Basic invariants
| Dimension: | $1$ |
| Group: | $C_6$ |
| Conductor: | \(91\)\(\medspace = 7 \cdot 13 \) |
| Artin field: | Galois closure of 6.6.5274997.1 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $C_6$ |
| Parity: | even |
| Dirichlet character: | \(\chi_{91}(25,\cdot)\) |
| Projective image: | $C_1$ |
| Projective field: | Galois closure of \(\Q\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - x^{5} - 14x^{4} + 9x^{3} + 35x^{2} - 16x - 1 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 41 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 41 }$:
\( x^{2} + 38x + 6 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 12 a + 40 + \left(29 a + 29\right)\cdot 41 + \left(40 a + 29\right)\cdot 41^{2} + 9\cdot 41^{3} + \left(37 a + 37\right)\cdot 41^{4} +O(41^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 12 a + 17 + \left(29 a + 33\right)\cdot 41 + \left(40 a + 11\right)\cdot 41^{2} + 25\cdot 41^{3} + \left(37 a + 15\right)\cdot 41^{4} +O(41^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 29 a + 28 + \left(11 a + 21\right)\cdot 41 + 14\cdot 41^{2} + \left(40 a + 4\right)\cdot 41^{3} + \left(3 a + 15\right)\cdot 41^{4} +O(41^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 29 a + 12 + \left(11 a + 27\right)\cdot 41 + 22\cdot 41^{2} + \left(40 a + 28\right)\cdot 41^{3} + \left(3 a + 2\right)\cdot 41^{4} +O(41^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 29 a + 35 + \left(11 a + 23\right)\cdot 41 + 40\cdot 41^{2} + \left(40 a + 12\right)\cdot 41^{3} + \left(3 a + 24\right)\cdot 41^{4} +O(41^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 12 a + 33 + \left(29 a + 27\right)\cdot 41 + \left(40 a + 3\right)\cdot 41^{2} + 41^{3} + \left(37 a + 28\right)\cdot 41^{4} +O(41^{5})\)
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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 |
| $1$ | $1$ | $()$ | $1$ |
| $1$ | $2$ | $(1,5)(2,4)(3,6)$ | $-1$ |
| $1$ | $3$ | $(1,2,6)(3,5,4)$ | $\zeta_{3}$ |
| $1$ | $3$ | $(1,6,2)(3,4,5)$ | $-\zeta_{3} - 1$ |
| $1$ | $6$ | $(1,3,2,5,6,4)$ | $\zeta_{3} + 1$ |
| $1$ | $6$ | $(1,4,6,5,2,3)$ | $-\zeta_{3}$ |
The blue line marks the conjugacy class containing complex conjugation.