Basic invariants
| Dimension: | $4$ |
| Group: | $C_3^2:D_4$ |
| Conductor: | \(204422512\)\(\medspace = 2^{4} \cdot 7^{3} \cdot 193^{2} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 6.0.12975004.2 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | 12T34 |
| Parity: | odd |
| Determinant: | 1.7.2t1.a.a |
| Projective image: | $\SOPlus(4,2)$ |
| Projective stem field: | Galois closure of 6.0.12975004.2 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - x^{5} + x^{4} - 8x^{3} + 8x^{2} + 20x + 8 \)
|
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:
| $r_{ 1 }$ | $=$ |
\( 19 a + 40 + \left(13 a + 13\right)\cdot 43 + \left(8 a + 19\right)\cdot 43^{2} + \left(6 a + 13\right)\cdot 43^{3} + \left(24 a + 36\right)\cdot 43^{4} +O(43^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 2 a + 23 + \left(2 a + 29\right)\cdot 43 + \left(23 a + 23\right)\cdot 43^{2} + \left(24 a + 30\right)\cdot 43^{3} + \left(10 a + 19\right)\cdot 43^{4} +O(43^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 12 + 33\cdot 43 + 40\cdot 43^{2} + 38\cdot 43^{3} + 6\cdot 43^{4} +O(43^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 14 + 14\cdot 43 + 29\cdot 43^{2} + 2\cdot 43^{3} + 6\cdot 43^{4} +O(43^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 41 a + 25 + \left(40 a + 29\right)\cdot 43 + \left(19 a + 1\right)\cdot 43^{2} + \left(18 a + 32\right)\cdot 43^{3} + \left(32 a + 5\right)\cdot 43^{4} +O(43^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 24 a + 16 + \left(29 a + 8\right)\cdot 43 + \left(34 a + 14\right)\cdot 43^{2} + \left(36 a + 11\right)\cdot 43^{3} + \left(18 a + 11\right)\cdot 43^{4} +O(43^{5})\)
|
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 |
| $1$ | $1$ | $()$ | $4$ |
| $6$ | $2$ | $(1,2)(3,4)(5,6)$ | $-2$ |
| $6$ | $2$ | $(1,3)$ | $0$ |
| $9$ | $2$ | $(1,3)(2,4)$ | $0$ |
| $4$ | $3$ | $(1,3,6)(2,4,5)$ | $1$ |
| $4$ | $3$ | $(2,4,5)$ | $-2$ |
| $18$ | $4$ | $(1,4,3,2)(5,6)$ | $0$ |
| $12$ | $6$ | $(1,2,3,4,6,5)$ | $1$ |
| $12$ | $6$ | $(1,3)(2,4,5)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.