Basic invariants
| Dimension: | $3$ |
| Group: | $S_4$ |
| Conductor: | \(870124\)\(\medspace = 2^{2} \cdot 19 \cdot 107^{2} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 4.2.870124.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $S_4$ |
| Parity: | odd |
| Determinant: | 1.19.2t1.a.a |
| Projective image: | $S_4$ |
| Projective stem field: | Galois closure of 4.2.870124.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{4} - x^{3} - 7x^{2} + 17x - 32 \)
|
The roots of $f$ are computed in $\Q_{ 157 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 35 + 107\cdot 157 + 151\cdot 157^{2} + 33\cdot 157^{3} + 87\cdot 157^{4} +O(157^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 81 + 70\cdot 157 + 83\cdot 157^{2} + 131\cdot 157^{3} + 13\cdot 157^{4} +O(157^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 94 + 83\cdot 157 + 17\cdot 157^{2} + 91\cdot 157^{3} + 132\cdot 157^{4} +O(157^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 105 + 52\cdot 157 + 61\cdot 157^{2} + 57\cdot 157^{3} + 80\cdot 157^{4} +O(157^{5})\)
|
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 | Complex conjugation |
| $1$ | $1$ | $()$ | $3$ | |
| $3$ | $2$ | $(1,2)(3,4)$ | $-1$ | |
| $6$ | $2$ | $(1,2)$ | $1$ | ✓ |
| $8$ | $3$ | $(1,2,3)$ | $0$ | |
| $6$ | $4$ | $(1,2,3,4)$ | $-1$ |