Basic invariants
| Dimension: | $2$ |
| Group: | $D_{6}$ |
| Conductor: | \(2912\)\(\medspace = 2^{5} \cdot 7 \cdot 13 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 6.0.8479744.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $D_{6}$ |
| Parity: | odd |
| Determinant: | 1.728.2t1.b.a |
| Projective image: | $S_3$ |
| Projective stem field: | Galois closure of 3.1.728.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - 8x^{3} + 25x^{2} - 40x + 32 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 17 }$ to precision 7.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 17 }$:
\( x^{2} + 16x + 3 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 9 + 16\cdot 17 + 15\cdot 17^{2} + 12\cdot 17^{3} + 4\cdot 17^{4} + 14\cdot 17^{5} +O(17^{7})\)
|
| $r_{ 2 }$ | $=$ |
\( 15 + 12\cdot 17^{3} + 12\cdot 17^{4} + 10\cdot 17^{5} + 5\cdot 17^{6} +O(17^{7})\)
|
| $r_{ 3 }$ | $=$ |
\( 2 a + 3 + \left(16 a + 10\right)\cdot 17 + \left(6 a + 13\right)\cdot 17^{2} + \left(14 a + 6\right)\cdot 17^{3} + \left(6 a + 1\right)\cdot 17^{4} + \left(6 a + 10\right)\cdot 17^{5} + \left(12 a + 13\right)\cdot 17^{6} +O(17^{7})\)
|
| $r_{ 4 }$ | $=$ |
\( 15 a + 5 + 7\cdot 17 + \left(10 a + 4\right)\cdot 17^{2} + \left(2 a + 14\right)\cdot 17^{3} + \left(10 a + 10\right)\cdot 17^{4} + \left(10 a + 9\right)\cdot 17^{5} + \left(4 a + 2\right)\cdot 17^{6} +O(17^{7})\)
|
| $r_{ 5 }$ | $=$ |
\( 9 a + 5 + \left(16 a + 4\right)\cdot 17 + \left(4 a + 14\right)\cdot 17^{2} + \left(12 a + 15\right)\cdot 17^{3} + \left(7 a + 12\right)\cdot 17^{4} + \left(15 a + 7\right)\cdot 17^{5} + \left(9 a + 8\right)\cdot 17^{6} +O(17^{7})\)
|
| $r_{ 6 }$ | $=$ |
\( 8 a + 14 + 11\cdot 17 + \left(12 a + 2\right)\cdot 17^{2} + \left(4 a + 6\right)\cdot 17^{3} + \left(9 a + 8\right)\cdot 17^{4} + \left(a + 15\right)\cdot 17^{5} + \left(7 a + 2\right)\cdot 17^{6} +O(17^{7})\)
|
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 |
| $1$ | $1$ | $()$ | $2$ | |
| $1$ | $2$ | $(1,2)(3,5)(4,6)$ | $-2$ | |
| $3$ | $2$ | $(1,3)(2,5)$ | $0$ | |
| $3$ | $2$ | $(1,5)(2,3)(4,6)$ | $0$ | ✓ |
| $2$ | $3$ | $(1,4,3)(2,6,5)$ | $-1$ | |
| $2$ | $6$ | $(1,6,3,2,4,5)$ | $1$ |