Basic invariants
| Dimension: | $2$ |
| Group: | $S_3\times C_3$ |
| Conductor: | \(2527\)\(\medspace = 7 \cdot 19^{2} \) |
| Artin stem field: | Galois closure of 6.0.121328851.1 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $S_3\times C_3$ |
| Parity: | odd |
| Determinant: | 1.133.6t1.b.b |
| Projective image: | $S_3$ |
| Projective stem field: | Galois closure of 3.1.931.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - 3x^{5} - x^{4} - 12x^{3} + 71x^{2} + 96x + 163 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 59 }$ to precision 6.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 59 }$:
\( x^{2} + 58x + 2 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 55 a + 53 + \left(28 a + 33\right)\cdot 59 + \left(30 a + 56\right)\cdot 59^{2} + \left(48 a + 10\right)\cdot 59^{3} + \left(47 a + 30\right)\cdot 59^{4} + \left(56 a + 3\right)\cdot 59^{5} +O(59^{6})\)
|
| $r_{ 2 }$ | $=$ |
\( 9 a + 14 + \left(7 a + 20\right)\cdot 59 + \left(23 a + 13\right)\cdot 59^{2} + \left(49 a + 42\right)\cdot 59^{3} + \left(11 a + 48\right)\cdot 59^{4} + \left(3 a + 47\right)\cdot 59^{5} +O(59^{6})\)
|
| $r_{ 3 }$ | $=$ |
\( 4 a + 49 + \left(30 a + 7\right)\cdot 59 + \left(28 a + 58\right)\cdot 59^{2} + \left(10 a + 28\right)\cdot 59^{3} + \left(11 a + 29\right)\cdot 59^{4} + \left(2 a + 12\right)\cdot 59^{5} +O(59^{6})\)
|
| $r_{ 4 }$ | $=$ |
\( 25 a + 8 + \left(31 a + 45\right)\cdot 59 + \left(32 a + 38\right)\cdot 59^{2} + \left(2 a + 57\right)\cdot 59^{3} + \left(16 a + 21\right)\cdot 59^{4} + \left(38 a + 55\right)\cdot 59^{5} +O(59^{6})\)
|
| $r_{ 5 }$ | $=$ |
\( 50 a + 23 + \left(51 a + 18\right)\cdot 59 + \left(35 a + 29\right)\cdot 59^{2} + \left(9 a + 9\right)\cdot 59^{3} + \left(47 a + 11\right)\cdot 59^{4} + \left(55 a + 39\right)\cdot 59^{5} +O(59^{6})\)
|
| $r_{ 6 }$ | $=$ |
\( 34 a + 33 + \left(27 a + 51\right)\cdot 59 + \left(26 a + 39\right)\cdot 59^{2} + \left(56 a + 27\right)\cdot 59^{3} + \left(42 a + 35\right)\cdot 59^{4} + \left(20 a + 18\right)\cdot 59^{5} +O(59^{6})\)
|
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 |
| $1$ | $1$ | $()$ | $2$ |
| $3$ | $2$ | $(1,4)(2,6)(3,5)$ | $0$ |
| $1$ | $3$ | $(1,5,6)(2,4,3)$ | $2 \zeta_{3}$ |
| $1$ | $3$ | $(1,6,5)(2,3,4)$ | $-2 \zeta_{3} - 2$ |
| $2$ | $3$ | $(2,3,4)$ | $-\zeta_{3}$ |
| $2$ | $3$ | $(2,4,3)$ | $\zeta_{3} + 1$ |
| $2$ | $3$ | $(1,6,5)(2,4,3)$ | $-1$ |
| $3$ | $6$ | $(1,2,5,4,6,3)$ | $0$ |
| $3$ | $6$ | $(1,3,6,4,5,2)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.