Basic invariants
| Dimension: | $1$ |
| Group: | $C_6$ |
| Conductor: | \(483\)\(\medspace = 3 \cdot 7 \cdot 23 \) |
| Artin field: | Galois closure of 6.6.788750109.1 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $C_6$ |
| Parity: | even |
| Dirichlet character: | \(\chi_{483}(137,\cdot)\) |
| Projective image: | $C_1$ |
| Projective field: | Galois closure of \(\Q\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - x^{5} - 56x^{4} + 37x^{3} + 889x^{2} - 324x - 3809 \)
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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:
| $r_{ 1 }$ | $=$ |
\( 25 a + 14 + \left(16 a + 2\right)\cdot 29 + \left(26 a + 14\right)\cdot 29^{2} + 4 a\cdot 29^{3} + \left(9 a + 5\right)\cdot 29^{4} +O(29^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 25 a + 28 + \left(16 a + 27\right)\cdot 29 + \left(26 a + 27\right)\cdot 29^{2} + \left(4 a + 1\right)\cdot 29^{3} + \left(9 a + 10\right)\cdot 29^{4} +O(29^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 4 a + 8 + 12 a\cdot 29 + \left(2 a + 28\right)\cdot 29^{2} + \left(24 a + 28\right)\cdot 29^{3} + \left(19 a + 21\right)\cdot 29^{4} +O(29^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 25 a + 3 + \left(16 a + 11\right)\cdot 29 + \left(26 a + 1\right)\cdot 29^{2} + \left(4 a + 15\right)\cdot 29^{3} + \left(9 a + 10\right)\cdot 29^{4} +O(29^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 4 a + 23 + \left(12 a + 3\right)\cdot 29 + \left(2 a + 14\right)\cdot 29^{2} + \left(24 a + 27\right)\cdot 29^{3} + \left(19 a + 16\right)\cdot 29^{4} +O(29^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 4 a + 12 + \left(12 a + 12\right)\cdot 29 + \left(2 a + 1\right)\cdot 29^{2} + \left(24 a + 13\right)\cdot 29^{3} + \left(19 a + 22\right)\cdot 29^{4} +O(29^{5})\)
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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 |
| $1$ | $1$ | $()$ | $1$ |
| $1$ | $2$ | $(1,5)(2,3)(4,6)$ | $-1$ |
| $1$ | $3$ | $(1,4,2)(3,5,6)$ | $-\zeta_{3} - 1$ |
| $1$ | $3$ | $(1,2,4)(3,6,5)$ | $\zeta_{3}$ |
| $1$ | $6$ | $(1,6,2,5,4,3)$ | $\zeta_{3} + 1$ |
| $1$ | $6$ | $(1,3,4,5,2,6)$ | $-\zeta_{3}$ |
The blue line marks the conjugacy class containing complex conjugation.