Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 487 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 44 + 330\cdot 487 + 215\cdot 487^{2} + 484\cdot 487^{3} + 312\cdot 487^{4} +O\left(487^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 227 + 96\cdot 487 + 73\cdot 487^{2} + 169\cdot 487^{3} + 142\cdot 487^{4} +O\left(487^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 315 + 455\cdot 487 + 213\cdot 487^{2} + 189\cdot 487^{3} + 249\cdot 487^{4} +O\left(487^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 407 + 252\cdot 487 + 468\cdot 487^{2} + 73\cdot 487^{3} + 62\cdot 487^{4} +O\left(487^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 475 + 88\cdot 487 + 355\cdot 487^{2} + 260\cdot 487^{3} + 259\cdot 487^{4} +O\left(487^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 483 + 236\cdot 487 + 134\cdot 487^{2} + 283\cdot 487^{3} + 434\cdot 487^{4} +O\left(487^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,2)$ |
| $(1,2,3,4,5,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $10$ |
| $15$ | $2$ | $(1,2)(3,4)(5,6)$ | $2$ |
| $15$ | $2$ | $(1,2)$ | $-2$ |
| $45$ | $2$ | $(1,2)(3,4)$ | $-2$ |
| $40$ | $3$ | $(1,2,3)(4,5,6)$ | $1$ |
| $40$ | $3$ | $(1,2,3)$ | $1$ |
| $90$ | $4$ | $(1,2,3,4)(5,6)$ | $0$ |
| $90$ | $4$ | $(1,2,3,4)$ | $0$ |
| $144$ | $5$ | $(1,2,3,4,5)$ | $0$ |
| $120$ | $6$ | $(1,2,3,4,5,6)$ | $-1$ |
| $120$ | $6$ | $(1,2,3)(4,5)$ | $1$ |
The blue line marks the conjugacy class containing complex conjugation.
