Basic invariants
| Dimension: | $5$ |
| Group: | $S_5$ |
| Conductor: | \(13236200869377149\)\(\medspace = 236549^{3} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin number field: | Galois closure of 5.5.236549.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $\PGL(2,5)$ |
| Parity: | even |
| Projective image: | $S_5$ |
| Projective field: | Galois closure of 5.5.236549.1 |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 461 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 98 + 99\cdot 461 + 277\cdot 461^{2} + 443\cdot 461^{3} + 317\cdot 461^{4} +O(461^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 203 + 279\cdot 461 + 175\cdot 461^{2} + 315\cdot 461^{3} + 430\cdot 461^{4} +O(461^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 247 + 346\cdot 461 + 218\cdot 461^{2} + 382\cdot 461^{3} + 138\cdot 461^{4} +O(461^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 386 + 441\cdot 461 + 9\cdot 461^{2} + 144\cdot 461^{3} + 38\cdot 461^{4} +O(461^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 450 + 215\cdot 461 + 240\cdot 461^{2} + 97\cdot 461^{3} + 457\cdot 461^{4} +O(461^{5})\)
|
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 |
| $c1$ | |||
| $1$ | $1$ | $()$ | $5$ |
| $10$ | $2$ | $(1,2)$ | $-1$ |
| $15$ | $2$ | $(1,2)(3,4)$ | $1$ |
| $20$ | $3$ | $(1,2,3)$ | $-1$ |
| $30$ | $4$ | $(1,2,3,4)$ | $1$ |
| $24$ | $5$ | $(1,2,3,4,5)$ | $0$ |
| $20$ | $6$ | $(1,2,3)(4,5)$ | $-1$ |