Basic invariants
| Dimension: | $3$ |
| Group: | $S_4$ |
| Conductor: | \(11453152\)\(\medspace = 2^{5} \cdot 71^{3} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 4.0.11453152.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $S_4$ |
| Parity: | even |
| Determinant: | 1.568.2t1.a.a |
| Projective image: | $S_4$ |
| Projective stem field: | Galois closure of 4.0.11453152.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{4} - 2x^{3} + 37x^{2} + 248x + 324 \)
|
The roots of $f$ are computed in $\Q_{ 127 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 7 + 84\cdot 127 + 42\cdot 127^{2} + 33\cdot 127^{3} + 33\cdot 127^{4} +O(127^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 12 + 103\cdot 127 + 11\cdot 127^{2} + 99\cdot 127^{3} + 127^{4} +O(127^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 20 + 47\cdot 127 + 104\cdot 127^{2} + 67\cdot 127^{3} + 9\cdot 127^{4} +O(127^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 90 + 19\cdot 127 + 95\cdot 127^{2} + 53\cdot 127^{3} + 82\cdot 127^{4} +O(127^{5})\)
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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 |
| $1$ | $1$ | $()$ | $3$ | |
| $3$ | $2$ | $(1,2)(3,4)$ | $-1$ | ✓ |
| $6$ | $2$ | $(1,2)$ | $1$ | |
| $8$ | $3$ | $(1,2,3)$ | $0$ | |
| $6$ | $4$ | $(1,2,3,4)$ | $-1$ |