Basic invariants
| Dimension: | $4$ |
| Group: | $S_5$ |
| Conductor: | \(18574970416\)\(\medspace = 2^{4} \cdot 1051^{3} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 5.3.16816.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $S_5$ |
| Parity: | odd |
| Determinant: | 1.1051.2t1.a.a |
| Projective image: | $S_5$ |
| Projective stem field: | Galois closure of 5.3.16816.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{5} - 2x^{3} - 2x^{2} - 4x - 2 \)
|
The roots of $f$ are computed in $\Q_{ 541 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 72 + 518\cdot 541 + 520\cdot 541^{2} + 441\cdot 541^{3} + 281\cdot 541^{4} +O(541^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 87 + 63\cdot 541 + 166\cdot 541^{2} + 541^{3} + 35\cdot 541^{4} +O(541^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 246 + 91\cdot 541 + 97\cdot 541^{2} + 177\cdot 541^{3} + 148\cdot 541^{4} +O(541^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 284 + 379\cdot 541 + 21\cdot 541^{2} + 301\cdot 541^{3} + 189\cdot 541^{4} +O(541^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 393 + 29\cdot 541 + 276\cdot 541^{2} + 160\cdot 541^{3} + 427\cdot 541^{4} +O(541^{5})\)
|
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 |
| $1$ | $1$ | $()$ | $4$ |
| $10$ | $2$ | $(1,2)$ | $-2$ |
| $15$ | $2$ | $(1,2)(3,4)$ | $0$ |
| $20$ | $3$ | $(1,2,3)$ | $1$ |
| $30$ | $4$ | $(1,2,3,4)$ | $0$ |
| $24$ | $5$ | $(1,2,3,4,5)$ | $-1$ |
| $20$ | $6$ | $(1,2,3)(4,5)$ | $1$ |
The blue line marks the conjugacy class containing complex conjugation.