Basic invariants
| Dimension: | $3$ |
| Group: | $S_4$ |
| Conductor: | \(9935104\)\(\medspace = 2^{8} \cdot 197^{2} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 4.4.50432.1 |
| 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 4.4.50432.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{4} - 10x^{2} - 12x - 1 \)
|
The roots of $f$ are computed in $\Q_{ 53 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 28 + 24\cdot 53 + 12\cdot 53^{2} + 36\cdot 53^{3} + 32\cdot 53^{4} +O(53^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 39 + 38\cdot 53^{2} + 38\cdot 53^{3} + 40\cdot 53^{4} +O(53^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 45 + 37\cdot 53 + 46\cdot 53^{2} + 18\cdot 53^{3} + 50\cdot 53^{4} +O(53^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 47 + 42\cdot 53 + 8\cdot 53^{2} + 12\cdot 53^{3} + 35\cdot 53^{4} +O(53^{5})\)
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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 | Complex conjugation |
| $1$ | $1$ | $()$ | $3$ | ✓ |
| $3$ | $2$ | $(1,2)(3,4)$ | $-1$ | |
| $6$ | $2$ | $(1,2)$ | $-1$ | |
| $8$ | $3$ | $(1,2,3)$ | $0$ | |
| $6$ | $4$ | $(1,2,3,4)$ | $1$ |