Basic invariants
| Dimension: | $2$ |
| Group: | $S_3\times C_3$ |
| Conductor: | \(975\)\(\medspace = 3 \cdot 5^{2} \cdot 13 \) |
| Artin stem field: | Galois closure of 6.0.2851875.1 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $S_3\times C_3$ |
| Parity: | odd |
| Determinant: | 1.39.6t1.a.a |
| Projective image: | $S_3$ |
| Projective stem field: | Galois closure of 3.1.12675.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - 3x^{5} + 4x^{4} + 2x^{3} - 2x^{2} - 2x + 13 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 47 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 47 }$:
\( x^{2} + 45x + 5 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 3 a + 16 + \left(9 a + 46\right)\cdot 47 + \left(41 a + 22\right)\cdot 47^{2} + \left(11 a + 35\right)\cdot 47^{3} + \left(45 a + 45\right)\cdot 47^{4} +O(47^{5})\)
| $r_{ 2 }$ |
$=$ |
\( 32 a + 42 + \left(26 a + 45\right)\cdot 47 + \left(43 a + 19\right)\cdot 47^{2} + \left(4 a + 2\right)\cdot 47^{3} + \left(6 a + 2\right)\cdot 47^{4} +O(47^{5})\)
| $r_{ 3 }$ |
$=$ |
\( 44 a + 22 + \left(37 a + 14\right)\cdot 47 + \left(5 a + 2\right)\cdot 47^{2} + \left(35 a + 18\right)\cdot 47^{3} + \left(a + 30\right)\cdot 47^{4} +O(47^{5})\)
| $r_{ 4 }$ |
$=$ |
\( 32 a + 41 + \left(21 a + 24\right)\cdot 47 + \left(14 a + 27\right)\cdot 47^{2} + \left(32 a + 9\right)\cdot 47^{3} + \left(8 a + 34\right)\cdot 47^{4} +O(47^{5})\)
| $r_{ 5 }$ |
$=$ |
\( 15 a + 11 + \left(25 a + 36\right)\cdot 47 + \left(32 a + 34\right)\cdot 47^{2} + \left(14 a + 12\right)\cdot 47^{3} + \left(38 a + 19\right)\cdot 47^{4} +O(47^{5})\)
| $r_{ 6 }$ |
$=$ |
\( 15 a + 12 + \left(20 a + 20\right)\cdot 47 + \left(3 a + 33\right)\cdot 47^{2} + \left(42 a + 15\right)\cdot 47^{3} + \left(40 a + 9\right)\cdot 47^{4} +O(47^{5})\)
| |
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 |
| $1$ | $1$ | $()$ | $2$ |
| $3$ | $2$ | $(1,3)(2,6)(4,5)$ | $0$ |
| $1$ | $3$ | $(1,4,6)(2,3,5)$ | $-2 \zeta_{3} - 2$ |
| $1$ | $3$ | $(1,6,4)(2,5,3)$ | $2 \zeta_{3}$ |
| $2$ | $3$ | $(1,6,4)$ | $\zeta_{3} + 1$ |
| $2$ | $3$ | $(1,4,6)$ | $-\zeta_{3}$ |
| $2$ | $3$ | $(1,4,6)(2,5,3)$ | $-1$ |
| $3$ | $6$ | $(1,2,4,3,6,5)$ | $0$ |
| $3$ | $6$ | $(1,5,6,3,4,2)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.