Basic invariants
| Dimension: | $1$ |
| Group: | $C_6$ |
| Conductor: | \(28\)\(\medspace = 2^{2} \cdot 7 \) |
| Artin field: | Galois closure of 6.0.153664.1 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $C_6$ |
| Parity: | odd |
| Dirichlet character: | \(\chi_{28}(11,\cdot)\) |
| Projective image: | $C_1$ |
| Projective field: | Galois closure of \(\Q\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} + 5x^{4} + 6x^{2} + 1 \)
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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:
| $r_{ 1 }$ | $=$ |
\( 32 a + 27 + \left(29 a + 22\right)\cdot 43 + \left(10 a + 9\right)\cdot 43^{2} + \left(18 a + 39\right)\cdot 43^{3} + \left(27 a + 16\right)\cdot 43^{4} +O(43^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 10 a + 38 + \left(19 a + 16\right)\cdot 43 + \left(38 a + 33\right)\cdot 43^{2} + \left(15 a + 32\right)\cdot 43^{3} + \left(6 a + 4\right)\cdot 43^{4} +O(43^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 26 a + 30 + \left(35 a + 16\right)\cdot 43 + \left(36 a + 42\right)\cdot 43^{2} + \left(28 a + 3\right)\cdot 43^{3} + \left(4 a + 12\right)\cdot 43^{4} +O(43^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 11 a + 16 + \left(13 a + 20\right)\cdot 43 + \left(32 a + 33\right)\cdot 43^{2} + \left(24 a + 3\right)\cdot 43^{3} + \left(15 a + 26\right)\cdot 43^{4} +O(43^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 33 a + 5 + \left(23 a + 26\right)\cdot 43 + \left(4 a + 9\right)\cdot 43^{2} + \left(27 a + 10\right)\cdot 43^{3} + \left(36 a + 38\right)\cdot 43^{4} +O(43^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 17 a + 13 + \left(7 a + 26\right)\cdot 43 + 6 a\cdot 43^{2} + \left(14 a + 39\right)\cdot 43^{3} + \left(38 a + 30\right)\cdot 43^{4} +O(43^{5})\)
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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 | Complex conjugation |
| $1$ | $1$ | $()$ | $1$ | |
| $1$ | $2$ | $(1,4)(2,5)(3,6)$ | $-1$ | ✓ |
| $1$ | $3$ | $(1,5,6)(2,3,4)$ | $\zeta_{3}$ | |
| $1$ | $3$ | $(1,6,5)(2,4,3)$ | $-\zeta_{3} - 1$ | |
| $1$ | $6$ | $(1,3,5,4,6,2)$ | $\zeta_{3} + 1$ | |
| $1$ | $6$ | $(1,2,6,4,5,3)$ | $-\zeta_{3}$ |