Basic invariants
| Dimension: | $4$ |
| Group: | $S_5$ |
| Conductor: | \(7367\)\(\medspace = 53 \cdot 139 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 5.3.7367.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $S_5$ |
| Parity: | odd |
| Determinant: | 1.7367.2t1.a.a |
| Projective image: | $S_5$ |
| Projective stem field: | Galois closure of 5.3.7367.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{5} - 2x^{4} + 3x^{2} - 2x - 1 \)
|
The roots of $f$ are computed in $\Q_{ 277 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 5 + 154\cdot 277 + 271\cdot 277^{2} + 170\cdot 277^{3} + 206\cdot 277^{4} +O(277^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 145 + 54\cdot 277 + 47\cdot 277^{2} + 208\cdot 277^{3} + 90\cdot 277^{4} +O(277^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 184 + 269\cdot 277 + 242\cdot 277^{2} + 209\cdot 277^{3} + 31\cdot 277^{4} +O(277^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 228 + 192\cdot 277 + 114\cdot 277^{2} + 122\cdot 277^{3} + 272\cdot 277^{4} +O(277^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 271 + 159\cdot 277 + 154\cdot 277^{2} + 119\cdot 277^{3} + 229\cdot 277^{4} +O(277^{5})\)
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Generators of the action on the roots $r_1, \ldots, r_{ 5 }$
| Cycle notation |
Character values on conjugacy classes
| Size | Order | Action on $r_1, \ldots, r_{ 5 }$ | Character value | Complex conjugation |
| $1$ | $1$ | $()$ | $4$ | |
| $10$ | $2$ | $(1,2)$ | $2$ | ✓ |
| $15$ | $2$ | $(1,2)(3,4)$ | $0$ | |
| $20$ | $3$ | $(1,2,3)$ | $1$ | |
| $30$ | $4$ | $(1,2,3,4)$ | $0$ | |
| $24$ | $5$ | $(1,2,3,4,5)$ | $-1$ | |
| $20$ | $6$ | $(1,2,3)(4,5)$ | $-1$ |