Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 433 }$ to precision 8.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 29 + 391\cdot 433 + 233\cdot 433^{2} + 337\cdot 433^{3} + 66\cdot 433^{4} + 168\cdot 433^{5} + 115\cdot 433^{6} + 83\cdot 433^{7} +O\left(433^{ 8 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 101 + 195\cdot 433 + 200\cdot 433^{2} + 54\cdot 433^{3} + 155\cdot 433^{4} + 96\cdot 433^{5} + 353\cdot 433^{6} + 335\cdot 433^{7} +O\left(433^{ 8 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 138 + 109\cdot 433 + 269\cdot 433^{2} + 91\cdot 433^{3} + 42\cdot 433^{4} + 126\cdot 433^{5} + 238\cdot 433^{6} + 389\cdot 433^{7} +O\left(433^{ 8 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 171 + 53\cdot 433 + 392\cdot 433^{2} + 410\cdot 433^{3} + 29\cdot 433^{4} + 31\cdot 433^{5} + 252\cdot 433^{6} + 63\cdot 433^{7} +O\left(433^{ 8 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 262 + 379\cdot 433 + 40\cdot 433^{2} + 22\cdot 433^{3} + 403\cdot 433^{4} + 401\cdot 433^{5} + 180\cdot 433^{6} + 369\cdot 433^{7} +O\left(433^{ 8 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 295 + 323\cdot 433 + 163\cdot 433^{2} + 341\cdot 433^{3} + 390\cdot 433^{4} + 306\cdot 433^{5} + 194\cdot 433^{6} + 43\cdot 433^{7} +O\left(433^{ 8 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 332 + 237\cdot 433 + 232\cdot 433^{2} + 378\cdot 433^{3} + 277\cdot 433^{4} + 336\cdot 433^{5} + 79\cdot 433^{6} + 97\cdot 433^{7} +O\left(433^{ 8 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 404 + 41\cdot 433 + 199\cdot 433^{2} + 95\cdot 433^{3} + 366\cdot 433^{4} + 264\cdot 433^{5} + 317\cdot 433^{6} + 349\cdot 433^{7} +O\left(433^{ 8 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,8)(2,7)(3,6)(4,5)$ |
| $(1,6)(3,8)(4,5)$ |
| $(1,3,8,6)(2,4,7,5)$ |
| $(1,2,8,7)(3,5,6,4)$ |
| $(1,8)(3,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$4$ |
| $1$ |
$2$ |
$(1,8)(2,7)(3,6)(4,5)$ |
$-4$ |
| $2$ |
$2$ |
$(1,8)(3,6)$ |
$0$ |
| $4$ |
$2$ |
$(1,6)(3,8)(4,5)$ |
$0$ |
| $4$ |
$2$ |
$(1,7)(2,8)(3,4)(5,6)$ |
$0$ |
| $4$ |
$2$ |
$(1,3)(4,5)(6,8)$ |
$0$ |
| $2$ |
$4$ |
$(1,3,8,6)(2,4,7,5)$ |
$0$ |
| $2$ |
$4$ |
$(1,6,8,3)(2,4,7,5)$ |
$0$ |
| $4$ |
$4$ |
$(1,2,8,7)(3,5,6,4)$ |
$0$ |
| $4$ |
$8$ |
$(1,2,3,4,8,7,6,5)$ |
$0$ |
| $4$ |
$8$ |
$(1,7,6,4,8,2,3,5)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
