Basic invariants
| Dimension: | $3$ |
| Group: | $S_4$ |
| Conductor: | \(2421136\)\(\medspace = 2^{4} \cdot 389^{2} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 4.4.6224.1 |
| 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 4.4.6224.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{4} - 6x^{2} - 2x + 5 \)
|
The roots of $f$ are computed in $\Q_{ 157 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 4 + 16\cdot 157 + 26\cdot 157^{2} + 56\cdot 157^{3} + 47\cdot 157^{4} +O(157^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 83 + 48\cdot 157 + 47\cdot 157^{2} + 137\cdot 157^{3} + 127\cdot 157^{4} +O(157^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 101 + 47\cdot 157 + 104\cdot 157^{2} + 16\cdot 157^{3} + 135\cdot 157^{4} +O(157^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 126 + 44\cdot 157 + 136\cdot 157^{2} + 103\cdot 157^{3} + 3\cdot 157^{4} +O(157^{5})\)
|
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.