Basic invariants
| Dimension: | $3$ |
| Group: | $S_4$ |
| Conductor: | \(408000\)\(\medspace = 2^{6} \cdot 3 \cdot 5^{3} \cdot 17 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 4.2.408000.2 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $S_4$ |
| Parity: | odd |
| Determinant: | 1.255.2t1.a.a |
| Projective image: | $S_4$ |
| Projective stem field: | Galois closure of 4.2.408000.2 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{4} - 2x^{3} + 14x^{2} + 32x + 6 \)
|
The roots of $f$ are computed in $\Q_{ 193 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 12 + 43\cdot 193 + 176\cdot 193^{2} + 141\cdot 193^{3} + 160\cdot 193^{4} +O(193^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 18 + 58\cdot 193 + 70\cdot 193^{2} + 171\cdot 193^{3} + 2\cdot 193^{4} +O(193^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 27 + 172\cdot 193 + 178\cdot 193^{2} + 193^{3} + 171\cdot 193^{4} +O(193^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 138 + 112\cdot 193 + 153\cdot 193^{2} + 70\cdot 193^{3} + 51\cdot 193^{4} +O(193^{5})\)
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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.