Basic invariants
| Dimension: | $2$ |
| Group: | $D_{6}$ |
| Conductor: | \(648\)\(\medspace = 2^{3} \cdot 3^{4} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 6.0.1259712.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $D_{6}$ |
| Parity: | odd |
| Determinant: | 1.8.2t1.b.a |
| Projective image: | $S_3$ |
| Projective stem field: | Galois closure of 3.1.648.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - 3x^{4} + 9x^{2} - 18x + 12 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 7 }$ to precision 8.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 7 }$:
\( x^{2} + 6x + 3 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 3 + 2\cdot 7 + 6\cdot 7^{2} + 6\cdot 7^{3} + 6\cdot 7^{4} + 6\cdot 7^{5} + 6\cdot 7^{6} + 7^{7} +O(7^{8})\)
|
| $r_{ 2 }$ | $=$ |
\( 2 + 5\cdot 7 + 4\cdot 7^{2} + 6\cdot 7^{3} + 2\cdot 7^{4} + 3\cdot 7^{5} + 7^{6} + 4\cdot 7^{7} +O(7^{8})\)
|
| $r_{ 3 }$ | $=$ |
\( 2 a + 1 + 3\cdot 7 + \left(2 a + 6\right)\cdot 7^{2} + \left(5 a + 1\right)\cdot 7^{4} + \left(5 a + 3\right)\cdot 7^{5} + \left(3 a + 4\right)\cdot 7^{6} +O(7^{8})\)
|
| $r_{ 4 }$ | $=$ |
\( 6 a + 3 + \left(a + 6\right)\cdot 7 + \left(5 a + 2\right)\cdot 7^{2} + \left(2 a + 1\right)\cdot 7^{3} + \left(a + 6\right)\cdot 7^{4} + 5\cdot 7^{5} + \left(a + 5\right)\cdot 7^{6} + \left(6 a + 5\right)\cdot 7^{7} +O(7^{8})\)
|
| $r_{ 5 }$ | $=$ |
\( 5 a + 3 + \left(6 a + 1\right)\cdot 7 + \left(4 a + 1\right)\cdot 7^{2} + \left(6 a + 6\right)\cdot 7^{3} + \left(a + 5\right)\cdot 7^{4} + \left(a + 3\right)\cdot 7^{5} + \left(3 a + 2\right)\cdot 7^{6} + \left(6 a + 4\right)\cdot 7^{7} +O(7^{8})\)
|
| $r_{ 6 }$ | $=$ |
\( a + 2 + \left(5 a + 2\right)\cdot 7 + \left(a + 6\right)\cdot 7^{2} + \left(4 a + 5\right)\cdot 7^{3} + \left(5 a + 4\right)\cdot 7^{4} + \left(6 a + 4\right)\cdot 7^{5} + \left(5 a + 6\right)\cdot 7^{6} + 3\cdot 7^{7} +O(7^{8})\)
|
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 |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,2)(3,4)(5,6)$ | $-2$ |
| $3$ | $2$ | $(1,3)(2,4)$ | $0$ |
| $3$ | $2$ | $(1,4)(2,3)(5,6)$ | $0$ |
| $2$ | $3$ | $(1,5,3)(2,6,4)$ | $-1$ |
| $2$ | $6$ | $(1,6,3,2,5,4)$ | $1$ |
The blue line marks the conjugacy class containing complex conjugation.