Basic invariants
| Dimension: | $5$ |
| Group: | $S_5$ |
| Conductor: | \(1075648000000\)\(\medspace = 2^{12} \cdot 5^{6} \cdot 7^{5} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 5.1.5488000000.2 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $\PGL(2,5)$ |
| Parity: | even |
| Determinant: | 1.28.2t1.a.a |
| Projective image: | $S_5$ |
| Projective stem field: | Galois closure of 5.1.5488000000.2 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{5} + 20x^{3} - 80x^{2} + 60x + 352 \)
|
The roots of $f$ are computed in $\Q_{ 523 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 15 + 277\cdot 523 + 386\cdot 523^{2} + 373\cdot 523^{3} + 328\cdot 523^{4} +O(523^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 150 + 162\cdot 523 + 520\cdot 523^{2} + 246\cdot 523^{3} + 109\cdot 523^{4} +O(523^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 216 + 123\cdot 523 + 308\cdot 523^{2} + 366\cdot 523^{3} + 339\cdot 523^{4} +O(523^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 307 + 171\cdot 523 + 330\cdot 523^{2} + 390\cdot 523^{3} + 270\cdot 523^{4} +O(523^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 358 + 311\cdot 523 + 23\cdot 523^{2} + 191\cdot 523^{3} + 520\cdot 523^{4} +O(523^{5})\)
|
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.