Basic invariants
| Dimension: | $3$ |
| Group: | $S_4$ |
| Conductor: | \(31936\)\(\medspace = 2^{6} \cdot 499 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 4.2.31936.2 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $S_4$ |
| Parity: | odd |
| Determinant: | 1.499.2t1.a.a |
| Projective image: | $S_4$ |
| Projective stem field: | Galois closure of 4.2.31936.2 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{4} - 2x^{3} - 4x^{2} - 4x + 6 \)
|
The roots of $f$ are computed in $\Q_{ 127 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 21 + 75\cdot 127 + 62\cdot 127^{2} + 7\cdot 127^{3} + 71\cdot 127^{4} +O(127^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 40 + 67\cdot 127 + 58\cdot 127^{2} + 92\cdot 127^{3} + 77\cdot 127^{4} +O(127^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 93 + 120\cdot 127 + 33\cdot 127^{2} + 6\cdot 127^{3} + 63\cdot 127^{4} +O(127^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 102 + 117\cdot 127 + 98\cdot 127^{2} + 20\cdot 127^{3} + 42\cdot 127^{4} +O(127^{5})\)
|
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.