Basic invariants
| Dimension: | $4$ |
| Group: | $S_5$ |
| Conductor: | \(4597\) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 5.1.4597.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $S_5$ |
| Parity: | even |
| Determinant: | 1.4597.2t1.a.a |
| Projective image: | $S_5$ |
| Projective stem field: | Galois closure of 5.1.4597.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{5} + x^{3} - 2x^{2} - x - 1 \)
|
The roots of $f$ are computed in $\Q_{ 353 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 113 + 146\cdot 353 + 304\cdot 353^{2} + 348\cdot 353^{3} + 275\cdot 353^{4} +O(353^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 152 + 112\cdot 353 + 231\cdot 353^{2} + 243\cdot 353^{3} + 275\cdot 353^{4} +O(353^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 164 + 221\cdot 353 + 253\cdot 353^{2} + 199\cdot 353^{3} + 171\cdot 353^{4} +O(353^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 292 + 197\cdot 353 + 331\cdot 353^{2} + 273\cdot 353^{3} + 3\cdot 353^{4} +O(353^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 338 + 27\cdot 353 + 291\cdot 353^{2} + 345\cdot 353^{3} + 331\cdot 353^{4} +O(353^{5})\)
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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 |
| $1$ | $1$ | $()$ | $4$ | |
| $10$ | $2$ | $(1,2)$ | $2$ | |
| $15$ | $2$ | $(1,2)(3,4)$ | $0$ | ✓ |
| $20$ | $3$ | $(1,2,3)$ | $1$ | |
| $30$ | $4$ | $(1,2,3,4)$ | $0$ | |
| $24$ | $5$ | $(1,2,3,4,5)$ | $-1$ | |
| $20$ | $6$ | $(1,2,3)(4,5)$ | $-1$ |