Basic invariants
| Dimension: | $1$ |
| Group: | $C_6$ |
| Conductor: | \(104\)\(\medspace = 2^{3} \cdot 13 \) |
| Artin field: | Galois closure of 6.6.14623232.1 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $C_6$ |
| Parity: | even |
| Dirichlet character: | \(\chi_{104}(61,\cdot)\) |
| Projective image: | $C_1$ |
| Projective field: | Galois closure of \(\Q\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - 2x^{5} - 13x^{4} + 14x^{3} + 26x^{2} - 28x + 1 \)
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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:
| $r_{ 1 }$ | $=$ |
\( 52 a + 16 + \left(25 a + 15\right)\cdot 53 + \left(29 a + 37\right)\cdot 53^{2} + \left(34 a + 49\right)\cdot 53^{3} + \left(38 a + 19\right)\cdot 53^{4} +O(53^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 52 a + 35 + \left(25 a + 29\right)\cdot 53 + \left(29 a + 52\right)\cdot 53^{2} + \left(34 a + 37\right)\cdot 53^{3} + \left(38 a + 50\right)\cdot 53^{4} +O(53^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( a + 5 + \left(27 a + 35\right)\cdot 53 + \left(23 a + 49\right)\cdot 53^{2} + \left(18 a + 16\right)\cdot 53^{3} + \left(14 a + 28\right)\cdot 53^{4} +O(53^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 52 a + 9 + \left(25 a + 36\right)\cdot 53 + \left(29 a + 10\right)\cdot 53^{2} + \left(34 a + 14\right)\cdot 53^{3} + \left(38 a + 14\right)\cdot 53^{4} +O(53^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( a + 31 + \left(27 a + 28\right)\cdot 53 + \left(23 a + 38\right)\cdot 53^{2} + \left(18 a + 40\right)\cdot 53^{3} + \left(14 a + 11\right)\cdot 53^{4} +O(53^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( a + 12 + \left(27 a + 14\right)\cdot 53 + \left(23 a + 23\right)\cdot 53^{2} + \left(18 a + 52\right)\cdot 53^{3} + \left(14 a + 33\right)\cdot 53^{4} +O(53^{5})\)
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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 |
| $1$ | $1$ | $()$ | $1$ |
| $1$ | $2$ | $(1,6)(2,5)(3,4)$ | $-1$ |
| $1$ | $3$ | $(1,4,2)(3,5,6)$ | $-\zeta_{3} - 1$ |
| $1$ | $3$ | $(1,2,4)(3,6,5)$ | $\zeta_{3}$ |
| $1$ | $6$ | $(1,5,4,6,2,3)$ | $-\zeta_{3}$ |
| $1$ | $6$ | $(1,3,2,6,4,5)$ | $\zeta_{3} + 1$ |
The blue line marks the conjugacy class containing complex conjugation.