Basic invariants
| Dimension: | $1$ |
| Group: | $C_6$ |
| Conductor: | \(532\)\(\medspace = 2^{2} \cdot 7 \cdot 19 \) |
| Artin field: | Galois closure of 6.0.7377869632.2 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $C_6$ |
| Parity: | odd |
| Dirichlet character: | \(\chi_{532}(75,\cdot)\) |
| Projective image: | $C_1$ |
| Projective field: | Galois closure of \(\Q\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} + 133x^{4} + 5054x^{2} + 48013 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 29 }$ to precision 6.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 29 }$:
\( x^{2} + 24x + 2 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 6 a + 14 + \left(8 a + 11\right)\cdot 29 + \left(14 a + 26\right)\cdot 29^{2} + \left(16 a + 23\right)\cdot 29^{3} + \left(23 a + 21\right)\cdot 29^{4} + \left(16 a + 27\right)\cdot 29^{5} +O(29^{6})\)
|
| $r_{ 2 }$ | $=$ |
\( 19 a + 25 + \left(18 a + 20\right)\cdot 29 + \left(11 a + 23\right)\cdot 29^{2} + \left(11 a + 20\right)\cdot 29^{3} + \left(23 a + 19\right)\cdot 29^{4} + \left(3 a + 16\right)\cdot 29^{5} +O(29^{6})\)
|
| $r_{ 3 }$ | $=$ |
\( 18 a + 13 + \left(21 a + 27\right)\cdot 29 + \left(3 a + 15\right)\cdot 29^{2} + \left(11 a + 17\right)\cdot 29^{3} + \left(7 a + 1\right)\cdot 29^{4} + \left(24 a + 1\right)\cdot 29^{5} +O(29^{6})\)
|
| $r_{ 4 }$ | $=$ |
\( 23 a + 15 + \left(20 a + 17\right)\cdot 29 + \left(14 a + 2\right)\cdot 29^{2} + \left(12 a + 5\right)\cdot 29^{3} + \left(5 a + 7\right)\cdot 29^{4} + \left(12 a + 1\right)\cdot 29^{5} +O(29^{6})\)
|
| $r_{ 5 }$ | $=$ |
\( 10 a + 4 + \left(10 a + 8\right)\cdot 29 + \left(17 a + 5\right)\cdot 29^{2} + \left(17 a + 8\right)\cdot 29^{3} + \left(5 a + 9\right)\cdot 29^{4} + \left(25 a + 12\right)\cdot 29^{5} +O(29^{6})\)
|
| $r_{ 6 }$ | $=$ |
\( 11 a + 16 + \left(7 a + 1\right)\cdot 29 + \left(25 a + 13\right)\cdot 29^{2} + \left(17 a + 11\right)\cdot 29^{3} + \left(21 a + 27\right)\cdot 29^{4} + \left(4 a + 27\right)\cdot 29^{5} +O(29^{6})\)
|
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 |
| $1$ | $1$ | $()$ | $1$ |
| $1$ | $2$ | $(1,4)(2,5)(3,6)$ | $-1$ |
| $1$ | $3$ | $(1,5,3)(2,6,4)$ | $\zeta_{3}$ |
| $1$ | $3$ | $(1,3,5)(2,4,6)$ | $-\zeta_{3} - 1$ |
| $1$ | $6$ | $(1,6,5,4,3,2)$ | $\zeta_{3} + 1$ |
| $1$ | $6$ | $(1,2,3,4,5,6)$ | $-\zeta_{3}$ |
The blue line marks the conjugacy class containing complex conjugation.