Basic invariants
| Dimension: | $1$ |
| Group: | $C_6$ |
| Conductor: | \(84\)\(\medspace = 2^{2} \cdot 3 \cdot 7 \) |
| Artin field: | Galois closure of 6.0.29042496.1 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $C_6$ |
| Parity: | odd |
| Dirichlet character: | \(\chi_{84}(47,\cdot)\) |
| Projective image: | $C_1$ |
| Projective field: | Galois closure of \(\Q\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} + 21x^{4} + 126x^{2} + 189 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 13 }$ to precision 6.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 13 }$:
\( x^{2} + 12x + 2 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 7 a + 3 + 3\cdot 13 + \left(7 a + 3\right)\cdot 13^{2} + \left(12 a + 10\right)\cdot 13^{3} + \left(5 a + 9\right)\cdot 13^{4} + \left(10 a + 10\right)\cdot 13^{5} +O(13^{6})\)
|
| $r_{ 2 }$ | $=$ |
\( 4 a + 11 + \left(10 a + 9\right)\cdot 13 + \left(a + 10\right)\cdot 13^{2} + \left(7 a + 3\right)\cdot 13^{3} + 6 a\cdot 13^{4} + \left(a + 9\right)\cdot 13^{5} +O(13^{6})\)
|
| $r_{ 3 }$ | $=$ |
\( 8 a + 9 + \left(2 a + 2\right)\cdot 13 + \left(5 a + 5\right)\cdot 13^{2} + \left(11 a + 3\right)\cdot 13^{3} + \left(a + 11\right)\cdot 13^{4} + \left(2 a + 12\right)\cdot 13^{5} +O(13^{6})\)
|
| $r_{ 4 }$ | $=$ |
\( 6 a + 10 + \left(12 a + 9\right)\cdot 13 + \left(5 a + 9\right)\cdot 13^{2} + 2\cdot 13^{3} + \left(7 a + 3\right)\cdot 13^{4} + \left(2 a + 2\right)\cdot 13^{5} +O(13^{6})\)
|
| $r_{ 5 }$ | $=$ |
\( 9 a + 2 + \left(2 a + 3\right)\cdot 13 + \left(11 a + 2\right)\cdot 13^{2} + \left(5 a + 9\right)\cdot 13^{3} + \left(6 a + 12\right)\cdot 13^{4} + \left(11 a + 3\right)\cdot 13^{5} +O(13^{6})\)
|
| $r_{ 6 }$ | $=$ |
\( 5 a + 4 + \left(10 a + 10\right)\cdot 13 + \left(7 a + 7\right)\cdot 13^{2} + \left(a + 9\right)\cdot 13^{3} + \left(11 a + 1\right)\cdot 13^{4} + 10 a\cdot 13^{5} +O(13^{6})\)
|
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 |
| $1$ | $1$ | $()$ | $1$ |
| $1$ | $2$ | $(1,4)(2,5)(3,6)$ | $-1$ |
| $1$ | $3$ | $(1,6,5)(2,4,3)$ | $-\zeta_{3} - 1$ |
| $1$ | $3$ | $(1,5,6)(2,3,4)$ | $\zeta_{3}$ |
| $1$ | $6$ | $(1,2,6,4,5,3)$ | $-\zeta_{3}$ |
| $1$ | $6$ | $(1,3,5,4,6,2)$ | $\zeta_{3} + 1$ |
The blue line marks the conjugacy class containing complex conjugation.