Basic invariants
| Dimension: | $1$ |
| Group: | $C_6$ |
| Conductor: | \(63\)\(\medspace = 3^{2} \cdot 7 \) |
| Artin field: | Galois closure of 6.0.110270727.1 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $C_6$ |
| Parity: | odd |
| Dirichlet character: | \(\chi_{63}(61,\cdot)\) |
| Projective image: | $C_1$ |
| Projective field: | Galois closure of \(\Q\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - 14x^{3} + 63x^{2} + 168x + 161 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 101 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 101 }$:
\( x^{2} + 97x + 2 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 100 a + 28 + \left(35 a + 30\right)\cdot 101 + \left(35 a + 92\right)\cdot 101^{2} + \left(41 a + 55\right)\cdot 101^{3} + \left(74 a + 52\right)\cdot 101^{4} +O(101^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( a + 24 + \left(65 a + 74\right)\cdot 101 + \left(65 a + 96\right)\cdot 101^{2} + \left(59 a + 84\right)\cdot 101^{3} + \left(26 a + 5\right)\cdot 101^{4} +O(101^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 10 a + \left(24 a + 94\right)\cdot 101 + \left(83 a + 47\right)\cdot 101^{2} + \left(56 a + 54\right)\cdot 101^{3} + \left(42 a + 43\right)\cdot 101^{4} +O(101^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 91 a + 40 + \left(76 a + 79\right)\cdot 101 + \left(17 a + 53\right)\cdot 101^{2} + \left(44 a + 97\right)\cdot 101^{3} + \left(58 a + 55\right)\cdot 101^{4} +O(101^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 11 a + 33 + \left(89 a + 92\right)\cdot 101 + \left(47 a + 55\right)\cdot 101^{2} + \left(15 a + 48\right)\cdot 101^{3} + \left(69 a + 93\right)\cdot 101^{4} +O(101^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 90 a + 77 + \left(11 a + 33\right)\cdot 101 + \left(53 a + 57\right)\cdot 101^{2} + \left(85 a + 62\right)\cdot 101^{3} + \left(31 a + 51\right)\cdot 101^{4} +O(101^{5})\)
|
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 |
| $1$ | $1$ | $()$ | $1$ |
| $1$ | $2$ | $(1,2)(3,4)(5,6)$ | $-1$ |
| $1$ | $3$ | $(1,4,5)(2,3,6)$ | $\zeta_{3}$ |
| $1$ | $3$ | $(1,5,4)(2,6,3)$ | $-\zeta_{3} - 1$ |
| $1$ | $6$ | $(1,6,4,2,5,3)$ | $\zeta_{3} + 1$ |
| $1$ | $6$ | $(1,3,5,2,4,6)$ | $-\zeta_{3}$ |
The blue line marks the conjugacy class containing complex conjugation.