Basic invariants
| Dimension: | $1$ |
| Group: | $C_6$ |
| Conductor: | \(344\)\(\medspace = 2^{3} \cdot 43 \) |
| Artin field: | Galois closure of 6.6.1750426112.1 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $C_6$ |
| Parity: | even |
| Dirichlet character: | \(\chi_{344}(165,\cdot)\) |
| Projective image: | $C_1$ |
| Projective field: | Galois closure of \(\Q\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - 2x^{5} - 33x^{4} + 20x^{3} + 220x^{2} + 64x - 188 \)
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The roots of $f$ are computed in an extension of $\Q_{ 59 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 59 }$:
\( x^{2} + 58x + 2 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 49 a + 56 + \left(14 a + 13\right)\cdot 59 + \left(41 a + 9\right)\cdot 59^{2} + \left(28 a + 5\right)\cdot 59^{3} + \left(16 a + 16\right)\cdot 59^{4} +O(59^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 10 a + 23 + \left(44 a + 45\right)\cdot 59 + \left(17 a + 7\right)\cdot 59^{2} + \left(30 a + 22\right)\cdot 59^{3} + \left(42 a + 21\right)\cdot 59^{4} +O(59^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 10 a + 35 + \left(44 a + 41\right)\cdot 59 + \left(17 a + 25\right)\cdot 59^{2} + \left(30 a + 25\right)\cdot 59^{3} + \left(42 a + 15\right)\cdot 59^{4} +O(59^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 49 a + 33 + \left(14 a + 20\right)\cdot 59 + \left(41 a + 40\right)\cdot 59^{2} + \left(28 a + 34\right)\cdot 59^{3} + \left(16 a + 33\right)\cdot 59^{4} +O(59^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 10 a + 46 + \left(44 a + 38\right)\cdot 59 + \left(17 a + 35\right)\cdot 59^{2} + \left(30 a + 51\right)\cdot 59^{3} + \left(42 a + 3\right)\cdot 59^{4} +O(59^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 49 a + 45 + \left(14 a + 16\right)\cdot 59 + \left(41 a + 58\right)\cdot 59^{2} + \left(28 a + 37\right)\cdot 59^{3} + \left(16 a + 27\right)\cdot 59^{4} +O(59^{5})\)
|
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 |
| $1$ | $1$ | $()$ | $1$ |
| $1$ | $2$ | $(1,5)(2,4)(3,6)$ | $-1$ |
| $1$ | $3$ | $(1,4,6)(2,3,5)$ | $\zeta_{3}$ |
| $1$ | $3$ | $(1,6,4)(2,5,3)$ | $-\zeta_{3} - 1$ |
| $1$ | $6$ | $(1,3,4,5,6,2)$ | $\zeta_{3} + 1$ |
| $1$ | $6$ | $(1,2,6,5,4,3)$ | $-\zeta_{3}$ |
The blue line marks the conjugacy class containing complex conjugation.