Basic invariants
| Dimension: | $2$ |
| Group: | $S_3\times C_3$ |
| Conductor: | \(2015\)\(\medspace = 5 \cdot 13 \cdot 31 \) |
| Artin stem field: | Galois closure of 6.0.629334875.1 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $S_3\times C_3$ |
| Parity: | odd |
| Determinant: | 1.2015.6t1.a.a |
| Projective image: | $S_3$ |
| Projective stem field: | Galois closure of 3.1.26195.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - 2x^{5} - 6x^{4} + 4x^{3} + 54x^{2} - 67x + 41 \)
|
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:
| $r_{ 1 }$ | $=$ |
\( 32 a + 23 + \left(31 a + 18\right)\cdot 47 + \left(14 a + 27\right)\cdot 47^{2} + \left(42 a + 42\right)\cdot 47^{3} + \left(38 a + 37\right)\cdot 47^{4} + \left(15 a + 39\right)\cdot 47^{5} +O(47^{6})\)
|
| $r_{ 2 }$ | $=$ |
\( 26 a + 35 + \left(24 a + 33\right)\cdot 47 + \left(46 a + 9\right)\cdot 47^{2} + \left(45 a + 15\right)\cdot 47^{3} + \left(14 a + 10\right)\cdot 47^{4} + \left(22 a + 26\right)\cdot 47^{5} +O(47^{6})\)
|
| $r_{ 3 }$ | $=$ |
\( 41 a + 32 + \left(39 a + 18\right)\cdot 47 + \left(31 a + 35\right)\cdot 47^{2} + \left(3 a + 37\right)\cdot 47^{3} + \left(23 a + 14\right)\cdot 47^{4} + \left(6 a + 45\right)\cdot 47^{5} +O(47^{6})\)
|
| $r_{ 4 }$ | $=$ |
\( 21 a + 40 + \left(22 a + 9\right)\cdot 47 + 31\cdot 47^{2} + \left(a + 13\right)\cdot 47^{3} + \left(32 a + 41\right)\cdot 47^{4} + \left(24 a + 8\right)\cdot 47^{5} +O(47^{6})\)
|
| $r_{ 5 }$ | $=$ |
\( 15 a + 40 + \left(15 a + 2\right)\cdot 47 + \left(32 a + 25\right)\cdot 47^{2} + \left(4 a + 18\right)\cdot 47^{3} + \left(8 a + 26\right)\cdot 47^{4} + \left(31 a + 32\right)\cdot 47^{5} +O(47^{6})\)
|
| $r_{ 6 }$ | $=$ |
\( 6 a + 20 + \left(7 a + 10\right)\cdot 47 + \left(15 a + 12\right)\cdot 47^{2} + \left(43 a + 13\right)\cdot 47^{3} + \left(23 a + 10\right)\cdot 47^{4} + \left(40 a + 35\right)\cdot 47^{5} +O(47^{6})\)
|
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 |
| $1$ | $1$ | $()$ | $2$ |
| $3$ | $2$ | $(1,2)(3,5)(4,6)$ | $0$ |
| $1$ | $3$ | $(1,3,4)(2,5,6)$ | $2 \zeta_{3}$ |
| $1$ | $3$ | $(1,4,3)(2,6,5)$ | $-2 \zeta_{3} - 2$ |
| $2$ | $3$ | $(1,4,3)(2,5,6)$ | $-1$ |
| $2$ | $3$ | $(2,5,6)$ | $\zeta_{3} + 1$ |
| $2$ | $3$ | $(2,6,5)$ | $-\zeta_{3}$ |
| $3$ | $6$ | $(1,5,3,6,4,2)$ | $0$ |
| $3$ | $6$ | $(1,2,4,6,3,5)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.