Basic invariants
| Dimension: | $2$ |
| Group: | $D_{6}$ |
| Conductor: | \(1840\)\(\medspace = 2^{4} \cdot 5 \cdot 23 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 6.0.16928000.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $D_{6}$ |
| Parity: | odd |
| Determinant: | 1.115.2t1.a.a |
| Projective image: | $S_3$ |
| Projective stem field: | Galois closure of 3.1.460.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - 3x^{4} - 6x^{3} + 11x^{2} - 6x + 9 \)
|
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:
| $r_{ 1 }$ | $=$ |
\( 5 + 18\cdot 47^{2} + 11\cdot 47^{3} + 6\cdot 47^{4} +O(47^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 27 + 38\cdot 47 + 37\cdot 47^{2} + 28\cdot 47^{3} + 16\cdot 47^{4} +O(47^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 43 a + 5 + \left(33 a + 4\right)\cdot 47 + \left(21 a + 28\right)\cdot 47^{2} + \left(5 a + 36\right)\cdot 47^{3} + \left(42 a + 31\right)\cdot 47^{4} +O(47^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 44 a + 33 + \left(38 a + 17\right)\cdot 47 + \left(2 a + 26\right)\cdot 47^{2} + \left(15 a + 5\right)\cdot 47^{3} + \left(36 a + 6\right)\cdot 47^{4} +O(47^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 3 a + 27 + \left(8 a + 4\right)\cdot 47 + \left(44 a + 40\right)\cdot 47^{2} + \left(31 a + 32\right)\cdot 47^{3} + \left(10 a + 16\right)\cdot 47^{4} +O(47^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 4 a + 44 + \left(13 a + 28\right)\cdot 47 + \left(25 a + 37\right)\cdot 47^{2} + \left(41 a + 25\right)\cdot 47^{3} + \left(4 a + 16\right)\cdot 47^{4} +O(47^{5})\)
|
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 |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,2)(3,4)(5,6)$ | $-2$ |
| $3$ | $2$ | $(1,3)(2,4)(5,6)$ | $0$ |
| $3$ | $2$ | $(1,4)(2,3)$ | $0$ |
| $2$ | $3$ | $(1,4,5)(2,3,6)$ | $-1$ |
| $2$ | $6$ | $(1,6,4,2,5,3)$ | $1$ |
The blue line marks the conjugacy class containing complex conjugation.