Basic invariants
| Dimension: | $2$ |
| Group: | $D_{6}$ |
| Conductor: | \(2856\)\(\medspace = 2^{3} \cdot 3 \cdot 7 \cdot 17 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 6.0.415993536.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $D_{6}$ |
| Parity: | odd |
| Determinant: | 1.2856.2t1.b.a |
| Projective image: | $S_3$ |
| Projective stem field: | Galois closure of 3.1.2856.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - x^{5} + 12x^{4} - 23x^{3} + 12x^{2} - x + 1 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 19 }$ to precision 8.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 19 }$:
\( x^{2} + 18x + 2 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 9 a + 6 + \left(6 a + 17\right)\cdot 19 + \left(18 a + 18\right)\cdot 19^{2} + \left(2 a + 13\right)\cdot 19^{3} + \left(7 a + 3\right)\cdot 19^{4} + 3 a\cdot 19^{5} + \left(2 a + 1\right)\cdot 19^{6} + \left(9 a + 17\right)\cdot 19^{7} +O(19^{8})\)
|
| $r_{ 2 }$ | $=$ |
\( 17 a + 5 + \left(11 a + 9\right)\cdot 19 + \left(13 a + 13\right)\cdot 19^{2} + \left(4 a + 12\right)\cdot 19^{3} + \left(9 a + 2\right)\cdot 19^{4} + \left(12 a + 7\right)\cdot 19^{5} + \left(6 a + 17\right)\cdot 19^{6} + \left(8 a + 11\right)\cdot 19^{7} +O(19^{8})\)
|
| $r_{ 3 }$ | $=$ |
\( 14 + 19 + 6\cdot 19^{2} + 19^{3} + 13\cdot 19^{4} + 2\cdot 19^{5} + 4\cdot 19^{6} + 15\cdot 19^{7} +O(19^{8})\)
|
| $r_{ 4 }$ | $=$ |
\( 15 + 9\cdot 19 + 10\cdot 19^{2} + 7\cdot 19^{3} + 3\cdot 19^{4} + 2\cdot 19^{5} + 4\cdot 19^{6} + 13\cdot 19^{7} +O(19^{8})\)
|
| $r_{ 5 }$ | $=$ |
\( 10 a + 15 + \left(12 a + 14\right)\cdot 19 + 11\cdot 19^{2} + \left(16 a + 17\right)\cdot 19^{3} + \left(11 a + 7\right)\cdot 19^{4} + \left(15 a + 15\right)\cdot 19^{5} + \left(16 a + 18\right)\cdot 19^{6} + \left(9 a + 4\right)\cdot 19^{7} +O(19^{8})\)
|
| $r_{ 6 }$ | $=$ |
\( 2 a + 3 + \left(7 a + 4\right)\cdot 19 + \left(5 a + 15\right)\cdot 19^{2} + \left(14 a + 3\right)\cdot 19^{3} + \left(9 a + 7\right)\cdot 19^{4} + \left(6 a + 10\right)\cdot 19^{5} + \left(12 a + 11\right)\cdot 19^{6} + \left(10 a + 13\right)\cdot 19^{7} +O(19^{8})\)
|
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 |
| $1$ | $1$ | $()$ | $2$ | |
| $1$ | $2$ | $(1,6)(2,5)(3,4)$ | $-2$ | |
| $3$ | $2$ | $(1,2)(3,4)(5,6)$ | $0$ | ✓ |
| $3$ | $2$ | $(1,4)(3,6)$ | $0$ | |
| $2$ | $3$ | $(1,5,4)(2,3,6)$ | $-1$ | |
| $2$ | $6$ | $(1,3,5,6,4,2)$ | $1$ |