Basic invariants
| Dimension: | $1$ |
| Group: | $C_6$ |
| Conductor: | \(104\)\(\medspace = 2^{3} \cdot 13 \) |
| Artin field: | Galois closure of 6.0.190102016.1 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $C_6$ |
| Parity: | odd |
| Dirichlet character: | \(\chi_{104}(75,\cdot)\) |
| Projective image: | $C_1$ |
| Projective field: | Galois closure of \(\Q\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} + 26x^{4} + 104x^{2} + 104 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 53 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 53 }$:
\( x^{2} + 49x + 2 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 52 a + 2 + 12 a\cdot 53 + \left(22 a + 15\right)\cdot 53^{2} + \left(46 a + 24\right)\cdot 53^{3} + \left(13 a + 48\right)\cdot 53^{4} +O(53^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 19 a + 15 + \left(15 a + 5\right)\cdot 53 + \left(21 a + 18\right)\cdot 53^{2} + \left(28 a + 33\right)\cdot 53^{3} + \left(5 a + 29\right)\cdot 53^{4} +O(53^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 46 a + 14 + \left(28 a + 18\right)\cdot 53 + \left(30 a + 6\right)\cdot 53^{2} + \left(51 a + 18\right)\cdot 53^{3} + \left(30 a + 43\right)\cdot 53^{4} +O(53^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( a + 51 + \left(40 a + 52\right)\cdot 53 + \left(30 a + 37\right)\cdot 53^{2} + \left(6 a + 28\right)\cdot 53^{3} + \left(39 a + 4\right)\cdot 53^{4} +O(53^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 34 a + 38 + \left(37 a + 47\right)\cdot 53 + \left(31 a + 34\right)\cdot 53^{2} + \left(24 a + 19\right)\cdot 53^{3} + \left(47 a + 23\right)\cdot 53^{4} +O(53^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 7 a + 39 + \left(24 a + 34\right)\cdot 53 + \left(22 a + 46\right)\cdot 53^{2} + \left(a + 34\right)\cdot 53^{3} + \left(22 a + 9\right)\cdot 53^{4} +O(53^{5})\)
|
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 |
| $1$ | $1$ | $()$ | $1$ |
| $1$ | $2$ | $(1,4)(2,5)(3,6)$ | $-1$ |
| $1$ | $3$ | $(1,3,5)(2,4,6)$ | $\zeta_{3}$ |
| $1$ | $3$ | $(1,5,3)(2,6,4)$ | $-\zeta_{3} - 1$ |
| $1$ | $6$ | $(1,2,3,4,5,6)$ | $\zeta_{3} + 1$ |
| $1$ | $6$ | $(1,6,5,4,3,2)$ | $-\zeta_{3}$ |
The blue line marks the conjugacy class containing complex conjugation.