Basic invariants
| Dimension: | $3$ |
| Group: | $S_4$ |
| Conductor: | \(179776\)\(\medspace = 2^{6} \cdot 53^{2} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 4.2.848.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $S_4$ |
| Parity: | even |
| Determinant: | 1.1.1t1.a.a |
| Projective image: | $S_4$ |
| Projective stem field: | Galois closure of 4.2.848.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{4} - x^{2} - 2x + 1 \)
|
The roots of $f$ are computed in $\Q_{ 67 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 3 + 2\cdot 67 + 28\cdot 67^{2} + 27\cdot 67^{3} + 65\cdot 67^{4} +O(67^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 12 + 16\cdot 67 + 27\cdot 67^{2} + 37\cdot 67^{3} + 33\cdot 67^{4} +O(67^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 58 + 57\cdot 67 + 49\cdot 67^{2} + 54\cdot 67^{3} + 10\cdot 67^{4} +O(67^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 61 + 57\cdot 67 + 28\cdot 67^{2} + 14\cdot 67^{3} + 24\cdot 67^{4} +O(67^{5})\)
|
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.