Basic invariants
| Dimension: | $1$ |
| Group: | $C_6$ |
| Conductor: | \(21\)\(\medspace = 3 \cdot 7 \) |
| Artin field: | Galois closure of 6.0.64827.1 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $C_6$ |
| Parity: | odd |
| Dirichlet character: | \(\chi_{21}(11,\cdot)\) |
| Projective image: | $C_1$ |
| Projective field: | Galois closure of \(\Q\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - x^{5} + 3x^{4} + 5x^{2} - 2x + 1 \)
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The roots of $f$ are computed in an extension of $\Q_{ 29 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 29 }$:
\( x^{2} + 24x + 2 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 9 a + 12 + \left(16 a + 27\right)\cdot 29 + \left(22 a + 9\right)\cdot 29^{2} + \left(21 a + 22\right)\cdot 29^{3} + 10\cdot 29^{4} +O(29^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 18 a + 24 + \left(13 a + 19\right)\cdot 29 + \left(13 a + 8\right)\cdot 29^{2} + \left(10 a + 24\right)\cdot 29^{3} + \left(12 a + 16\right)\cdot 29^{4} +O(29^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 13 a + 24 + \left(13 a + 23\right)\cdot 29 + \left(21 a + 18\right)\cdot 29^{2} + \left(21 a + 6\right)\cdot 29^{3} + \left(28 a + 25\right)\cdot 29^{4} +O(29^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 20 a + 28 + \left(12 a + 12\right)\cdot 29 + \left(6 a + 19\right)\cdot 29^{2} + \left(7 a + 21\right)\cdot 29^{3} + \left(28 a + 21\right)\cdot 29^{4} +O(29^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 11 a + 27 + \left(15 a + 11\right)\cdot 29 + \left(15 a + 4\right)\cdot 29^{2} + \left(18 a + 5\right)\cdot 29^{3} + \left(16 a + 10\right)\cdot 29^{4} +O(29^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 16 a + 2 + \left(15 a + 20\right)\cdot 29 + \left(7 a + 25\right)\cdot 29^{2} + \left(7 a + 6\right)\cdot 29^{3} + 2\cdot 29^{4} +O(29^{5})\)
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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 | Complex conjugation |
| $1$ | $1$ | $()$ | $1$ | |
| $1$ | $2$ | $(1,4)(2,5)(3,6)$ | $-1$ | ✓ |
| $1$ | $3$ | $(1,6,2)(3,5,4)$ | $\zeta_{3}$ | |
| $1$ | $3$ | $(1,2,6)(3,4,5)$ | $-\zeta_{3} - 1$ | |
| $1$ | $6$ | $(1,5,6,4,2,3)$ | $\zeta_{3} + 1$ | |
| $1$ | $6$ | $(1,3,2,4,6,5)$ | $-\zeta_{3}$ |