Basic invariants
| Dimension: | $4$ |
| Group: | $C_3^2:D_4$ |
| Conductor: | \(17856\)\(\medspace = 2^{6} \cdot 3^{2} \cdot 31 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 6.0.53568.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $C_3^2:D_4$ |
| Parity: | even |
| Determinant: | 1.124.2t1.a.a |
| Projective image: | $\SOPlus(4,2)$ |
| Projective stem field: | Galois closure of 6.0.53568.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - 2x^{5} + 2x^{4} + 2x + 1 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 43 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 43 }$:
\( x^{2} + 42x + 3 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 28 + 29\cdot 43 + 15\cdot 43^{2} + 16\cdot 43^{3} + 4\cdot 43^{4} +O(43^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 2 a + 27 + \left(29 a + 8\right)\cdot 43 + \left(33 a + 27\right)\cdot 43^{2} + \left(21 a + 13\right)\cdot 43^{3} + \left(38 a + 2\right)\cdot 43^{4} +O(43^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 41 a + 9 + \left(39 a + 7\right)\cdot 43 + \left(37 a + 36\right)\cdot 43^{2} + \left(25 a + 40\right)\cdot 43^{3} + \left(33 a + 36\right)\cdot 43^{4} +O(43^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 41 a + 29 + \left(13 a + 35\right)\cdot 43 + \left(9 a + 31\right)\cdot 43^{2} + \left(21 a + 1\right)\cdot 43^{3} + \left(4 a + 19\right)\cdot 43^{4} +O(43^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 31 + 41\cdot 43 + 26\cdot 43^{2} + 27\cdot 43^{3} + 21\cdot 43^{4} +O(43^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 2 a + 7 + \left(3 a + 6\right)\cdot 43 + \left(5 a + 34\right)\cdot 43^{2} + \left(17 a + 28\right)\cdot 43^{3} + \left(9 a + 1\right)\cdot 43^{4} +O(43^{5})\)
|
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 |
| $1$ | $1$ | $()$ | $4$ |
| $6$ | $2$ | $(1,2)(3,4)(5,6)$ | $0$ |
| $6$ | $2$ | $(3,6)$ | $2$ |
| $9$ | $2$ | $(3,6)(4,5)$ | $0$ |
| $4$ | $3$ | $(1,3,6)$ | $1$ |
| $4$ | $3$ | $(1,3,6)(2,4,5)$ | $-2$ |
| $18$ | $4$ | $(1,2)(3,5,6,4)$ | $0$ |
| $12$ | $6$ | $(1,4,3,5,6,2)$ | $0$ |
| $12$ | $6$ | $(2,4,5)(3,6)$ | $-1$ |
The blue line marks the conjugacy class containing complex conjugation.