Basic invariants
| Dimension: | $1$ |
| Group: | $C_6$ |
| Conductor: | \(273\)\(\medspace = 3 \cdot 7 \cdot 13 \) |
| Artin field: | Galois closure of 6.0.142424919.1 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $C_6$ |
| Parity: | odd |
| Dirichlet character: | \(\chi_{273}(233,\cdot)\) |
| Projective image: | $C_1$ |
| Projective field: | Galois closure of \(\Q\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - x^{5} + 25x^{4} - 17x^{3} + 295x^{2} - 81x + 1429 \)
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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:
| $r_{ 1 }$ | $=$ |
\( 5 a + 20 + \left(14 a + 24\right)\cdot 29 + \left(22 a + 7\right)\cdot 29^{2} + \left(17 a + 11\right)\cdot 29^{3} + \left(3 a + 1\right)\cdot 29^{4} +O(29^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 24 a + 20 + \left(14 a + 15\right)\cdot 29 + \left(6 a + 21\right)\cdot 29^{2} + \left(11 a + 3\right)\cdot 29^{3} + \left(25 a + 2\right)\cdot 29^{4} +O(29^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 24 a + 2 + \left(14 a + 7\right)\cdot 29 + \left(6 a + 5\right)\cdot 29^{2} + \left(11 a + 18\right)\cdot 29^{3} + \left(25 a + 25\right)\cdot 29^{4} +O(29^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 5 a + 6 + \left(14 a + 28\right)\cdot 29 + \left(22 a + 22\right)\cdot 29^{2} + \left(17 a + 9\right)\cdot 29^{3} + \left(3 a + 25\right)\cdot 29^{4} +O(29^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 24 a + 16 + \left(14 a + 3\right)\cdot 29 + \left(6 a + 19\right)\cdot 29^{2} + \left(11 a + 19\right)\cdot 29^{3} + \left(25 a + 1\right)\cdot 29^{4} +O(29^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 5 a + 24 + \left(14 a + 7\right)\cdot 29 + \left(22 a + 10\right)\cdot 29^{2} + \left(17 a + 24\right)\cdot 29^{3} + \left(3 a + 1\right)\cdot 29^{4} +O(29^{5})\)
|
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 |
| $1$ | $1$ | $()$ | $1$ | |
| $1$ | $2$ | $(1,5)(2,6)(3,4)$ | $-1$ | ✓ |
| $1$ | $3$ | $(1,4,6)(2,5,3)$ | $\zeta_{3}$ | |
| $1$ | $3$ | $(1,6,4)(2,3,5)$ | $-\zeta_{3} - 1$ | |
| $1$ | $6$ | $(1,2,4,5,6,3)$ | $\zeta_{3} + 1$ | |
| $1$ | $6$ | $(1,3,6,5,4,2)$ | $-\zeta_{3}$ |