Basic invariants
| Dimension: | $1$ |
| Group: | $C_6$ |
| Conductor: | \(1260\)\(\medspace = 2^{2} \cdot 3^{2} \cdot 5 \cdot 7 \) |
| Artin field: | Galois closure of 6.0.126023688000.14 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $C_6$ |
| Parity: | odd |
| Dirichlet character: | \(\chi_{1260}(1159,\cdot)\) |
| Projective image: | $C_1$ |
| Projective field: | Galois closure of \(\Q\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - 27x^{4} - 56x^{3} + 516x^{2} + 2016x + 4164 \)
|
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:
| $r_{ 1 }$ | $=$ |
\( 7 a + 16 + \left(7 a + 4\right)\cdot 31 + \left(20 a + 26\right)\cdot 31^{2} + \left(25 a + 23\right)\cdot 31^{3} + \left(28 a + 4\right)\cdot 31^{4} +O(31^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 24 a + 9 + \left(23 a + 5\right)\cdot 31 + \left(10 a + 30\right)\cdot 31^{2} + \left(5 a + 14\right)\cdot 31^{3} + \left(2 a + 16\right)\cdot 31^{4} +O(31^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 7 a + 30 + \left(7 a + 1\right)\cdot 31 + \left(20 a + 20\right)\cdot 31^{2} + \left(25 a + 7\right)\cdot 31^{3} + \left(28 a + 9\right)\cdot 31^{4} +O(31^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 24 a + 30 + \left(23 a + 11\right)\cdot 31 + \left(10 a + 28\right)\cdot 31^{2} + \left(5 a + 23\right)\cdot 31^{3} + \left(2 a + 5\right)\cdot 31^{4} +O(31^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 24 a + 13 + \left(23 a + 9\right)\cdot 31 + \left(10 a + 22\right)\cdot 31^{2} + \left(5 a + 7\right)\cdot 31^{3} + \left(2 a + 10\right)\cdot 31^{4} +O(31^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 7 a + 26 + \left(7 a + 28\right)\cdot 31 + \left(20 a + 27\right)\cdot 31^{2} + \left(25 a + 14\right)\cdot 31^{3} + \left(28 a + 15\right)\cdot 31^{4} +O(31^{5})\)
|
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 |
| $1$ | $1$ | $()$ | $1$ |
| $1$ | $2$ | $(1,4)(2,6)(3,5)$ | $-1$ |
| $1$ | $3$ | $(1,6,3)(2,5,4)$ | $\zeta_{3}$ |
| $1$ | $3$ | $(1,3,6)(2,4,5)$ | $-\zeta_{3} - 1$ |
| $1$ | $6$ | $(1,5,6,4,3,2)$ | $\zeta_{3} + 1$ |
| $1$ | $6$ | $(1,2,3,4,6,5)$ | $-\zeta_{3}$ |
The blue line marks the conjugacy class containing complex conjugation.