Basic invariants
| Dimension: | $3$ |
| Group: | $S_4$ |
| Conductor: | \(59428681\)\(\medspace = 13^{2} \cdot 593^{2} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 4.0.7709.1 |
| 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 4.0.7709.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{4} - 2x^{3} + 6x^{2} - 3x + 2 \)
|
The roots of $f$ are computed in $\Q_{ 239 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 29 + 197\cdot 239 + 232\cdot 239^{2} + 189\cdot 239^{3} + 139\cdot 239^{4} +O(239^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 102 + 54\cdot 239 + 44\cdot 239^{2} + 224\cdot 239^{3} + 123\cdot 239^{4} +O(239^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 171 + 176\cdot 239 + 30\cdot 239^{2} + 104\cdot 239^{3} + 71\cdot 239^{4} +O(239^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 178 + 49\cdot 239 + 170\cdot 239^{2} + 198\cdot 239^{3} + 142\cdot 239^{4} +O(239^{5})\)
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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 |
| $1$ | $1$ | $()$ | $3$ | |
| $3$ | $2$ | $(1,2)(3,4)$ | $-1$ | ✓ |
| $6$ | $2$ | $(1,2)$ | $-1$ | |
| $8$ | $3$ | $(1,2,3)$ | $0$ | |
| $6$ | $4$ | $(1,2,3,4)$ | $1$ |