Basic invariants
| Dimension: | $3$ |
| Group: | $S_4\times C_2$ |
| Conductor: | \(9775\)\(\medspace = 5^{2} \cdot 17 \cdot 23 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 6.2.224825.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $S_4\times C_2$ |
| Parity: | odd |
| Determinant: | 1.391.2t1.a.a |
| Projective image: | $S_4$ |
| Projective stem field: | Galois closure of 4.2.166175.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - 2x^{5} - 3x^{4} + 7x^{3} + 6x^{2} - 8x - 8 \)
|
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:
| $r_{ 1 }$ | $=$ |
\( 9 + 5\cdot 43 + 6\cdot 43^{2} + 40\cdot 43^{3} + 40\cdot 43^{4} + 34\cdot 43^{5} + 13\cdot 43^{6} + 12\cdot 43^{7} +O(43^{8})\)
|
| $r_{ 2 }$ | $=$ |
\( 38 + 21\cdot 43 + 5\cdot 43^{2} + 14\cdot 43^{3} + 23\cdot 43^{4} + 8\cdot 43^{5} + 27\cdot 43^{6} + 10\cdot 43^{7} +O(43^{8})\)
|
| $r_{ 3 }$ | $=$ |
\( 33 a + 16 + \left(29 a + 35\right)\cdot 43 + \left(30 a + 29\right)\cdot 43^{2} + \left(32 a + 1\right)\cdot 43^{3} + \left(14 a + 38\right)\cdot 43^{4} + \left(26 a + 1\right)\cdot 43^{5} + \left(10 a + 2\right)\cdot 43^{6} + \left(35 a + 39\right)\cdot 43^{7} +O(43^{8})\)
|
| $r_{ 4 }$ | $=$ |
\( 7 a + 6 + \left(22 a + 31\right)\cdot 43 + \left(15 a + 31\right)\cdot 43^{2} + \left(22 a + 9\right)\cdot 43^{3} + \left(32 a + 41\right)\cdot 43^{4} + \left(24 a + 38\right)\cdot 43^{5} + \left(39 a + 20\right)\cdot 43^{6} + \left(6 a + 39\right)\cdot 43^{7} +O(43^{8})\)
|
| $r_{ 5 }$ | $=$ |
\( 10 a + 6 + \left(13 a + 32\right)\cdot 43 + \left(12 a + 30\right)\cdot 43^{2} + \left(10 a + 3\right)\cdot 43^{3} + \left(28 a + 20\right)\cdot 43^{4} + \left(16 a + 13\right)\cdot 43^{5} + \left(32 a + 29\right)\cdot 43^{6} + \left(7 a + 20\right)\cdot 43^{7} +O(43^{8})\)
|
| $r_{ 6 }$ | $=$ |
\( 36 a + 13 + \left(20 a + 3\right)\cdot 43 + \left(27 a + 25\right)\cdot 43^{2} + \left(20 a + 16\right)\cdot 43^{3} + \left(10 a + 8\right)\cdot 43^{4} + \left(18 a + 31\right)\cdot 43^{5} + \left(3 a + 35\right)\cdot 43^{6} + \left(36 a + 6\right)\cdot 43^{7} +O(43^{8})\)
|
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 |
| $1$ | $1$ | $()$ | $3$ |
| $1$ | $2$ | $(1,2)(3,6)(4,5)$ | $-3$ |
| $3$ | $2$ | $(1,2)$ | $1$ |
| $3$ | $2$ | $(1,2)(3,6)$ | $-1$ |
| $6$ | $2$ | $(3,4)(5,6)$ | $1$ |
| $6$ | $2$ | $(1,2)(3,4)(5,6)$ | $-1$ |
| $8$ | $3$ | $(1,3,4)(2,6,5)$ | $0$ |
| $6$ | $4$ | $(1,6,2,3)$ | $1$ |
| $6$ | $4$ | $(1,5,2,4)(3,6)$ | $-1$ |
| $8$ | $6$ | $(1,6,5,2,3,4)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.