Basic invariants
| Dimension: | $2$ |
| Group: | $S_3\times C_3$ |
| Conductor: | \(3240\)\(\medspace = 2^{3} \cdot 3^{4} \cdot 5 \) |
| Artin stem field: | Galois closure of 6.0.1259712000.3 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $S_3\times C_3$ |
| Parity: | odd |
| Determinant: | 1.360.6t1.b.b |
| Projective image: | $S_3$ |
| Projective stem field: | Galois closure of 3.1.1080.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} + 6x^{4} - 6x^{3} + 9x^{2} - 18x + 39 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 19 }$ to precision 6.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 19 }$:
\( x^{2} + 18x + 2 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 17 a + 11 + \left(12 a + 7\right)\cdot 19 + \left(13 a + 9\right)\cdot 19^{2} + \left(3 a + 4\right)\cdot 19^{3} + \left(11 a + 16\right)\cdot 19^{4} + 16\cdot 19^{5} +O(19^{6})\)
|
| $r_{ 2 }$ | $=$ |
\( 13 a + 1 + \left(8 a + 2\right)\cdot 19 + \left(18 a + 9\right)\cdot 19^{2} + \left(6 a + 13\right)\cdot 19^{3} + \left(13 a + 10\right)\cdot 19^{4} + \left(16 a + 17\right)\cdot 19^{5} +O(19^{6})\)
|
| $r_{ 3 }$ | $=$ |
\( 15 a + 13 + \left(14 a + 13\right)\cdot 19 + \left(4 a + 9\right)\cdot 19^{2} + \left(3 a + 12\right)\cdot 19^{3} + \left(2 a + 4\right)\cdot 19^{4} + 16 a\cdot 19^{5} +O(19^{6})\)
|
| $r_{ 4 }$ | $=$ |
\( 2 a + 9 + \left(6 a + 3\right)\cdot 19 + \left(5 a + 10\right)\cdot 19^{2} + \left(15 a + 13\right)\cdot 19^{3} + \left(7 a + 4\right)\cdot 19^{4} + \left(18 a + 6\right)\cdot 19^{5} +O(19^{6})\)
|
| $r_{ 5 }$ | $=$ |
\( 6 a + 14 + \left(10 a + 16\right)\cdot 19 + 18\cdot 19^{2} + \left(12 a + 1\right)\cdot 19^{3} + \left(5 a + 17\right)\cdot 19^{4} + \left(2 a + 1\right)\cdot 19^{5} +O(19^{6})\)
|
| $r_{ 6 }$ | $=$ |
\( 4 a + 9 + \left(4 a + 13\right)\cdot 19 + \left(14 a + 18\right)\cdot 19^{2} + \left(15 a + 10\right)\cdot 19^{3} + \left(16 a + 3\right)\cdot 19^{4} + \left(2 a + 14\right)\cdot 19^{5} +O(19^{6})\)
|
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 |
| $1$ | $1$ | $()$ | $2$ |
| $3$ | $2$ | $(1,6)(2,3)(4,5)$ | $0$ |
| $1$ | $3$ | $(1,3,5)(2,4,6)$ | $-2 \zeta_{3} - 2$ |
| $1$ | $3$ | $(1,5,3)(2,6,4)$ | $2 \zeta_{3}$ |
| $2$ | $3$ | $(1,3,5)$ | $-\zeta_{3}$ |
| $2$ | $3$ | $(1,5,3)$ | $\zeta_{3} + 1$ |
| $2$ | $3$ | $(1,5,3)(2,4,6)$ | $-1$ |
| $3$ | $6$ | $(1,4,3,6,5,2)$ | $0$ |
| $3$ | $6$ | $(1,2,5,6,3,4)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.