Basic invariants
| Dimension: | $2$ |
| Group: | $D_{6}$ |
| Conductor: | \(212\)\(\medspace = 2^{2} \cdot 53 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 6.0.179776.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $D_{6}$ |
| Parity: | odd |
| Determinant: | 1.212.2t1.a.a |
| Projective image: | $S_3$ |
| Projective stem field: | Galois closure of 3.1.212.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - 2x^{5} - 2x^{4} + 6x^{3} - 2x^{2} - 4x + 5 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 41 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 41 }$:
\( x^{2} + 38x + 6 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 10 + 26\cdot 41 + 20\cdot 41^{2} + 16\cdot 41^{3} + 36\cdot 41^{4} +O(41^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 2 a + 29 + \left(36 a + 31\right)\cdot 41 + \left(39 a + 12\right)\cdot 41^{2} + 40 a\cdot 41^{3} + \left(11 a + 13\right)\cdot 41^{4} +O(41^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 39 a + 35 + \left(4 a + 14\right)\cdot 41 + \left(a + 14\right)\cdot 41^{2} + 41^{3} + \left(29 a + 8\right)\cdot 41^{4} +O(41^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 36 a + 20 + \left(21 a + 31\right)\cdot 41 + \left(37 a + 4\right)\cdot 41^{2} + 14 a\cdot 41^{3} + \left(4 a + 33\right)\cdot 41^{4} +O(41^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 26 + 39\cdot 41 + 15\cdot 41^{2} + 15\cdot 41^{3} + 41^{4} +O(41^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 5 a + 5 + \left(19 a + 20\right)\cdot 41 + \left(3 a + 13\right)\cdot 41^{2} + \left(26 a + 7\right)\cdot 41^{3} + \left(36 a + 31\right)\cdot 41^{4} +O(41^{5})\)
|
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 |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,5)(2,6)(3,4)$ | $-2$ |
| $3$ | $2$ | $(1,2)(5,6)$ | $0$ |
| $3$ | $2$ | $(1,4)(2,6)(3,5)$ | $0$ |
| $2$ | $3$ | $(1,3,2)(4,6,5)$ | $-1$ |
| $2$ | $6$ | $(1,6,3,5,2,4)$ | $1$ |
The blue line marks the conjugacy class containing complex conjugation.