Basic invariants
| Dimension: | $3$ |
| Group: | $S_4$ |
| Conductor: | \(17856\)\(\medspace = 2^{6} \cdot 3^{2} \cdot 31 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 4.2.17856.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $S_4$ |
| Parity: | odd |
| Determinant: | 1.31.2t1.a.a |
| Projective image: | $S_4$ |
| Projective stem field: | Galois closure of 4.2.17856.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{4} - 2x^{3} + 2x^{2} + 2x - 5 \)
|
The roots of $f$ are computed in $\Q_{ 131 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 39 + 79\cdot 131 + 21\cdot 131^{2} + 128\cdot 131^{3} + 91\cdot 131^{4} +O(131^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 51 + 97\cdot 131 + 51\cdot 131^{2} + 95\cdot 131^{3} + 113\cdot 131^{4} +O(131^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 79 + 39\cdot 131 + 5\cdot 131^{2} + 86\cdot 131^{3} + 91\cdot 131^{4} +O(131^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 95 + 45\cdot 131 + 52\cdot 131^{2} + 83\cdot 131^{3} + 95\cdot 131^{4} +O(131^{5})\)
|
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.