Basic invariants
| Dimension: | $2$ |
| Group: | $S_3\times C_3$ |
| Conductor: | \(2268\)\(\medspace = 2^{2} \cdot 3^{4} \cdot 7 \) |
| Artin stem field: | Galois closure of 6.0.15431472.2 |
| 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 3.1.588.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - 8x^{3} + 28 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 47 }$ to precision 7.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 47 }$:
\( x^{2} + 45x + 5 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 31 a + 12 + \left(45 a + 27\right)\cdot 47 + \left(22 a + 1\right)\cdot 47^{2} + \left(39 a + 3\right)\cdot 47^{3} + 4 a\cdot 47^{4} + 45\cdot 47^{5} + \left(12 a + 29\right)\cdot 47^{6} +O(47^{7})\)
|
| $r_{ 2 }$ | $=$ |
\( 27 a + 18 + \left(14 a + 14\right)\cdot 47 + \left(10 a + 37\right)\cdot 47^{2} + \left(18 a + 32\right)\cdot 47^{3} + \left(15 a + 23\right)\cdot 47^{4} + \left(23 a + 26\right)\cdot 47^{5} + \left(35 a + 25\right)\cdot 47^{6} +O(47^{7})\)
|
| $r_{ 3 }$ | $=$ |
\( 20 a + 25 + \left(32 a + 16\right)\cdot 47 + \left(36 a + 43\right)\cdot 47^{2} + \left(28 a + 11\right)\cdot 47^{3} + \left(31 a + 36\right)\cdot 47^{4} + \left(23 a + 10\right)\cdot 47^{5} + \left(11 a + 26\right)\cdot 47^{6} +O(47^{7})\)
|
| $r_{ 4 }$ | $=$ |
\( 16 a + 27 + \left(a + 40\right)\cdot 47 + \left(24 a + 1\right)\cdot 47^{2} + \left(7 a + 12\right)\cdot 47^{3} + \left(42 a + 17\right)\cdot 47^{4} + \left(46 a + 40\right)\cdot 47^{5} + \left(34 a + 6\right)\cdot 47^{6} +O(47^{7})\)
|
| $r_{ 5 }$ | $=$ |
\( 43 a + 10 + \left(15 a + 3\right)\cdot 47 + \left(34 a + 2\right)\cdot 47^{2} + \left(25 a + 32\right)\cdot 47^{3} + \left(10 a + 10\right)\cdot 47^{4} + \left(23 a + 38\right)\cdot 47^{5} + \left(23 a + 37\right)\cdot 47^{6} +O(47^{7})\)
|
| $r_{ 6 }$ | $=$ |
\( 4 a + 2 + \left(31 a + 39\right)\cdot 47 + \left(12 a + 7\right)\cdot 47^{2} + \left(21 a + 2\right)\cdot 47^{3} + \left(36 a + 6\right)\cdot 47^{4} + \left(23 a + 27\right)\cdot 47^{5} + \left(23 a + 14\right)\cdot 47^{6} +O(47^{7})\)
|
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 | Complex conjugation |
| $1$ | $1$ | $()$ | $2$ | |
| $3$ | $2$ | $(1,2)(3,6)(4,5)$ | $0$ | ✓ |
| $1$ | $3$ | $(1,3,5)(2,6,4)$ | $2 \zeta_{3}$ | |
| $1$ | $3$ | $(1,5,3)(2,4,6)$ | $-2 \zeta_{3} - 2$ | |
| $2$ | $3$ | $(1,5,3)$ | $-\zeta_{3}$ | |
| $2$ | $3$ | $(1,3,5)$ | $\zeta_{3} + 1$ | |
| $2$ | $3$ | $(1,3,5)(2,4,6)$ | $-1$ | |
| $3$ | $6$ | $(1,6,5,2,3,4)$ | $0$ | |
| $3$ | $6$ | $(1,4,3,2,5,6)$ | $0$ |