Basic invariants
| Dimension: | $1$ |
| Group: | $C_6$ |
| Conductor: | \(36\)\(\medspace = 2^{2} \cdot 3^{2} \) |
| Artin field: | Galois closure of \(\Q(\zeta_{36})^+\) |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $C_6$ |
| Parity: | even |
| Dirichlet character: | \(\chi_{36}(11,\cdot)\) |
| Projective image: | $C_1$ |
| Projective field: | Galois closure of \(\Q\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - 6x^{4} + 9x^{2} - 3 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 17 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 17 }$:
\( x^{2} + 16x + 3 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 14 a + 10 + \left(5 a + 12\right)\cdot 17 + \left(11 a + 5\right)\cdot 17^{2} + 5\cdot 17^{3} + \left(12 a + 11\right)\cdot 17^{4} +O(17^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 12 a + 11 + \left(3 a + 12\right)\cdot 17 + \left(9 a + 5\right)\cdot 17^{2} + \left(2 a + 3\right)\cdot 17^{3} + \left(6 a + 15\right)\cdot 17^{4} +O(17^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 2 a + 16 + \left(2 a + 16\right)\cdot 17 + \left(2 a + 16\right)\cdot 17^{2} + \left(15 a + 1\right)\cdot 17^{3} + \left(5 a + 13\right)\cdot 17^{4} +O(17^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 3 a + 7 + \left(11 a + 4\right)\cdot 17 + \left(5 a + 11\right)\cdot 17^{2} + \left(16 a + 11\right)\cdot 17^{3} + \left(4 a + 5\right)\cdot 17^{4} +O(17^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 5 a + 6 + \left(13 a + 4\right)\cdot 17 + \left(7 a + 11\right)\cdot 17^{2} + \left(14 a + 13\right)\cdot 17^{3} + \left(10 a + 1\right)\cdot 17^{4} +O(17^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 15 a + 1 + 14 a\cdot 17 + 14 a\cdot 17^{2} + \left(a + 15\right)\cdot 17^{3} + \left(11 a + 3\right)\cdot 17^{4} +O(17^{5})\)
|
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 |
| $1$ | $1$ | $()$ | $1$ |
| $1$ | $2$ | $(1,4)(2,5)(3,6)$ | $-1$ |
| $1$ | $3$ | $(1,5,6)(2,3,4)$ | $-\zeta_{3} - 1$ |
| $1$ | $3$ | $(1,6,5)(2,4,3)$ | $\zeta_{3}$ |
| $1$ | $6$ | $(1,3,5,4,6,2)$ | $-\zeta_{3}$ |
| $1$ | $6$ | $(1,2,6,4,5,3)$ | $\zeta_{3} + 1$ |
The blue line marks the conjugacy class containing complex conjugation.