Basic invariants
| Dimension: | $3$ |
| Group: | $S_4$ |
| Conductor: | \(3888\)\(\medspace = 2^{4} \cdot 3^{5} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 4.2.3888.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $S_4$ |
| Parity: | odd |
| Determinant: | 1.3.2t1.a.a |
| Projective image: | $S_4$ |
| Projective stem field: | Galois closure of 4.2.3888.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{4} - 2x^{3} - 6x + 3 \)
|
The roots of $f$ are computed in $\Q_{ 379 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 87 + 178\cdot 379 + 58\cdot 379^{2} + 322\cdot 379^{3} + 369\cdot 379^{4} +O(379^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 149 + 144\cdot 379 + 5\cdot 379^{2} + 107\cdot 379^{3} + 158\cdot 379^{4} +O(379^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 225 + 115\cdot 379 + 220\cdot 379^{2} + 184\cdot 379^{3} + 160\cdot 379^{4} +O(379^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 299 + 319\cdot 379 + 94\cdot 379^{2} + 144\cdot 379^{3} + 69\cdot 379^{4} +O(379^{5})\)
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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.