Basic invariants
| Dimension: | $2$ |
| Group: | $D_{6}$ |
| Conductor: | \(135\)\(\medspace = 3^{3} \cdot 5 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 6.2.91125.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $D_{6}$ |
| Parity: | odd |
| Determinant: | 1.15.2t1.a.a |
| Projective image: | $S_3$ |
| Projective stem field: | Galois closure of 3.1.135.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - x^{3} - 1 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 11 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 11 }$:
\( x^{2} + 7x + 2 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 5 a + 4 + 10\cdot 11 + \left(5 a + 10\right)\cdot 11^{2} + \left(5 a + 2\right)\cdot 11^{3} + \left(8 a + 2\right)\cdot 11^{4} +O(11^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 2 + 7\cdot 11 + 6\cdot 11^{2} + 7\cdot 11^{3} + 3\cdot 11^{4} +O(11^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 5 + 4\cdot 11 + 2\cdot 11^{2} + 10\cdot 11^{3} + 10\cdot 11^{4} +O(11^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 9 a + 3 + \left(9 a + 3\right)\cdot 11 + \left(a + 3\right)\cdot 11^{2} + \left(8 a + 8\right)\cdot 11^{3} + \left(4 a + 3\right)\cdot 11^{4} +O(11^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 6 a + 2 + \left(10 a + 7\right)\cdot 11 + \left(5 a + 8\right)\cdot 11^{2} + \left(5 a + 8\right)\cdot 11^{3} + \left(2 a + 8\right)\cdot 11^{4} +O(11^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 2 a + 6 + a\cdot 11 + \left(9 a + 1\right)\cdot 11^{2} + \left(2 a + 6\right)\cdot 11^{3} + \left(6 a + 3\right)\cdot 11^{4} +O(11^{5})\)
|
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 |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,4)(2,3)(5,6)$ | $-2$ |
| $3$ | $2$ | $(1,2)(3,4)(5,6)$ | $0$ |
| $3$ | $2$ | $(1,5)(4,6)$ | $0$ |
| $2$ | $3$ | $(1,3,5)(2,6,4)$ | $-1$ |
| $2$ | $6$ | $(1,6,3,4,5,2)$ | $1$ |
The blue line marks the conjugacy class containing complex conjugation.