Basic invariants
| Dimension: | $1$ |
| Group: | $C_6$ |
| Conductor: | \(333\)\(\medspace = 3^{2} \cdot 37 \) |
| Artin field: | Galois closure of 6.6.332334333.1 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $C_6$ |
| Parity: | even |
| Dirichlet character: | \(\chi_{333}(184,\cdot)\) |
| Projective image: | $C_1$ |
| Projective field: | Galois closure of \(\Q\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - 3x^{5} - 30x^{4} + 63x^{3} + 219x^{2} - 300x - 271 \)
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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:
| $r_{ 1 }$ | $=$ |
\( 14 a + 5 + 3 a\cdot 17 + \left(7 a + 13\right)\cdot 17^{2} + \left(14 a + 2\right)\cdot 17^{3} + \left(7 a + 8\right)\cdot 17^{4} +O(17^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 14 a + 6 + \left(3 a + 14\right)\cdot 17 + \left(7 a + 11\right)\cdot 17^{2} + \left(14 a + 15\right)\cdot 17^{3} + \left(7 a + 1\right)\cdot 17^{4} +O(17^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 3 a + 9 + \left(13 a + 7\right)\cdot 17 + \left(9 a + 7\right)\cdot 17^{2} + \left(2 a + 3\right)\cdot 17^{3} + \left(9 a + 10\right)\cdot 17^{4} +O(17^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 3 a + 2 + \left(13 a + 7\right)\cdot 17 + \left(9 a + 16\right)\cdot 17^{2} + \left(2 a + 9\right)\cdot 17^{3} + \left(9 a + 1\right)\cdot 17^{4} +O(17^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 3 a + 3 + \left(13 a + 4\right)\cdot 17 + \left(9 a + 15\right)\cdot 17^{2} + \left(2 a + 5\right)\cdot 17^{3} + \left(9 a + 12\right)\cdot 17^{4} +O(17^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 14 a + 12 + 3 a\cdot 17 + \left(7 a + 4\right)\cdot 17^{2} + \left(14 a + 13\right)\cdot 17^{3} + \left(7 a + 16\right)\cdot 17^{4} +O(17^{5})\)
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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 |
| $1$ | $1$ | $()$ | $1$ |
| $1$ | $2$ | $(1,4)(2,5)(3,6)$ | $-1$ |
| $1$ | $3$ | $(1,6,2)(3,5,4)$ | $\zeta_{3}$ |
| $1$ | $3$ | $(1,2,6)(3,4,5)$ | $-\zeta_{3} - 1$ |
| $1$ | $6$ | $(1,3,2,4,6,5)$ | $-\zeta_{3}$ |
| $1$ | $6$ | $(1,5,6,4,2,3)$ | $\zeta_{3} + 1$ |
The blue line marks the conjugacy class containing complex conjugation.