Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 433 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 7 + 264\cdot 433 + 109\cdot 433^{2} + 287\cdot 433^{3} + 162\cdot 433^{4} +O\left(433^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 46 + 341\cdot 433 + 239\cdot 433^{2} + 20\cdot 433^{3} + 427\cdot 433^{4} +O\left(433^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 56 + 135\cdot 433 + 212\cdot 433^{2} + 356\cdot 433^{3} + 117\cdot 433^{4} +O\left(433^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 65 + 180\cdot 433 + 180\cdot 433^{2} + 205\cdot 433^{3} + 352\cdot 433^{4} +O\left(433^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 368 + 252\cdot 433 + 252\cdot 433^{2} + 227\cdot 433^{3} + 80\cdot 433^{4} +O\left(433^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 377 + 297\cdot 433 + 220\cdot 433^{2} + 76\cdot 433^{3} + 315\cdot 433^{4} +O\left(433^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 387 + 91\cdot 433 + 193\cdot 433^{2} + 412\cdot 433^{3} + 5\cdot 433^{4} +O\left(433^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 426 + 168\cdot 433 + 323\cdot 433^{2} + 145\cdot 433^{3} + 270\cdot 433^{4} +O\left(433^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,8)(2,7)(3,6)(4,5)$ |
| $(1,7)(2,8)(4,5)$ |
| $(1,7,8,2)(3,4,6,5)$ |
| $(1,8)(2,7)$ |
| $(1,6,8,3)(2,5,7,4)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $4$ |
| $1$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $-4$ |
| $2$ | $2$ | $(1,8)(2,7)$ | $0$ |
| $4$ | $2$ | $(1,7)(2,8)(4,5)$ | $0$ |
| $4$ | $2$ | $(1,3)(2,4)(5,7)(6,8)$ | $0$ |
| $4$ | $2$ | $(1,2)(4,5)(7,8)$ | $0$ |
| $2$ | $4$ | $(1,7,8,2)(3,4,6,5)$ | $0$ |
| $2$ | $4$ | $(1,2,8,7)(3,4,6,5)$ | $0$ |
| $4$ | $4$ | $(1,6,8,3)(2,5,7,4)$ | $0$ |
| $4$ | $8$ | $(1,6,2,4,8,3,7,5)$ | $0$ |
| $4$ | $8$ | $(1,3,7,4,8,6,2,5)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
