Basic invariants
| Dimension: | $1$ |
| Group: | $C_6$ |
| Conductor: | \(168\)\(\medspace = 2^{3} \cdot 3 \cdot 7 \) |
| Artin field: | Galois closure of 6.6.232339968.1 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $C_6$ |
| Parity: | even |
| Dirichlet character: | \(\chi_{168}(101,\cdot)\) |
| Projective image: | $C_1$ |
| Projective field: | Galois closure of \(\Q\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - 42x^{4} + 504x^{2} - 1512 \)
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The roots of $f$ are computed in an extension of $\Q_{ 43 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 43 }$:
\( x^{2} + 42x + 3 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 27 a + 8 + \left(12 a + 7\right)\cdot 43 + \left(15 a + 20\right)\cdot 43^{2} + \left(40 a + 30\right)\cdot 43^{3} + \left(28 a + 5\right)\cdot 43^{4} +O(43^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 19 a + 12 + \left(42 a + 31\right)\cdot 43 + 20\cdot 43^{2} + \left(30 a + 28\right)\cdot 43^{3} + \left(27 a + 22\right)\cdot 43^{4} +O(43^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 42 a + 22 + \left(31 a + 26\right)\cdot 43 + \left(7 a + 33\right)\cdot 43^{2} + \left(31 a + 9\right)\cdot 43^{3} + \left(8 a + 11\right)\cdot 43^{4} +O(43^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 16 a + 35 + \left(30 a + 35\right)\cdot 43 + \left(27 a + 22\right)\cdot 43^{2} + \left(2 a + 12\right)\cdot 43^{3} + \left(14 a + 37\right)\cdot 43^{4} +O(43^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 24 a + 31 + 11\cdot 43 + \left(42 a + 22\right)\cdot 43^{2} + \left(12 a + 14\right)\cdot 43^{3} + \left(15 a + 20\right)\cdot 43^{4} +O(43^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( a + 21 + \left(11 a + 16\right)\cdot 43 + \left(35 a + 9\right)\cdot 43^{2} + \left(11 a + 33\right)\cdot 43^{3} + \left(34 a + 31\right)\cdot 43^{4} +O(43^{5})\)
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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 |
| $1$ | $1$ | $()$ | $1$ |
| $1$ | $2$ | $(1,4)(2,5)(3,6)$ | $-1$ |
| $1$ | $3$ | $(1,6,2)(3,5,4)$ | $\zeta_{3}$ |
| $1$ | $3$ | $(1,2,6)(3,4,5)$ | $-\zeta_{3} - 1$ |
| $1$ | $6$ | $(1,5,6,4,2,3)$ | $\zeta_{3} + 1$ |
| $1$ | $6$ | $(1,3,2,4,6,5)$ | $-\zeta_{3}$ |
The blue line marks the conjugacy class containing complex conjugation.