Basic invariants
| Dimension: | $3$ |
| Group: | $S_4\times C_2$ |
| Conductor: | \(804\)\(\medspace = 2^{2} \cdot 3 \cdot 67 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 6.0.215472.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $S_4\times C_2$ |
| Parity: | even |
| Determinant: | 1.201.2t1.a.a |
| Projective image: | $S_4$ |
| Projective stem field: | Galois closure of 4.2.2412.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - x^{5} + 3x^{3} - x + 1 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 43 }$ to precision 7.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 43 }$:
\( x^{2} + 42x + 3 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 24 a + 21 + \left(20 a + 36\right)\cdot 43 + \left(31 a + 33\right)\cdot 43^{2} + \left(42 a + 33\right)\cdot 43^{3} + 41 a\cdot 43^{4} + \left(37 a + 3\right)\cdot 43^{5} + \left(22 a + 1\right)\cdot 43^{6} +O(43^{7})\)
|
| $r_{ 2 }$ | $=$ |
\( 12 + 17\cdot 43 + 34\cdot 43^{2} + 28\cdot 43^{3} + 28\cdot 43^{4} + 18\cdot 43^{5} + 34\cdot 43^{6} +O(43^{7})\)
|
| $r_{ 3 }$ | $=$ |
\( 28 a + 3 + \left(38 a + 41\right)\cdot 43 + \left(12 a + 6\right)\cdot 43^{2} + \left(22 a + 34\right)\cdot 43^{3} + 14 a\cdot 43^{4} + \left(7 a + 20\right)\cdot 43^{5} + \left(18 a + 19\right)\cdot 43^{6} +O(43^{7})\)
|
| $r_{ 4 }$ | $=$ |
\( 19 a + 2 + \left(22 a + 33\right)\cdot 43 + \left(11 a + 1\right)\cdot 43^{2} + 2\cdot 43^{3} + a\cdot 43^{4} + \left(5 a + 42\right)\cdot 43^{5} + \left(20 a + 28\right)\cdot 43^{6} +O(43^{7})\)
|
| $r_{ 5 }$ | $=$ |
\( 18 + 35\cdot 43 + 27\cdot 43^{2} + 29\cdot 43^{3} + 19\cdot 43^{4} + 32\cdot 43^{5} + 14\cdot 43^{6} +O(43^{7})\)
|
| $r_{ 6 }$ | $=$ |
\( 15 a + 31 + \left(4 a + 8\right)\cdot 43 + \left(30 a + 24\right)\cdot 43^{2} + 20 a\cdot 43^{3} + \left(28 a + 36\right)\cdot 43^{4} + \left(35 a + 12\right)\cdot 43^{5} + \left(24 a + 30\right)\cdot 43^{6} +O(43^{7})\)
|
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 |
| $1$ | $1$ | $()$ | $3$ |
| $1$ | $2$ | $(1,6)(2,5)(3,4)$ | $-3$ |
| $3$ | $2$ | $(1,6)$ | $1$ |
| $3$ | $2$ | $(1,6)(2,5)$ | $-1$ |
| $6$ | $2$ | $(2,3)(4,5)$ | $1$ |
| $6$ | $2$ | $(1,6)(2,3)(4,5)$ | $-1$ |
| $8$ | $3$ | $(1,2,3)(4,6,5)$ | $0$ |
| $6$ | $4$ | $(1,5,6,2)$ | $1$ |
| $6$ | $4$ | $(1,4,6,3)(2,5)$ | $-1$ |
| $8$ | $6$ | $(1,5,4,6,2,3)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.