Basic invariants
| Dimension: | $1$ |
| Group: | $C_6$ |
| Conductor: | \(344\)\(\medspace = 2^{3} \cdot 43 \) |
| Artin field: | Galois closure of 6.6.75268322816.1 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $C_6$ |
| Parity: | even |
| Dirichlet character: | \(\chi_{344}(179,\cdot)\) |
| Projective image: | $C_1$ |
| Projective field: | Galois closure of \(\Q\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - 86x^{4} + 1376x^{2} - 5504 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 47 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 47 }$:
\( x^{2} + 45x + 5 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 32 a + 15 + \left(a + 14\right)\cdot 47 + \left(22 a + 2\right)\cdot 47^{2} + \left(28 a + 6\right)\cdot 47^{3} + \left(29 a + 8\right)\cdot 47^{4} +O(47^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 19 a + 28 + \left(24 a + 8\right)\cdot 47 + \left(28 a + 7\right)\cdot 47^{2} + \left(18 a + 19\right)\cdot 47^{3} + \left(40 a + 39\right)\cdot 47^{4} +O(47^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 11 a + 36 + \left(46 a + 29\right)\cdot 47 + \left(29 a + 16\right)\cdot 47^{2} + \left(5 a + 9\right)\cdot 47^{3} + \left(a + 25\right)\cdot 47^{4} +O(47^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 15 a + 32 + \left(45 a + 32\right)\cdot 47 + \left(24 a + 44\right)\cdot 47^{2} + \left(18 a + 40\right)\cdot 47^{3} + \left(17 a + 38\right)\cdot 47^{4} +O(47^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 28 a + 19 + \left(22 a + 38\right)\cdot 47 + \left(18 a + 39\right)\cdot 47^{2} + \left(28 a + 27\right)\cdot 47^{3} + \left(6 a + 7\right)\cdot 47^{4} +O(47^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 36 a + 11 + 17\cdot 47 + \left(17 a + 30\right)\cdot 47^{2} + \left(41 a + 37\right)\cdot 47^{3} + \left(45 a + 21\right)\cdot 47^{4} +O(47^{5})\)
|
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 |
| $1$ | $1$ | $()$ | $1$ |
| $1$ | $2$ | $(1,4)(2,5)(3,6)$ | $-1$ |
| $1$ | $3$ | $(1,6,2)(3,5,4)$ | $\zeta_{3}$ |
| $1$ | $3$ | $(1,2,6)(3,4,5)$ | $-\zeta_{3} - 1$ |
| $1$ | $6$ | $(1,3,2,4,6,5)$ | $-\zeta_{3}$ |
| $1$ | $6$ | $(1,5,6,4,2,3)$ | $\zeta_{3} + 1$ |
The blue line marks the conjugacy class containing complex conjugation.