Basic invariants
| Dimension: | $2$ |
| Group: | $S_3\times C_3$ |
| Conductor: | \(567\)\(\medspace = 3^{4} \cdot 7 \) |
| Artin stem field: | Galois closure of 6.0.964467.3 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $S_3\times C_3$ |
| Parity: | odd |
| Determinant: | 1.63.6t1.b.a |
| Projective image: | $S_3$ |
| Projective stem field: | Galois closure of 3.1.1323.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - 4x^{3} + 7 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 47 }$ to precision 6.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 47 }$:
\( x^{2} + 45x + 5 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 44 a + 14 + \left(18 a + 22\right)\cdot 47 + \left(35 a + 4\right)\cdot 47^{2} + \left(32 a + 22\right)\cdot 47^{3} + \left(25 a + 35\right)\cdot 47^{4} + \left(43 a + 16\right)\cdot 47^{5} +O(47^{6})\)
|
| $r_{ 2 }$ | $=$ |
\( 8 a + 21 + \left(17 a + 6\right)\cdot 47 + \left(12 a + 36\right)\cdot 47^{2} + \left(16 a + 37\right)\cdot 47^{3} + \left(24 a + 26\right)\cdot 47^{4} + \left(30 a + 13\right)\cdot 47^{5} +O(47^{6})\)
|
| $r_{ 3 }$ | $=$ |
\( 39 a + 37 + \left(29 a + 32\right)\cdot 47 + \left(34 a + 43\right)\cdot 47^{2} + \left(30 a + 10\right)\cdot 47^{3} + \left(22 a + 12\right)\cdot 47^{4} + \left(16 a + 3\right)\cdot 47^{5} +O(47^{6})\)
|
| $r_{ 4 }$ | $=$ |
\( 3 a + 8 + \left(28 a + 16\right)\cdot 47 + \left(11 a + 9\right)\cdot 47^{2} + \left(14 a + 5\right)\cdot 47^{3} + \left(21 a + 7\right)\cdot 47^{4} + \left(3 a + 31\right)\cdot 47^{5} +O(47^{6})\)
|
| $r_{ 5 }$ | $=$ |
\( 11 a + 43 + \left(45 a + 38\right)\cdot 47 + \left(23 a + 45\right)\cdot 47^{2} + \left(30 a + 13\right)\cdot 47^{3} + \left(45 a + 46\right)\cdot 47^{4} + \left(33 a + 26\right)\cdot 47^{5} +O(47^{6})\)
|
| $r_{ 6 }$ | $=$ |
\( 36 a + 18 + \left(a + 24\right)\cdot 47 + \left(23 a + 1\right)\cdot 47^{2} + \left(16 a + 4\right)\cdot 47^{3} + \left(a + 13\right)\cdot 47^{4} + \left(13 a + 2\right)\cdot 47^{5} +O(47^{6})\)
|
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 |
| $1$ | $1$ | $()$ | $2$ |
| $3$ | $2$ | $(1,6)(2,5)(3,4)$ | $0$ |
| $1$ | $3$ | $(1,3,5)(2,6,4)$ | $-2 \zeta_{3} - 2$ |
| $1$ | $3$ | $(1,5,3)(2,4,6)$ | $2 \zeta_{3}$ |
| $2$ | $3$ | $(1,5,3)$ | $\zeta_{3} + 1$ |
| $2$ | $3$ | $(1,3,5)$ | $-\zeta_{3}$ |
| $2$ | $3$ | $(1,3,5)(2,4,6)$ | $-1$ |
| $3$ | $6$ | $(1,2,3,6,5,4)$ | $0$ |
| $3$ | $6$ | $(1,4,5,6,3,2)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.