Basic invariants
| Dimension: | $1$ |
| Group: | $C_6$ |
| Conductor: | \(252\)\(\medspace = 2^{2} \cdot 3^{2} \cdot 7 \) |
| Artin field: | Galois closure of 6.0.21171979584.1 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $C_6$ |
| Parity: | odd |
| Dirichlet character: | \(\chi_{252}(59,\cdot)\) |
| Projective image: | $C_1$ |
| Projective field: | Galois closure of \(\Q\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} + 42x^{4} + 441x^{2} + 525 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 127 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 127 }$:
\( x^{2} + 126x + 3 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 61 a + 33 + \left(58 a + 1\right)\cdot 127 + \left(69 a + 58\right)\cdot 127^{2} + \left(34 a + 17\right)\cdot 127^{3} + \left(123 a + 19\right)\cdot 127^{4} +O(127^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 62 a + 96 + \left(117 a + 35\right)\cdot 127 + \left(77 a + 83\right)\cdot 127^{2} + \left(119 a + 42\right)\cdot 127^{3} + \left(45 a + 100\right)\cdot 127^{4} +O(127^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( a + 63 + \left(59 a + 34\right)\cdot 127 + \left(8 a + 25\right)\cdot 127^{2} + \left(85 a + 25\right)\cdot 127^{3} + \left(49 a + 81\right)\cdot 127^{4} +O(127^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 66 a + 94 + \left(68 a + 125\right)\cdot 127 + \left(57 a + 68\right)\cdot 127^{2} + \left(92 a + 109\right)\cdot 127^{3} + \left(3 a + 107\right)\cdot 127^{4} +O(127^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 65 a + 31 + \left(9 a + 91\right)\cdot 127 + \left(49 a + 43\right)\cdot 127^{2} + \left(7 a + 84\right)\cdot 127^{3} + \left(81 a + 26\right)\cdot 127^{4} +O(127^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 126 a + 64 + \left(67 a + 92\right)\cdot 127 + \left(118 a + 101\right)\cdot 127^{2} + \left(41 a + 101\right)\cdot 127^{3} + \left(77 a + 45\right)\cdot 127^{4} +O(127^{5})\)
|
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 |
| $1$ | $1$ | $()$ | $1$ |
| $1$ | $2$ | $(1,4)(2,5)(3,6)$ | $-1$ |
| $1$ | $3$ | $(1,5,3)(2,6,4)$ | $\zeta_{3}$ |
| $1$ | $3$ | $(1,3,5)(2,4,6)$ | $-\zeta_{3} - 1$ |
| $1$ | $6$ | $(1,6,5,4,3,2)$ | $\zeta_{3} + 1$ |
| $1$ | $6$ | $(1,2,3,4,5,6)$ | $-\zeta_{3}$ |
The blue line marks the conjugacy class containing complex conjugation.