Basic invariants
| Dimension: | $1$ |
| Group: | $C_6$ |
| Conductor: | \(312\)\(\medspace = 2^{3} \cdot 3 \cdot 13 \) |
| Artin field: | Galois closure of 6.6.5132754432.1 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $C_6$ |
| Parity: | even |
| Dirichlet character: | \(\chi_{312}(179,\cdot)\) |
| Projective image: | $C_1$ |
| Projective field: | Galois closure of \(\Q\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - 78x^{4} + 936x^{2} - 2808 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 47 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 47 }$:
\( x^{2} + 45x + 5 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 9 a + 38 + \left(43 a + 31\right)\cdot 47 + \left(3 a + 17\right)\cdot 47^{2} + \left(9 a + 16\right)\cdot 47^{3} + \left(9 a + 42\right)\cdot 47^{4} +O(47^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 13 a + 34 + 29 a\cdot 47 + \left(45 a + 16\right)\cdot 47^{2} + \left(12 a + 33\right)\cdot 47^{3} + \left(42 a + 34\right)\cdot 47^{4} +O(47^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 41 a + 6 + \left(31 a + 12\right)\cdot 47 + \left(37 a + 25\right)\cdot 47^{2} + \left(31 a + 10\right)\cdot 47^{3} + \left(34 a + 28\right)\cdot 47^{4} +O(47^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 38 a + 9 + \left(3 a + 15\right)\cdot 47 + \left(43 a + 29\right)\cdot 47^{2} + \left(37 a + 30\right)\cdot 47^{3} + \left(37 a + 4\right)\cdot 47^{4} +O(47^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 34 a + 13 + \left(17 a + 46\right)\cdot 47 + \left(a + 30\right)\cdot 47^{2} + \left(34 a + 13\right)\cdot 47^{3} + \left(4 a + 12\right)\cdot 47^{4} +O(47^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 6 a + 41 + \left(15 a + 34\right)\cdot 47 + \left(9 a + 21\right)\cdot 47^{2} + \left(15 a + 36\right)\cdot 47^{3} + \left(12 a + 18\right)\cdot 47^{4} +O(47^{5})\)
|
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 | Complex conjugation |
| $1$ | $1$ | $()$ | $1$ | ✓ |
| $1$ | $2$ | $(1,4)(2,5)(3,6)$ | $-1$ | |
| $1$ | $3$ | $(1,6,5)(2,4,3)$ | $-\zeta_{3} - 1$ | |
| $1$ | $3$ | $(1,5,6)(2,3,4)$ | $\zeta_{3}$ | |
| $1$ | $6$ | $(1,2,6,4,5,3)$ | $-\zeta_{3}$ | |
| $1$ | $6$ | $(1,3,5,4,6,2)$ | $\zeta_{3} + 1$ |