Basic invariants
| Dimension: | $3$ |
| Group: | $S_4$ |
| Conductor: | \(182183\)\(\medspace = 23 \cdot 89^{2} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 4.2.182183.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $S_4$ |
| Parity: | odd |
| Determinant: | 1.23.2t1.a.a |
| Projective image: | $S_4$ |
| Projective stem field: | Galois closure of 4.2.182183.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{4} - x^{3} - 4x^{2} - 9x - 8 \)
|
The roots of $f$ are computed in $\Q_{ 101 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 48 + 35\cdot 101 + 27\cdot 101^{2} + 95\cdot 101^{3} + 55\cdot 101^{4} +O(101^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 75 + 43\cdot 101 + 61\cdot 101^{2} + 22\cdot 101^{3} + 90\cdot 101^{4} +O(101^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 86 + 12\cdot 101 + 43\cdot 101^{2} + 65\cdot 101^{3} + 94\cdot 101^{4} +O(101^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 95 + 8\cdot 101 + 70\cdot 101^{2} + 18\cdot 101^{3} + 62\cdot 101^{4} +O(101^{5})\)
|
Generators of the action on the roots $r_1, \ldots, r_{ 4 }$
| Cycle notation |
Character values on conjugacy classes
| Size | Order | Action on $r_1, \ldots, r_{ 4 }$ | Character value |
| $1$ | $1$ | $()$ | $3$ |
| $3$ | $2$ | $(1,2)(3,4)$ | $-1$ |
| $6$ | $2$ | $(1,2)$ | $1$ |
| $8$ | $3$ | $(1,2,3)$ | $0$ |
| $6$ | $4$ | $(1,2,3,4)$ | $-1$ |
The blue line marks the conjugacy class containing complex conjugation.