Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 467 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 198 + 203\cdot 467 + 185\cdot 467^{2} + 314\cdot 467^{3} + 409\cdot 467^{4} +O\left(467^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 338 + 186\cdot 467 + 79\cdot 467^{2} + 398\cdot 467^{3} + 270\cdot 467^{4} +O\left(467^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 433 + 369\cdot 467 + 305\cdot 467^{2} + 344\cdot 467^{3} + 48\cdot 467^{4} +O\left(467^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 437 + 388\cdot 467 + 81\cdot 467^{2} + 126\cdot 467^{3} + 349\cdot 467^{4} +O\left(467^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 464 + 251\cdot 467 + 281\cdot 467^{2} + 217\cdot 467^{3} + 322\cdot 467^{4} +O\left(467^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 5 }$
| Cycle notation |
| $(1,2)$ |
| $(1,2,3,4,5)$ |
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.
