Basic invariants
| Dimension: | $6$ |
| Group: | $S_5$ |
| Conductor: | \(9515998270871\)\(\medspace = 21191^{3} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 5.3.21191.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | 20T30 |
| Parity: | odd |
| Determinant: | 1.21191.2t1.a.a |
| Projective image: | $S_5$ |
| Projective stem field: | Galois closure of 5.3.21191.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{5} - x^{4} - x^{2} - 3x - 1 \)
|
The roots of $f$ are computed in $\Q_{ 257 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 6 + 112\cdot 257 + 39\cdot 257^{2} + 171\cdot 257^{3} + 199\cdot 257^{4} +O(257^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 20 + 213\cdot 257 + 152\cdot 257^{2} + 58\cdot 257^{3} + 44\cdot 257^{4} +O(257^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 55 + 106\cdot 257 + 180\cdot 257^{2} + 38\cdot 257^{3} + 134\cdot 257^{4} +O(257^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 181 + 87\cdot 257 + 92\cdot 257^{2} + 133\cdot 257^{3} + 165\cdot 257^{4} +O(257^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 253 + 251\cdot 257 + 48\cdot 257^{2} + 112\cdot 257^{3} + 227\cdot 257^{4} +O(257^{5})\)
|
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 |
| $1$ | $1$ | $()$ | $6$ |
| $10$ | $2$ | $(1,2)$ | $0$ |
| $15$ | $2$ | $(1,2)(3,4)$ | $-2$ |
| $20$ | $3$ | $(1,2,3)$ | $0$ |
| $30$ | $4$ | $(1,2,3,4)$ | $0$ |
| $24$ | $5$ | $(1,2,3,4,5)$ | $1$ |
| $20$ | $6$ | $(1,2,3)(4,5)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.