Basic invariants
| Dimension: | $2$ |
| Group: | $S_3\times C_3$ |
| Conductor: | \(648\)\(\medspace = 2^{3} \cdot 3^{4} \) |
| Artin stem field: | Galois closure of 6.0.10077696.3 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $S_3\times C_3$ |
| Parity: | odd |
| Determinant: | 1.72.6t1.c.a |
| Projective image: | $S_3$ |
| Projective stem field: | Galois closure of 3.1.216.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} + 6x^{4} - 2x^{3} + 9x^{2} - 6x + 7 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 17 }$ to precision 6.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 17 }$:
\( x^{2} + 16x + 3 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 5 a + 2 + \left(2 a + 16\right)\cdot 17 + \left(7 a + 15\right)\cdot 17^{2} + \left(2 a + 4\right)\cdot 17^{3} + \left(2 a + 15\right)\cdot 17^{4} + \left(6 a + 8\right)\cdot 17^{5} +O(17^{6})\)
|
| $r_{ 2 }$ | $=$ |
\( 12 a + 7 + \left(14 a + 13\right)\cdot 17 + \left(9 a + 3\right)\cdot 17^{2} + 14 a\cdot 17^{3} + \left(14 a + 15\right)\cdot 17^{4} + \left(10 a + 12\right)\cdot 17^{5} +O(17^{6})\)
|
| $r_{ 3 }$ | $=$ |
\( 15 a + 16 + \left(15 a + 3\right)\cdot 17 + \left(16 a + 14\right)\cdot 17^{2} + \left(7 a + 4\right)\cdot 17^{3} + 9\cdot 17^{4} + \left(9 a + 14\right)\cdot 17^{5} +O(17^{6})\)
|
| $r_{ 4 }$ | $=$ |
\( 3 a + 13 + \left(a + 15\right)\cdot 17 + \left(7 a + 14\right)\cdot 17^{2} + \left(10 a + 3\right)\cdot 17^{3} + 2 a\cdot 17^{4} + \left(15 a + 15\right)\cdot 17^{5} +O(17^{6})\)
|
| $r_{ 5 }$ | $=$ |
\( 2 a + 14 + \left(a + 4\right)\cdot 17 + 15\cdot 17^{2} + \left(9 a + 12\right)\cdot 17^{3} + \left(16 a + 1\right)\cdot 17^{4} + \left(7 a + 6\right)\cdot 17^{5} +O(17^{6})\)
|
| $r_{ 6 }$ | $=$ |
\( 14 a + 16 + \left(15 a + 13\right)\cdot 17 + \left(9 a + 3\right)\cdot 17^{2} + \left(6 a + 7\right)\cdot 17^{3} + \left(14 a + 9\right)\cdot 17^{4} + \left(a + 10\right)\cdot 17^{5} +O(17^{6})\)
|
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 |
| $1$ | $1$ | $()$ | $2$ |
| $3$ | $2$ | $(1,5)(2,6)(3,4)$ | $0$ |
| $1$ | $3$ | $(1,6,3)(2,4,5)$ | $2 \zeta_{3}$ |
| $1$ | $3$ | $(1,3,6)(2,5,4)$ | $-2 \zeta_{3} - 2$ |
| $2$ | $3$ | $(1,6,3)$ | $\zeta_{3} + 1$ |
| $2$ | $3$ | $(1,3,6)$ | $-\zeta_{3}$ |
| $2$ | $3$ | $(1,3,6)(2,4,5)$ | $-1$ |
| $3$ | $6$ | $(1,2,6,4,3,5)$ | $0$ |
| $3$ | $6$ | $(1,5,3,4,6,2)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.