Basic invariants
| Dimension: | $3$ |
| Group: | $S_4$ |
| Conductor: | \(1257\)\(\medspace = 3 \cdot 419 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 4.0.1257.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $S_4$ |
| Parity: | even |
| Determinant: | 1.1257.2t1.a.a |
| Projective image: | $S_4$ |
| Projective stem field: | Galois closure of 4.0.1257.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{4} - x^{2} - x + 2 \)
|
The roots of $f$ are computed in $\Q_{ 101 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 13 + 38\cdot 101 + 41\cdot 101^{2} + 56\cdot 101^{3} + 43\cdot 101^{4} +O(101^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 18 + 36\cdot 101 + 36\cdot 101^{2} + 24\cdot 101^{3} + 32\cdot 101^{4} +O(101^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 21 + 49\cdot 101 + 56\cdot 101^{2} + 36\cdot 101^{3} + 18\cdot 101^{4} +O(101^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 49 + 78\cdot 101 + 67\cdot 101^{2} + 84\cdot 101^{3} + 6\cdot 101^{4} +O(101^{5})\)
|
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 |
| $1$ | $1$ | $()$ | $3$ | |
| $3$ | $2$ | $(1,2)(3,4)$ | $-1$ | ✓ |
| $6$ | $2$ | $(1,2)$ | $1$ | |
| $8$ | $3$ | $(1,2,3)$ | $0$ | |
| $6$ | $4$ | $(1,2,3,4)$ | $-1$ |