Basic invariants
| Dimension: | $5$ |
| Group: | $S_5$ |
| Conductor: | \(509991889\)\(\medspace = 11^{2} \cdot 2053^{2} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 5.3.22583.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $S_5$ |
| Parity: | even |
| Determinant: | 1.1.1t1.a.a |
| Projective image: | $S_5$ |
| Projective stem field: | Galois closure of 5.3.22583.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{5} - x^{2} - 2x + 1 \)
|
The roots of $f$ are computed in $\Q_{ 157 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 16 + 103\cdot 157 + 13\cdot 157^{2} + 151\cdot 157^{3} + 13\cdot 157^{4} +O(157^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 53 + 6\cdot 157 + 17\cdot 157^{2} + 135\cdot 157^{3} + 45\cdot 157^{4} +O(157^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 55 + 69\cdot 157 + 73\cdot 157^{2} + 21\cdot 157^{3} + 114\cdot 157^{4} +O(157^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 64 + 7\cdot 157 + 106\cdot 157^{2} + 21\cdot 157^{3} + 138\cdot 157^{4} +O(157^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 126 + 127\cdot 157 + 103\cdot 157^{2} + 141\cdot 157^{3} + 157^{4} +O(157^{5})\)
|
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 |
| $1$ | $1$ | $()$ | $5$ |
| $10$ | $2$ | $(1,2)$ | $1$ |
| $15$ | $2$ | $(1,2)(3,4)$ | $1$ |
| $20$ | $3$ | $(1,2,3)$ | $-1$ |
| $30$ | $4$ | $(1,2,3,4)$ | $-1$ |
| $24$ | $5$ | $(1,2,3,4,5)$ | $0$ |
| $20$ | $6$ | $(1,2,3)(4,5)$ | $1$ |
The blue line marks the conjugacy class containing complex conjugation.