Basic invariants
| Dimension: | $3$ |
| Group: | $S_4$ |
| Conductor: | \(250120\)\(\medspace = 2^{3} \cdot 5 \cdot 13^{2} \cdot 37 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 4.2.250120.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $S_4$ |
| Parity: | odd |
| Determinant: | 1.1480.2t1.b.a |
| Projective image: | $S_4$ |
| Projective stem field: | Galois closure of 4.2.250120.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{4} - 2x^{3} - 14x^{2} + 2x + 53 \)
|
The roots of $f$ are computed in $\Q_{ 347 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 43 + 113\cdot 347 + 71\cdot 347^{2} + 38\cdot 347^{3} + 12\cdot 347^{4} +O(347^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 60 + 286\cdot 347 + 17\cdot 347^{2} + 156\cdot 347^{3} + 81\cdot 347^{4} +O(347^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 70 + 63\cdot 347 + 149\cdot 347^{2} + 104\cdot 347^{3} + 328\cdot 347^{4} +O(347^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 176 + 231\cdot 347 + 108\cdot 347^{2} + 48\cdot 347^{3} + 272\cdot 347^{4} +O(347^{5})\)
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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.