Basic invariants
| Dimension: | $3$ |
| Group: | $S_4$ |
| Conductor: | \(556516\)\(\medspace = 2^{2} \cdot 373^{2} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 4.0.1492.1 |
| 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 4.0.1492.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{4} - x^{3} + 2x^{2} + 2 \)
|
The roots of $f$ are computed in $\Q_{ 229 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 34 + 159\cdot 229 + 67\cdot 229^{2} + 206\cdot 229^{3} + 32\cdot 229^{4} +O(229^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 64 + 155\cdot 229 + 73\cdot 229^{2} + 162\cdot 229^{3} + 92\cdot 229^{4} +O(229^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 65 + 160\cdot 229 + 52\cdot 229^{2} + 60\cdot 229^{3} + 66\cdot 229^{4} +O(229^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 67 + 212\cdot 229 + 34\cdot 229^{2} + 29\cdot 229^{3} + 37\cdot 229^{4} +O(229^{5})\)
|
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.