Basic invariants
| Dimension: | $2$ |
| Group: | $D_{6}$ |
| Conductor: | \(10975\)\(\medspace = 5^{2} \cdot 439 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 6.0.10575564875.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $D_{6}$ |
| Parity: | odd |
| Determinant: | 1.439.2t1.a.a |
| Projective image: | $S_3$ |
| Projective stem field: | Galois closure of 3.1.439.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - 2x^{5} - 17x^{4} + 71x^{3} + 28x^{2} - 477x + 1251 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 17 }$ to precision 8.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 17 }$:
\( x^{2} + 16x + 3 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 16 a + 7 + \left(7 a + 13\right)\cdot 17 + \left(7 a + 7\right)\cdot 17^{2} + 9\cdot 17^{3} + \left(3 a + 4\right)\cdot 17^{4} + \left(5 a + 13\right)\cdot 17^{5} + \left(15 a + 4\right)\cdot 17^{6} + \left(11 a + 6\right)\cdot 17^{7} +O(17^{8})\)
|
| $r_{ 2 }$ | $=$ |
\( 10 + 10\cdot 17 + 11\cdot 17^{2} + 10\cdot 17^{4} + 2\cdot 17^{5} + 10\cdot 17^{6} + 16\cdot 17^{7} +O(17^{8})\)
|
| $r_{ 3 }$ | $=$ |
\( a + 6 + \left(9 a + 5\right)\cdot 17 + \left(9 a + 7\right)\cdot 17^{2} + \left(16 a + 2\right)\cdot 17^{3} + \left(13 a + 7\right)\cdot 17^{4} + \left(11 a + 15\right)\cdot 17^{5} + \left(a + 14\right)\cdot 17^{6} + \left(5 a + 2\right)\cdot 17^{7} +O(17^{8})\)
|
| $r_{ 4 }$ | $=$ |
\( 5 + 15\cdot 17 + 17^{2} + 5\cdot 17^{3} + 5\cdot 17^{4} + 5\cdot 17^{5} + 14\cdot 17^{6} + 7\cdot 17^{7} +O(17^{8})\)
|
| $r_{ 5 }$ | $=$ |
\( 6 a + 1 + \left(7 a + 11\right)\cdot 17 + \left(16 a + 6\right)\cdot 17^{2} + 7\cdot 17^{3} + \left(10 a + 7\right)\cdot 17^{4} + \left(16 a + 12\right)\cdot 17^{5} + \left(7 a + 7\right)\cdot 17^{6} + \left(10 a + 7\right)\cdot 17^{7} +O(17^{8})\)
|
| $r_{ 6 }$ | $=$ |
\( 11 a + 7 + \left(9 a + 12\right)\cdot 17 + 15\cdot 17^{2} + \left(16 a + 8\right)\cdot 17^{3} + \left(6 a + 16\right)\cdot 17^{4} + 17^{5} + \left(9 a + 16\right)\cdot 17^{6} + \left(6 a + 9\right)\cdot 17^{7} +O(17^{8})\)
|
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 |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,6)(2,4)(3,5)$ | $-2$ |
| $3$ | $2$ | $(1,2)(3,5)(4,6)$ | $0$ |
| $3$ | $2$ | $(1,3)(5,6)$ | $0$ |
| $2$ | $3$ | $(1,4,3)(2,5,6)$ | $-1$ |
| $2$ | $6$ | $(1,5,4,6,3,2)$ | $1$ |
The blue line marks the conjugacy class containing complex conjugation.