Basic invariants
| Dimension: | $1$ |
| Group: | $C_6$ |
| Conductor: | \(936\)\(\medspace = 2^{3} \cdot 3^{2} \cdot 13 \) |
| Artin field: | Galois closure of 6.0.7380232704.12 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $C_6$ |
| Parity: | odd |
| Dirichlet character: | \(\chi_{936}(259,\cdot)\) |
| Projective image: | $C_1$ |
| Projective field: | Galois closure of \(\Q\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} + 72x^{4} - 2x^{3} + 2037x^{2} + 162x + 21867 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 19 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 19 }$:
\( x^{2} + 18x + 2 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 2 a + 15 + 8 a\cdot 19 + \left(14 a + 2\right)\cdot 19^{2} + 17\cdot 19^{3} + \left(a + 10\right)\cdot 19^{4} +O(19^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 17 a + 17 + \left(10 a + 6\right)\cdot 19 + \left(4 a + 8\right)\cdot 19^{2} + \left(18 a + 3\right)\cdot 19^{3} + \left(17 a + 11\right)\cdot 19^{4} +O(19^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 17 a + 11 + \left(10 a + 13\right)\cdot 19 + \left(4 a + 1\right)\cdot 19^{2} + 18 a\cdot 19^{3} + \left(17 a + 11\right)\cdot 19^{4} +O(19^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 17 a + 13 + \left(10 a + 7\right)\cdot 19 + \left(4 a + 18\right)\cdot 19^{2} + \left(18 a + 13\right)\cdot 19^{3} + \left(17 a + 6\right)\cdot 19^{4} +O(19^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 2 a + 9 + \left(8 a + 7\right)\cdot 19 + \left(14 a + 14\right)\cdot 19^{2} + 13\cdot 19^{3} + \left(a + 10\right)\cdot 19^{4} +O(19^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 2 a + 11 + \left(8 a + 1\right)\cdot 19 + \left(14 a + 12\right)\cdot 19^{2} + 8\cdot 19^{3} + \left(a + 6\right)\cdot 19^{4} +O(19^{5})\)
|
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 |
| $1$ | $1$ | $()$ | $1$ |
| $1$ | $2$ | $(1,2)(3,5)(4,6)$ | $-1$ |
| $1$ | $3$ | $(1,6,5)(2,4,3)$ | $-\zeta_{3} - 1$ |
| $1$ | $3$ | $(1,5,6)(2,3,4)$ | $\zeta_{3}$ |
| $1$ | $6$ | $(1,3,6,2,5,4)$ | $-\zeta_{3}$ |
| $1$ | $6$ | $(1,4,5,2,6,3)$ | $\zeta_{3} + 1$ |
The blue line marks the conjugacy class containing complex conjugation.