Basic invariants
| Dimension: | $3$ |
| Group: | $S_4\times C_2$ |
| Conductor: | \(963\)\(\medspace = 3^{2} \cdot 107 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 6.4.309123.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $S_4\times C_2$ |
| Parity: | odd |
| Determinant: | 1.107.2t1.a.a |
| Projective image: | $S_4$ |
| Projective stem field: | Galois closure of 4.0.2889.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - 2x^{5} - 2x^{4} + 5x^{3} - 2x^{2} - 2x + 1 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 43 }$ to precision 8.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 43 }$:
\( x^{2} + 42x + 3 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 35 a + 2 + \left(40 a + 10\right)\cdot 43 + \left(14 a + 31\right)\cdot 43^{2} + \left(36 a + 8\right)\cdot 43^{3} + \left(22 a + 27\right)\cdot 43^{4} + \left(9 a + 37\right)\cdot 43^{5} + \left(30 a + 15\right)\cdot 43^{6} + \left(18 a + 6\right)\cdot 43^{7} +O(43^{8})\)
|
| $r_{ 2 }$ | $=$ |
\( 42 a + 11 + \left(40 a + 31\right)\cdot 43 + \left(20 a + 24\right)\cdot 43^{2} + \left(28 a + 7\right)\cdot 43^{3} + \left(24 a + 17\right)\cdot 43^{4} + \left(42 a + 4\right)\cdot 43^{5} + \left(23 a + 42\right)\cdot 43^{6} + \left(33 a + 4\right)\cdot 43^{7} +O(43^{8})\)
|
| $r_{ 3 }$ | $=$ |
\( 32 + 28\cdot 43 + 26\cdot 43^{2} + 5\cdot 43^{3} + 34\cdot 43^{4} + 3\cdot 43^{5} + 43^{6} + 3\cdot 43^{7} +O(43^{8})\)
|
| $r_{ 4 }$ | $=$ |
\( 39 + 12\cdot 43 + 36\cdot 43^{2} + 18\cdot 43^{3} + 23\cdot 43^{4} + 36\cdot 43^{5} + 9\cdot 43^{6} + 19\cdot 43^{7} +O(43^{8})\)
|
| $r_{ 5 }$ | $=$ |
\( 8 a + 37 + \left(2 a + 15\right)\cdot 43 + \left(28 a + 5\right)\cdot 43^{2} + \left(6 a + 30\right)\cdot 43^{3} + \left(20 a + 13\right)\cdot 43^{4} + \left(33 a + 24\right)\cdot 43^{5} + \left(12 a + 36\right)\cdot 43^{6} + \left(24 a + 37\right)\cdot 43^{7} +O(43^{8})\)
|
| $r_{ 6 }$ | $=$ |
\( a + 10 + \left(2 a + 30\right)\cdot 43 + \left(22 a + 4\right)\cdot 43^{2} + \left(14 a + 15\right)\cdot 43^{3} + \left(18 a + 13\right)\cdot 43^{4} + 22\cdot 43^{5} + \left(19 a + 23\right)\cdot 43^{6} + \left(9 a + 14\right)\cdot 43^{7} +O(43^{8})\)
|
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 |
| $1$ | $1$ | $()$ | $3$ |
| $1$ | $2$ | $(1,6)(2,5)(3,4)$ | $-3$ |
| $3$ | $2$ | $(1,6)(2,5)$ | $-1$ |
| $3$ | $2$ | $(2,5)$ | $1$ |
| $6$ | $2$ | $(1,2)(5,6)$ | $1$ |
| $6$ | $2$ | $(1,6)(2,3)(4,5)$ | $-1$ |
| $8$ | $3$ | $(1,3,2)(4,5,6)$ | $0$ |
| $6$ | $4$ | $(1,5,6,2)$ | $1$ |
| $6$ | $4$ | $(1,6)(2,4,5,3)$ | $-1$ |
| $8$ | $6$ | $(1,3,2,6,4,5)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.