Basic invariants
| Dimension: | $1$ |
| Group: | $C_6$ |
| Conductor: | \(840\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \cdot 7 \) |
| Artin field: | Galois closure of 6.0.4148928000.4 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $C_6$ |
| Parity: | odd |
| Dirichlet character: | \(\chi_{840}(149,\cdot)\) |
| Projective image: | $C_1$ |
| Projective field: | Galois closure of \(\Q\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - 2x^{5} + 87x^{4} - 114x^{3} + 2762x^{2} - 1864x + 31681 \)
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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:
| $r_{ 1 }$ | $=$ |
\( 34 a + 1 + \left(28 a + 31\right)\cdot 41 + \left(38 a + 8\right)\cdot 41^{2} + \left(30 a + 11\right)\cdot 41^{3} + \left(14 a + 33\right)\cdot 41^{4} +O(41^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 7 a + 21 + \left(12 a + 1\right)\cdot 41 + \left(2 a + 14\right)\cdot 41^{2} + \left(10 a + 24\right)\cdot 41^{3} + \left(26 a + 5\right)\cdot 41^{4} +O(41^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 7 a + 37 + \left(12 a + 36\right)\cdot 41 + \left(2 a + 5\right)\cdot 41^{2} + 10 a\cdot 41^{3} + \left(26 a + 18\right)\cdot 41^{4} +O(41^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 7 a + 14 + \left(12 a + 40\right)\cdot 41 + \left(2 a + 28\right)\cdot 41^{2} + \left(10 a + 15\right)\cdot 41^{3} + \left(26 a + 37\right)\cdot 41^{4} +O(41^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 34 a + 17 + \left(28 a + 25\right)\cdot 41 + 38 a\cdot 41^{2} + \left(30 a + 28\right)\cdot 41^{3} + \left(14 a + 4\right)\cdot 41^{4} +O(41^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 34 a + 35 + \left(28 a + 28\right)\cdot 41 + \left(38 a + 23\right)\cdot 41^{2} + \left(30 a + 2\right)\cdot 41^{3} + \left(14 a + 24\right)\cdot 41^{4} +O(41^{5})\)
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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 |
| $1$ | $1$ | $()$ | $1$ |
| $1$ | $2$ | $(1,2)(3,5)(4,6)$ | $-1$ |
| $1$ | $3$ | $(1,6,5)(2,4,3)$ | $-\zeta_{3} - 1$ |
| $1$ | $3$ | $(1,5,6)(2,3,4)$ | $\zeta_{3}$ |
| $1$ | $6$ | $(1,3,6,2,5,4)$ | $-\zeta_{3}$ |
| $1$ | $6$ | $(1,4,5,2,6,3)$ | $\zeta_{3} + 1$ |
The blue line marks the conjugacy class containing complex conjugation.