Basic invariants
| Dimension: | $2$ |
| Group: | $D_{6}$ |
| Conductor: | \(87\)\(\medspace = 3 \cdot 29 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 6.0.22707.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $D_{6}$ |
| Parity: | odd |
| Determinant: | 1.87.2t1.a.a |
| Projective image: | $S_3$ |
| Projective stem field: | Galois closure of 3.1.87.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - x^{5} + 4x^{4} - 4x^{3} + 5x^{2} - 3x + 1 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 19 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 19 }$:
\( x^{2} + 18x + 2 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 6 a + 8 + \left(7 a + 12\right)\cdot 19 + \left(15 a + 9\right)\cdot 19^{2} + \left(3 a + 10\right)\cdot 19^{3} + \left(9 a + 18\right)\cdot 19^{4} +O(19^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 11 a + 1 + \left(2 a + 8\right)\cdot 19 + 6 a\cdot 19^{2} + \left(16 a + 17\right)\cdot 19^{3} + \left(16 a + 13\right)\cdot 19^{4} +O(19^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 18 + 14\cdot 19 + 6\cdot 19^{2} + 12\cdot 19^{3} + 8\cdot 19^{4} +O(19^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 5 + 8\cdot 19 + 18\cdot 19^{2} + 9\cdot 19^{3} + 15\cdot 19^{4} +O(19^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 13 a + 14 + \left(11 a + 13\right)\cdot 19 + \left(3 a + 17\right)\cdot 19^{2} + \left(15 a + 17\right)\cdot 19^{3} + \left(9 a + 4\right)\cdot 19^{4} +O(19^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 8 a + 12 + \left(16 a + 18\right)\cdot 19 + \left(12 a + 3\right)\cdot 19^{2} + \left(2 a + 8\right)\cdot 19^{3} + \left(2 a + 14\right)\cdot 19^{4} +O(19^{5})\)
|
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 |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,2)(3,4)(5,6)$ | $-2$ |
| $3$ | $2$ | $(3,6)(4,5)$ | $0$ |
| $3$ | $2$ | $(1,2)(3,5)(4,6)$ | $0$ |
| $2$ | $3$ | $(1,5,4)(2,6,3)$ | $-1$ |
| $2$ | $6$ | $(1,3,5,2,4,6)$ | $1$ |
The blue line marks the conjugacy class containing complex conjugation.