Basic invariants
| Dimension: | $1$ |
| Group: | $C_6$ |
| Conductor: | \(252\)\(\medspace = 2^{2} \cdot 3^{2} \cdot 7 \) |
| Artin field: | Galois closure of 6.0.432081216.1 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $C_6$ |
| Parity: | odd |
| Dirichlet character: | \(\chi_{252}(167,\cdot)\) |
| Projective image: | $C_1$ |
| Projective field: | Galois closure of \(\Q\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} + 42x^{4} + 441x^{2} + 1029 \)
|
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:
| $r_{ 1 }$ | $=$ |
\( 31 a + 44 + \left(36 a + 21\right)\cdot 53 + \left(51 a + 47\right)\cdot 53^{2} + \left(22 a + 32\right)\cdot 53^{3} + \left(32 a + 52\right)\cdot 53^{4} +O(53^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 25 a + 3 + \left(15 a + 8\right)\cdot 53 + \left(49 a + 15\right)\cdot 53^{2} + \left(22 a + 5\right)\cdot 53^{3} + \left(38 a + 14\right)\cdot 53^{4} +O(53^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 47 a + 12 + \left(31 a + 39\right)\cdot 53 + \left(50 a + 20\right)\cdot 53^{2} + \left(52 a + 25\right)\cdot 53^{3} + \left(5 a + 14\right)\cdot 53^{4} +O(53^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 22 a + 9 + \left(16 a + 31\right)\cdot 53 + \left(a + 5\right)\cdot 53^{2} + \left(30 a + 20\right)\cdot 53^{3} + 20 a\cdot 53^{4} +O(53^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 28 a + 50 + \left(37 a + 44\right)\cdot 53 + \left(3 a + 37\right)\cdot 53^{2} + \left(30 a + 47\right)\cdot 53^{3} + \left(14 a + 38\right)\cdot 53^{4} +O(53^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 6 a + 41 + \left(21 a + 13\right)\cdot 53 + \left(2 a + 32\right)\cdot 53^{2} + 27\cdot 53^{3} + \left(47 a + 38\right)\cdot 53^{4} +O(53^{5})\)
|
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 |
| $1$ | $1$ | $()$ | $1$ |
| $1$ | $2$ | $(1,4)(2,5)(3,6)$ | $-1$ |
| $1$ | $3$ | $(1,3,5)(2,4,6)$ | $-\zeta_{3} - 1$ |
| $1$ | $3$ | $(1,5,3)(2,6,4)$ | $\zeta_{3}$ |
| $1$ | $6$ | $(1,2,3,4,5,6)$ | $-\zeta_{3}$ |
| $1$ | $6$ | $(1,6,5,4,3,2)$ | $\zeta_{3} + 1$ |
The blue line marks the conjugacy class containing complex conjugation.