Basic invariants
| Dimension: | $3$ |
| Group: | $S_4$ |
| Conductor: | \(5744\)\(\medspace = 2^{4} \cdot 359 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 4.4.5744.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $S_4$ |
| Parity: | even |
| Determinant: | 1.1436.2t1.a.a |
| Projective image: | $S_4$ |
| Projective stem field: | Galois closure of 4.4.5744.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{4} - 5x^{2} - 2x + 1 \)
|
The roots of $f$ are computed in $\Q_{ 163 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 25 + 123\cdot 163 + 149\cdot 163^{3} + 146\cdot 163^{4} +O(163^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 29 + 12\cdot 163 + 78\cdot 163^{2} + 132\cdot 163^{3} + 83\cdot 163^{4} +O(163^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 43 + 160\cdot 163 + 114\cdot 163^{2} + 121\cdot 163^{3} + 52\cdot 163^{4} +O(163^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 66 + 30\cdot 163 + 132\cdot 163^{2} + 85\cdot 163^{3} + 42\cdot 163^{4} +O(163^{5})\)
|
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.