Basic invariants
| Dimension: | $3$ |
| Group: | $S_4$ |
| Conductor: | \(18225\)\(\medspace = 3^{6} \cdot 5^{2} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 4.2.6075.1 |
| 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 4.2.6075.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{4} - x^{3} + 4x - 1 \)
|
The roots of $f$ are computed in $\Q_{ 223 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 38 + 69\cdot 223 + 123\cdot 223^{2} + 59\cdot 223^{3} + 61\cdot 223^{4} +O(223^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 83 + 56\cdot 223 + 15\cdot 223^{2} + 123\cdot 223^{3} + 208\cdot 223^{4} +O(223^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 123 + 222\cdot 223 + 177\cdot 223^{2} + 193\cdot 223^{3} + 130\cdot 223^{4} +O(223^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 203 + 97\cdot 223 + 129\cdot 223^{2} + 69\cdot 223^{3} + 45\cdot 223^{4} +O(223^{5})\)
|
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.