Basic invariants
| Dimension: | $1$ |
| Group: | $C_6$ |
| Conductor: | \(2933\)\(\medspace = 7 \cdot 419 \) |
| Artin field: | Galois closure of 6.0.176617701659.2 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $C_6$ |
| Parity: | odd |
| Dirichlet character: | \(\chi_{2933}(837,\cdot)\) |
| Projective image: | $C_1$ |
| Projective field: | Galois closure of \(\Q\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - x^{5} + 310x^{4} - 207x^{3} + 32975x^{2} - 10816x + 1202039 \)
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The roots of $f$ are computed in an extension of $\Q_{ 71 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 71 }$:
\( x^{2} + 69x + 7 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 60 a + 61 + \left(2 a + 4\right)\cdot 71 + \left(33 a + 1\right)\cdot 71^{2} + \left(32 a + 9\right)\cdot 71^{3} + \left(38 a + 59\right)\cdot 71^{4} +O(71^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 11 a + 6 + \left(68 a + 20\right)\cdot 71 + \left(37 a + 15\right)\cdot 71^{2} + 38 a\cdot 71^{3} + \left(32 a + 47\right)\cdot 71^{4} +O(71^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 60 a + 51 + \left(2 a + 37\right)\cdot 71 + \left(33 a + 58\right)\cdot 71^{2} + \left(32 a + 45\right)\cdot 71^{3} + \left(38 a + 13\right)\cdot 71^{4} +O(71^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 11 a + 29 + \left(68 a + 54\right)\cdot 71 + \left(37 a + 50\right)\cdot 71^{2} + \left(38 a + 6\right)\cdot 71^{3} + \left(32 a + 58\right)\cdot 71^{4} +O(71^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 11 a + 39 + \left(68 a + 21\right)\cdot 71 + \left(37 a + 64\right)\cdot 71^{2} + \left(38 a + 40\right)\cdot 71^{3} + \left(32 a + 32\right)\cdot 71^{4} +O(71^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 60 a + 28 + \left(2 a + 3\right)\cdot 71 + \left(33 a + 23\right)\cdot 71^{2} + \left(32 a + 39\right)\cdot 71^{3} + \left(38 a + 2\right)\cdot 71^{4} +O(71^{5})\)
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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 |
| $1$ | $1$ | $()$ | $1$ |
| $1$ | $2$ | $(1,5)(2,6)(3,4)$ | $-1$ |
| $1$ | $3$ | $(1,3,6)(2,5,4)$ | $-\zeta_{3} - 1$ |
| $1$ | $3$ | $(1,6,3)(2,4,5)$ | $\zeta_{3}$ |
| $1$ | $6$ | $(1,2,3,5,6,4)$ | $-\zeta_{3}$ |
| $1$ | $6$ | $(1,4,6,5,3,2)$ | $\zeta_{3} + 1$ |
The blue line marks the conjugacy class containing complex conjugation.