Basic invariants
| Dimension: | $3$ |
| Group: | $S_4$ |
| Conductor: | \(1765\)\(\medspace = 5 \cdot 353 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 4.0.1765.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $S_4$ |
| Parity: | even |
| Determinant: | 1.1765.2t1.a.a |
| Projective image: | $S_4$ |
| Projective stem field: | Galois closure of 4.0.1765.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{4} - x^{3} - x^{2} - x + 3 \)
|
The roots of $f$ are computed in $\Q_{ 173 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 14 + 18\cdot 173 + 26\cdot 173^{2} + 168\cdot 173^{3} + 58\cdot 173^{4} +O(173^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 52 + 100\cdot 173 + 46\cdot 173^{2} + 123\cdot 173^{3} + 162\cdot 173^{4} +O(173^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 131 + 120\cdot 173 + 45\cdot 173^{2} + 56\cdot 173^{3} + 155\cdot 173^{4} +O(173^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 150 + 106\cdot 173 + 54\cdot 173^{2} + 171\cdot 173^{3} + 141\cdot 173^{4} +O(173^{5})\)
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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.