Basic invariants
| Dimension: | $1$ |
| Group: | $C_6$ |
| Conductor: | \(79\) |
| Artin field: | Galois closure of 6.0.3077056399.1 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $C_6$ |
| Parity: | odd |
| Dirichlet character: | \(\chi_{79}(24,\cdot)\) |
| Projective image: | $C_1$ |
| Projective field: | Galois closure of \(\Q\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - x^{5} + 7x^{4} - 63x^{3} - 81x^{2} + 353x + 541 \)
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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:
| $r_{ 1 }$ | $=$ |
\( 39 a + 32 + \left(39 a + 37\right)\cdot 41 + \left(5 a + 12\right)\cdot 41^{2} + \left(27 a + 38\right)\cdot 41^{3} + \left(13 a + 7\right)\cdot 41^{4} +O(41^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 3 a + 16 + \left(37 a + 33\right)\cdot 41 + \left(31 a + 18\right)\cdot 41^{2} + \left(36 a + 17\right)\cdot 41^{3} + \left(18 a + 26\right)\cdot 41^{4} +O(41^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 6 a + 24 + \left(25 a + 4\right)\cdot 41 + \left(21 a + 12\right)\cdot 41^{2} + \left(11 a + 4\right)\cdot 41^{3} + \left(40 a + 17\right)\cdot 41^{4} +O(41^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 2 a + 26 + \left(a + 36\right)\cdot 41 + \left(35 a + 31\right)\cdot 41^{2} + \left(13 a + 31\right)\cdot 41^{3} + \left(27 a + 21\right)\cdot 41^{4} +O(41^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 35 a + 1 + \left(15 a + 33\right)\cdot 41 + \left(19 a + 10\right)\cdot 41^{2} + \left(29 a + 17\right)\cdot 41^{3} + 3\cdot 41^{4} +O(41^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 38 a + 25 + \left(3 a + 18\right)\cdot 41 + \left(9 a + 36\right)\cdot 41^{2} + \left(4 a + 13\right)\cdot 41^{3} + \left(22 a + 5\right)\cdot 41^{4} +O(41^{5})\)
|
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 |
| $1$ | $1$ | $()$ | $1$ |
| $1$ | $2$ | $(1,4)(2,6)(3,5)$ | $-1$ |
| $1$ | $3$ | $(1,3,2)(4,5,6)$ | $\zeta_{3}$ |
| $1$ | $3$ | $(1,2,3)(4,6,5)$ | $-\zeta_{3} - 1$ |
| $1$ | $6$ | $(1,5,2,4,3,6)$ | $-\zeta_{3}$ |
| $1$ | $6$ | $(1,6,3,4,2,5)$ | $\zeta_{3} + 1$ |
The blue line marks the conjugacy class containing complex conjugation.