Basic invariants
| Dimension: | $1$ |
| Group: | $C_6$ |
| Conductor: | \(36\)\(\medspace = 2^{2} \cdot 3^{2} \) |
| Artin field: | Galois closure of 6.0.419904.1 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $C_6$ |
| Parity: | odd |
| Dirichlet character: | \(\chi_{36}(7,\cdot)\) |
| Projective image: | $C_1$ |
| Projective field: | Galois closure of \(\Q\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} + 6x^{4} + 9x^{2} + 1 \)
|
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:
| $r_{ 1 }$ | $=$ |
\( 16 a + 11 + \left(10 a + 2\right)\cdot 19 + \left(3 a + 13\right)\cdot 19^{2} + \left(3 a + 9\right)\cdot 19^{3} + \left(11 a + 5\right)\cdot 19^{4} +O(19^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 7 a + 6 + \left(7 a + 9\right)\cdot 19 + \left(12 a + 16\right)\cdot 19^{2} + \left(16 a + 16\right)\cdot 19^{3} + \left(3 a + 15\right)\cdot 19^{4} +O(19^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 15 a + 2 + 7\cdot 19 + \left(3 a + 8\right)\cdot 19^{2} + \left(18 a + 11\right)\cdot 19^{3} + \left(3 a + 16\right)\cdot 19^{4} +O(19^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 3 a + 8 + \left(8 a + 16\right)\cdot 19 + \left(15 a + 5\right)\cdot 19^{2} + \left(15 a + 9\right)\cdot 19^{3} + \left(7 a + 13\right)\cdot 19^{4} +O(19^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 12 a + 13 + \left(11 a + 9\right)\cdot 19 + \left(6 a + 2\right)\cdot 19^{2} + \left(2 a + 2\right)\cdot 19^{3} + \left(15 a + 3\right)\cdot 19^{4} +O(19^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 4 a + 17 + \left(18 a + 11\right)\cdot 19 + \left(15 a + 10\right)\cdot 19^{2} + 7\cdot 19^{3} + \left(15 a + 2\right)\cdot 19^{4} +O(19^{5})\)
|
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 | Complex conjugation |
| $1$ | $1$ | $()$ | $1$ | |
| $1$ | $2$ | $(1,4)(2,5)(3,6)$ | $-1$ | ✓ |
| $1$ | $3$ | $(1,3,2)(4,6,5)$ | $\zeta_{3}$ | |
| $1$ | $3$ | $(1,2,3)(4,5,6)$ | $-\zeta_{3} - 1$ | |
| $1$ | $6$ | $(1,6,2,4,3,5)$ | $-\zeta_{3}$ | |
| $1$ | $6$ | $(1,5,3,4,2,6)$ | $\zeta_{3} + 1$ |