Basic invariants
| Dimension: | $3$ |
| Group: | $S_4$ |
| Conductor: | \(1593\)\(\medspace = 3^{3} \cdot 59 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 4.0.1593.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $S_4$ |
| Parity: | even |
| Determinant: | 1.177.2t1.a.a |
| Projective image: | $S_4$ |
| Projective stem field: | Galois closure of 4.0.1593.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{4} - 2x^{3} + 3x^{2} - x + 2 \)
|
The roots of $f$ are computed in $\Q_{ 149 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 5 + 11\cdot 149 + 90\cdot 149^{2} + 23\cdot 149^{3} + 41\cdot 149^{4} +O(149^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 32 + 96\cdot 149 + 102\cdot 149^{2} + 23\cdot 149^{3} + 49\cdot 149^{4} +O(149^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 52 + 115\cdot 149 + 21\cdot 149^{2} + 92\cdot 149^{3} + 14\cdot 149^{4} +O(149^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 62 + 75\cdot 149 + 83\cdot 149^{2} + 9\cdot 149^{3} + 44\cdot 149^{4} +O(149^{5})\)
|
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 | Complex conjugation |
| $1$ | $1$ | $()$ | $3$ | |
| $3$ | $2$ | $(1,2)(3,4)$ | $-1$ | ✓ |
| $6$ | $2$ | $(1,2)$ | $1$ | |
| $8$ | $3$ | $(1,2,3)$ | $0$ | |
| $6$ | $4$ | $(1,2,3,4)$ | $-1$ |