Basic invariants
| Dimension: | $1$ |
| Group: | $C_6$ |
| Conductor: | \(63\)\(\medspace = 3^{2} \cdot 7 \) |
| Artin field: | Galois closure of 6.6.330812181.2 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $C_6$ |
| Parity: | even |
| Dirichlet character: | \(\chi_{63}(5,\cdot)\) |
| Projective image: | $C_1$ |
| Projective field: | Galois closure of \(\Q\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - 21x^{4} - 14x^{3} + 63x^{2} + 84x + 28 \)
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The roots of $f$ are computed in an extension of $\Q_{ 31 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 31 }$:
\( x^{2} + 29x + 3 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 4 a + 26 + \left(28 a + 11\right)\cdot 31 + \left(16 a + 21\right)\cdot 31^{2} + \left(3 a + 20\right)\cdot 31^{3} + \left(15 a + 9\right)\cdot 31^{4} +O(31^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 9 a + 26 + \left(19 a + 19\right)\cdot 31 + \left(25 a + 24\right)\cdot 31^{2} + \left(9 a + 29\right)\cdot 31^{3} + \left(25 a + 7\right)\cdot 31^{4} +O(31^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 13 a + 15 + \left(16 a + 10\right)\cdot 31 + \left(11 a + 9\right)\cdot 31^{2} + \left(13 a + 27\right)\cdot 31^{3} + \left(9 a + 7\right)\cdot 31^{4} +O(31^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 18 a + 10 + \left(14 a + 30\right)\cdot 31 + \left(19 a + 15\right)\cdot 31^{2} + \left(17 a + 11\right)\cdot 31^{3} + \left(21 a + 13\right)\cdot 31^{4} +O(31^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 27 a + 3 + \left(2 a + 2\right)\cdot 31 + \left(14 a + 27\right)\cdot 31^{2} + \left(27 a + 10\right)\cdot 31^{3} + \left(15 a + 5\right)\cdot 31^{4} +O(31^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 22 a + 13 + \left(11 a + 18\right)\cdot 31 + \left(5 a + 25\right)\cdot 31^{2} + \left(21 a + 23\right)\cdot 31^{3} + \left(5 a + 17\right)\cdot 31^{4} +O(31^{5})\)
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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 |
| $1$ | $1$ | $()$ | $1$ |
| $1$ | $2$ | $(1,5)(2,6)(3,4)$ | $-1$ |
| $1$ | $3$ | $(1,2,4)(3,5,6)$ | $\zeta_{3}$ |
| $1$ | $3$ | $(1,4,2)(3,6,5)$ | $-\zeta_{3} - 1$ |
| $1$ | $6$ | $(1,6,4,5,2,3)$ | $-\zeta_{3}$ |
| $1$ | $6$ | $(1,3,2,5,4,6)$ | $\zeta_{3} + 1$ |
The blue line marks the conjugacy class containing complex conjugation.