Basic invariants
| Dimension: | $5$ |
| Group: | $S_5$ |
| Conductor: | \(15579283489\)\(\medspace = 7^{2} \cdot 11^{2} \cdot 1621^{2} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin number field: | Galois closure of 5.5.124817.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $S_5$ |
| Parity: | even |
| Projective image: | $S_5$ |
| Projective field: | Galois closure of 5.5.124817.1 |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 457 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 44 + 244\cdot 457 + 322\cdot 457^{2} + 104\cdot 457^{3} + 437\cdot 457^{4} +O(457^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 268 + 134\cdot 457 + 199\cdot 457^{2} + 181\cdot 457^{3} + 337\cdot 457^{4} +O(457^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 331 + 223\cdot 457 + 448\cdot 457^{2} + 121\cdot 457^{3} + 278\cdot 457^{4} +O(457^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 361 + 161\cdot 457 + 409\cdot 457^{2} + 40\cdot 457^{3} + 119\cdot 457^{4} +O(457^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 367 + 149\cdot 457 + 448\cdot 457^{2} + 7\cdot 457^{3} + 199\cdot 457^{4} +O(457^{5})\)
|
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 values |
| $c1$ | |||
| $1$ | $1$ | $()$ | $5$ |
| $10$ | $2$ | $(1,2)$ | $1$ |
| $15$ | $2$ | $(1,2)(3,4)$ | $1$ |
| $20$ | $3$ | $(1,2,3)$ | $-1$ |
| $30$ | $4$ | $(1,2,3,4)$ | $-1$ |
| $24$ | $5$ | $(1,2,3,4,5)$ | $0$ |
| $20$ | $6$ | $(1,2,3)(4,5)$ | $1$ |