Basic invariants
| Dimension: | $2$ |
| Group: | $D_{6}$ |
| Conductor: | \(684\)\(\medspace = 2^{2} \cdot 3^{2} \cdot 19 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 6.0.155952.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $D_{6}$ |
| Parity: | odd |
| Determinant: | 1.19.2t1.a.a |
| Projective image: | $S_3$ |
| Projective stem field: | Galois closure of 3.1.76.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} + 2x^{4} - 4x^{3} + 4x^{2} - 4x + 4 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 13 }$ to precision 6.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 13 }$:
\( x^{2} + 12x + 2 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 7 a + 4 + \left(a + 2\right)\cdot 13 + \left(5 a + 1\right)\cdot 13^{2} + \left(5 a + 11\right)\cdot 13^{3} + \left(2 a + 3\right)\cdot 13^{4} + \left(11 a + 4\right)\cdot 13^{5} +O(13^{6})\)
|
| $r_{ 2 }$ | $=$ |
\( 6 a + 11 + \left(11 a + 9\right)\cdot 13 + \left(7 a + 4\right)\cdot 13^{2} + \left(7 a + 11\right)\cdot 13^{3} + 10 a\cdot 13^{4} + a\cdot 13^{5} +O(13^{6})\)
|
| $r_{ 3 }$ | $=$ |
\( 8 a + 12 + \left(3 a + 11\right)\cdot 13 + 9 a\cdot 13^{2} + \left(a + 5\right)\cdot 13^{3} + \left(4 a + 7\right)\cdot 13^{4} + 7 a\cdot 13^{5} +O(13^{6})\)
|
| $r_{ 4 }$ | $=$ |
\( 11 + 7\cdot 13^{2} + 3\cdot 13^{3} + 8\cdot 13^{4} + 8\cdot 13^{5} +O(13^{6})\)
|
| $r_{ 5 }$ | $=$ |
\( 5 a + 7 + \left(9 a + 7\right)\cdot 13 + \left(3 a + 6\right)\cdot 13^{2} + \left(11 a + 10\right)\cdot 13^{3} + \left(8 a + 9\right)\cdot 13^{4} + \left(5 a + 3\right)\cdot 13^{5} +O(13^{6})\)
|
| $r_{ 6 }$ | $=$ |
\( 7 + 6\cdot 13 + 5\cdot 13^{2} + 10\cdot 13^{3} + 8\cdot 13^{4} + 8\cdot 13^{5} +O(13^{6})\)
|
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 |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,3)(2,5)(4,6)$ | $-2$ |
| $3$ | $2$ | $(2,4)(5,6)$ | $0$ |
| $3$ | $2$ | $(1,3)(2,6)(4,5)$ | $0$ |
| $2$ | $3$ | $(1,2,4)(3,5,6)$ | $-1$ |
| $2$ | $6$ | $(1,5,4,3,2,6)$ | $1$ |
The blue line marks the conjugacy class containing complex conjugation.