Basic invariants
| Dimension: | $2$ |
| Group: | $D_{6}$ |
| Conductor: | \(2352\)\(\medspace = 2^{4} \cdot 3 \cdot 7^{2} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 6.2.38723328.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $D_{6}$ |
| Parity: | odd |
| Determinant: | 1.3.2t1.a.a |
| Projective image: | $S_3$ |
| Projective stem field: | Galois closure of 3.1.588.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - 28 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 29 }$ to precision 6.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 29 }$:
\( x^{2} + 24x + 2 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 6 a + 20 + \left(19 a + 12\right)\cdot 29 + \left(6 a + 27\right)\cdot 29^{2} + \left(14 a + 13\right)\cdot 29^{3} + 7\cdot 29^{4} + \left(27 a + 6\right)\cdot 29^{5} +O(29^{6})\)
|
| $r_{ 2 }$ | $=$ |
\( 12 + 28\cdot 29 + 10\cdot 29^{2} + 5\cdot 29^{3} + 3\cdot 29^{4} + 2\cdot 29^{5} +O(29^{6})\)
|
| $r_{ 3 }$ | $=$ |
\( 23 a + 21 + \left(9 a + 15\right)\cdot 29 + \left(22 a + 12\right)\cdot 29^{2} + \left(14 a + 20\right)\cdot 29^{3} + \left(28 a + 24\right)\cdot 29^{4} + \left(a + 24\right)\cdot 29^{5} +O(29^{6})\)
|
| $r_{ 4 }$ | $=$ |
\( 23 a + 9 + \left(9 a + 16\right)\cdot 29 + \left(22 a + 1\right)\cdot 29^{2} + \left(14 a + 15\right)\cdot 29^{3} + \left(28 a + 21\right)\cdot 29^{4} + \left(a + 22\right)\cdot 29^{5} +O(29^{6})\)
|
| $r_{ 5 }$ | $=$ |
\( 17 + 18\cdot 29^{2} + 23\cdot 29^{3} + 25\cdot 29^{4} + 26\cdot 29^{5} +O(29^{6})\)
|
| $r_{ 6 }$ | $=$ |
\( 6 a + 8 + \left(19 a + 13\right)\cdot 29 + \left(6 a + 16\right)\cdot 29^{2} + \left(14 a + 8\right)\cdot 29^{3} + 4\cdot 29^{4} + \left(27 a + 4\right)\cdot 29^{5} +O(29^{6})\)
|
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 |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,4)(2,5)(3,6)$ | $-2$ |
| $3$ | $2$ | $(1,2)(3,6)(4,5)$ | $0$ |
| $3$ | $2$ | $(1,3)(4,6)$ | $0$ |
| $2$ | $3$ | $(1,5,3)(2,6,4)$ | $-1$ |
| $2$ | $6$ | $(1,6,5,4,3,2)$ | $1$ |
The blue line marks the conjugacy class containing complex conjugation.