Basic invariants
| Dimension: | $3$ |
| Group: | $S_4$ |
| Conductor: | \(73167\)\(\medspace = 3 \cdot 29^{3} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 4.2.73167.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $S_4$ |
| Parity: | odd |
| Determinant: | 1.87.2t1.a.a |
| Projective image: | $S_4$ |
| Projective stem field: | Galois closure of 4.2.73167.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{4} - x^{3} + 4x^{2} + 9x - 6 \)
|
The roots of $f$ are computed in $\Q_{ 359 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 47 + 284\cdot 359 + 259\cdot 359^{2} + 305\cdot 359^{3} + 153\cdot 359^{4} +O(359^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 154 + 157\cdot 359 + 308\cdot 359^{2} + 156\cdot 359^{3} + 305\cdot 359^{4} +O(359^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 199 + 220\cdot 359 + 163\cdot 359^{2} + 63\cdot 359^{3} + 250\cdot 359^{4} +O(359^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 319 + 55\cdot 359 + 345\cdot 359^{2} + 191\cdot 359^{3} + 8\cdot 359^{4} +O(359^{5})\)
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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.