Basic invariants
| Dimension: | $2$ |
| Group: | $S_3\times C_3$ |
| Conductor: | \(1260\)\(\medspace = 2^{2} \cdot 3^{2} \cdot 5 \cdot 7 \) |
| Artin stem field: | Galois closure of 6.0.55566000.2 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $S_3\times C_3$ |
| Parity: | odd |
| Determinant: | 1.315.6t1.h.b |
| Projective image: | $S_3$ |
| Projective stem field: | Galois closure of 3.1.11340.2 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - x^{5} + 2x^{4} + 19x^{3} + 22x^{2} + 7x + 1 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 19 }$ to precision 7.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 19 }$:
\( x^{2} + 18x + 2 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 16 a + 11 + \left(6 a + 6\right)\cdot 19 + \left(16 a + 6\right)\cdot 19^{2} + \left(16 a + 12\right)\cdot 19^{3} + \left(9 a + 16\right)\cdot 19^{4} + 19^{5} + \left(4 a + 4\right)\cdot 19^{6} +O(19^{7})\)
|
| $r_{ 2 }$ | $=$ |
\( 14 a + 4 + \left(16 a + 1\right)\cdot 19 + \left(7 a + 17\right)\cdot 19^{2} + \left(9 a + 17\right)\cdot 19^{3} + \left(7 a + 4\right)\cdot 19^{4} + \left(5 a + 7\right)\cdot 19^{5} + \left(7 a + 14\right)\cdot 19^{6} +O(19^{7})\)
|
| $r_{ 3 }$ | $=$ |
\( 17 a + \left(14 a + 6\right)\cdot 19 + \left(11 a + 6\right)\cdot 19^{2} + \left(a + 2\right)\cdot 19^{3} + \left(11 a + 16\right)\cdot 19^{4} + \left(5 a + 8\right)\cdot 19^{5} + \left(10 a + 14\right)\cdot 19^{6} +O(19^{7})\)
|
| $r_{ 4 }$ | $=$ |
\( 3 a + 8 + \left(12 a + 16\right)\cdot 19 + \left(2 a + 15\right)\cdot 19^{2} + \left(2 a + 12\right)\cdot 19^{3} + \left(9 a + 9\right)\cdot 19^{4} + \left(18 a + 11\right)\cdot 19^{5} + \left(14 a + 7\right)\cdot 19^{6} +O(19^{7})\)
|
| $r_{ 5 }$ | $=$ |
\( 2 a + 17 + \left(4 a + 3\right)\cdot 19 + \left(7 a + 3\right)\cdot 19^{2} + \left(17 a + 11\right)\cdot 19^{3} + \left(7 a + 6\right)\cdot 19^{4} + \left(13 a + 3\right)\cdot 19^{5} + 8 a\cdot 19^{6} +O(19^{7})\)
|
| $r_{ 6 }$ | $=$ |
\( 5 a + 18 + \left(2 a + 3\right)\cdot 19 + \left(11 a + 8\right)\cdot 19^{2} + 9 a\cdot 19^{3} + \left(11 a + 3\right)\cdot 19^{4} + \left(13 a + 5\right)\cdot 19^{5} + \left(11 a + 16\right)\cdot 19^{6} +O(19^{7})\)
|
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 |
| $1$ | $1$ | $()$ | $2$ |
| $3$ | $2$ | $(1,5)(2,4)(3,6)$ | $0$ |
| $1$ | $3$ | $(1,3,2)(4,5,6)$ | $-2 \zeta_{3} - 2$ |
| $1$ | $3$ | $(1,2,3)(4,6,5)$ | $2 \zeta_{3}$ |
| $2$ | $3$ | $(1,3,2)$ | $-\zeta_{3}$ |
| $2$ | $3$ | $(1,2,3)$ | $\zeta_{3} + 1$ |
| $2$ | $3$ | $(1,2,3)(4,5,6)$ | $-1$ |
| $3$ | $6$ | $(1,6,3,4,2,5)$ | $0$ |
| $3$ | $6$ | $(1,5,2,4,3,6)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.