Basic invariants
| Dimension: | $1$ |
| Group: | $C_6$ |
| Conductor: | \(1001\)\(\medspace = 7 \cdot 11 \cdot 13 \) |
| Artin field: | Galois closure of 6.0.7021021007.3 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $C_6$ |
| Parity: | odd |
| Dirichlet character: | \(\chi_{1001}(142,\cdot)\) |
| Projective image: | $C_1$ |
| Projective field: | Galois closure of \(\Q\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - x^{5} + 103x^{4} - 69x^{3} + 3857x^{2} - 1225x + 51947 \)
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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:
| $r_{ 1 }$ | $=$ |
\( 16 a + 22 + 21 a\cdot 29 + \left(18 a + 7\right)\cdot 29^{2} + \left(12 a + 6\right)\cdot 29^{3} + \left(23 a + 2\right)\cdot 29^{4} +O(29^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 16 a + 11 + \left(21 a + 9\right)\cdot 29 + \left(18 a + 23\right)\cdot 29^{2} + \left(12 a + 20\right)\cdot 29^{3} + \left(23 a + 7\right)\cdot 29^{4} +O(29^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 13 a + 4 + \left(7 a + 14\right)\cdot 29 + \left(10 a + 8\right)\cdot 29^{2} + \left(16 a + 7\right)\cdot 29^{3} + \left(5 a + 25\right)\cdot 29^{4} +O(29^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 13 a + \left(7 a + 2\right)\cdot 29 + \left(10 a + 6\right)\cdot 29^{2} + \left(16 a + 23\right)\cdot 29^{3} + \left(5 a + 24\right)\cdot 29^{4} +O(29^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 16 a + 7 + \left(21 a + 26\right)\cdot 29 + \left(18 a + 20\right)\cdot 29^{2} + \left(12 a + 7\right)\cdot 29^{3} + \left(23 a + 7\right)\cdot 29^{4} +O(29^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 13 a + 15 + \left(7 a + 5\right)\cdot 29 + \left(10 a + 21\right)\cdot 29^{2} + \left(16 a + 21\right)\cdot 29^{3} + \left(5 a + 19\right)\cdot 29^{4} +O(29^{5})\)
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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 |
| $1$ | $1$ | $()$ | $1$ |
| $1$ | $2$ | $(1,6)(2,3)(4,5)$ | $-1$ |
| $1$ | $3$ | $(1,2,5)(3,4,6)$ | $\zeta_{3}$ |
| $1$ | $3$ | $(1,5,2)(3,6,4)$ | $-\zeta_{3} - 1$ |
| $1$ | $6$ | $(1,3,5,6,2,4)$ | $-\zeta_{3}$ |
| $1$ | $6$ | $(1,4,2,6,5,3)$ | $\zeta_{3} + 1$ |
The blue line marks the conjugacy class containing complex conjugation.