^{4} +O(101^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 17 a + 57 + \left(63 a + 55\right)\cdot 101 + \left(62 a + 58\right)\cdot 101^{2} + \left(9 a + 12\right)\cdot 101^{3} + \left(14 a + 76\right)\cdot 101^{4} +O(101^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 84 a + 94 + \left(37 a + 41\right)\cdot 101 + \left(38 a + 55\right)\cdot 101^{2} + \left(91 a + 48\right)\cdot 101^{3} + \left(86 a + 67\right)\cdot 101^{4} +O(101^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 17 a + 26 + \left(63 a + 8\right)\cdot 101 + \left(62 a + 70\right)\cdot 101^{2} + \left(9 a + 72\right)\cdot 101^{3} + \left(14 a + 20\right)\cdot 101^{4} +O(101^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 84 a + 24 + \left(37 a + 89\right)\cdot 101 + \left(38 a + 43\right)\cdot 101^{2} + \left(91 a + 89\right)\cdot 101^{3} + \left(86 a + 21\right)\cdot 101^{4} +O(101^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 17 a + 69 + \left(63 a + 87\right)\cdot 101 + \left(62 a + 44\right)\cdot 101^{2} + \left(9 a + 1\right)\cdot 101^{3} + \left(14 a + 35\right)\cdot 101^{4} +O(101^{5})\)
|
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 |
| $1$ | $1$ | $()$ | $1$ | |
| $1$ | $2$ | $(1,6)(2,5)(3,4)$ | $-1$ | ✓ |
| $1$ | $3$ | $(1,3,5)(2,6,4)$ | $-\zeta_{3} - 1$ | |
| $1$ | $3$ | $(1,5,3)(2,4,6)$ | $\zeta_{3}$ | |
| $1$ | $6$ | $(1,2,3,6,5,4)$ | $-\zeta_{3}$ | |
| $1$ | $6$ | $(1,4,5,6,3,2)$ | $\zeta_{3} + 1$ |