Basic invariants
| Dimension: | $4$ |
| Group: | $S_5$ |
| Conductor: | \(1649\)\(\medspace = 17 \cdot 97 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 5.1.1649.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $S_5$ |
| Parity: | even |
| Determinant: | 1.1649.2t1.a.a |
| Projective image: | $S_5$ |
| Projective stem field: | Galois closure of 5.1.1649.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{5} - x^{4} + x^{2} - x + 1 \)
|
The roots of $f$ are computed in $\Q_{ 499 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 121 + 359\cdot 499 + 431\cdot 499^{2} + 56\cdot 499^{3} + 376\cdot 499^{4} +O(499^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 123 + 266\cdot 499 + 309\cdot 499^{2} + 381\cdot 499^{3} + 84\cdot 499^{4} +O(499^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 163 + 125\cdot 499 + 219\cdot 499^{2} + 146\cdot 499^{3} + 173\cdot 499^{4} +O(499^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 180 + 69\cdot 499 + 489\cdot 499^{2} + 150\cdot 499^{3} +O(499^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 412 + 177\cdot 499 + 47\cdot 499^{2} + 262\cdot 499^{3} + 363\cdot 499^{4} +O(499^{5})\)
|
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 |
| $1$ | $1$ | $()$ | $4$ | |
| $10$ | $2$ | $(1,2)$ | $2$ | |
| $15$ | $2$ | $(1,2)(3,4)$ | $0$ | ✓ |
| $20$ | $3$ | $(1,2,3)$ | $1$ | |
| $30$ | $4$ | $(1,2,3,4)$ | $0$ | |
| $24$ | $5$ | $(1,2,3,4,5)$ | $-1$ | |
| $20$ | $6$ | $(1,2,3)(4,5)$ | $-1$ |