Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 463 }$ to precision 10.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 43 + 329\cdot 463 + 227\cdot 463^{2} + 446\cdot 463^{3} + 173\cdot 463^{4} + 172\cdot 463^{5} + 266\cdot 463^{6} + 128\cdot 463^{7} + 461\cdot 463^{8} + 175\cdot 463^{9} +O\left(463^{ 10 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 84 + 263\cdot 463 + 248\cdot 463^{2} + 77\cdot 463^{3} + 267\cdot 463^{4} + 296\cdot 463^{5} + 90\cdot 463^{6} + 37\cdot 463^{7} + 211\cdot 463^{8} + 358\cdot 463^{9} +O\left(463^{ 10 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 107 + 89\cdot 463 + 35\cdot 463^{2} + 428\cdot 463^{3} + 179\cdot 463^{4} + 338\cdot 463^{5} + 296\cdot 463^{6} + 292\cdot 463^{7} + 54\cdot 463^{8} + 440\cdot 463^{9} +O\left(463^{ 10 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 155 + 356\cdot 463 + 220\cdot 463^{2} + 327\cdot 463^{3} + 23\cdot 463^{4} + 348\cdot 463^{5} + 392\cdot 463^{6} + 49\cdot 463^{7} + 218\cdot 463^{8} + 183\cdot 463^{9} +O\left(463^{ 10 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 309 + 106\cdot 463 + 242\cdot 463^{2} + 135\cdot 463^{3} + 439\cdot 463^{4} + 114\cdot 463^{5} + 70\cdot 463^{6} + 413\cdot 463^{7} + 244\cdot 463^{8} + 279\cdot 463^{9} +O\left(463^{ 10 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 357 + 373\cdot 463 + 427\cdot 463^{2} + 34\cdot 463^{3} + 283\cdot 463^{4} + 124\cdot 463^{5} + 166\cdot 463^{6} + 170\cdot 463^{7} + 408\cdot 463^{8} + 22\cdot 463^{9} +O\left(463^{ 10 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 380 + 199\cdot 463 + 214\cdot 463^{2} + 385\cdot 463^{3} + 195\cdot 463^{4} + 166\cdot 463^{5} + 372\cdot 463^{6} + 425\cdot 463^{7} + 251\cdot 463^{8} + 104\cdot 463^{9} +O\left(463^{ 10 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 421 + 133\cdot 463 + 235\cdot 463^{2} + 16\cdot 463^{3} + 289\cdot 463^{4} + 290\cdot 463^{5} + 196\cdot 463^{6} + 334\cdot 463^{7} + 463^{8} + 287\cdot 463^{9} +O\left(463^{ 10 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,8)(2,7)(3,6)(4,5)$ |
| $(1,4,8,5)(2,6,7,3)$ |
| $(1,2)(4,5)(7,8)$ |
| $(1,2,8,7)(3,5,6,4)$ |
| $(1,8)(2,7)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $4$ |
| $1$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $-4$ |
| $2$ | $2$ | $(1,8)(2,7)$ | $0$ |
| $4$ | $2$ | $(1,2)(4,5)(7,8)$ | $0$ |
| $4$ | $2$ | $(1,5)(2,3)(4,8)(6,7)$ | $0$ |
| $4$ | $2$ | $(1,7)(2,8)(4,5)$ | $0$ |
| $2$ | $4$ | $(1,2,8,7)(3,5,6,4)$ | $0$ |
| $2$ | $4$ | $(1,7,8,2)(3,5,6,4)$ | $0$ |
| $4$ | $4$ | $(1,4,8,5)(2,6,7,3)$ | $0$ |
| $4$ | $8$ | $(1,5,2,6,8,4,7,3)$ | $0$ |
| $4$ | $8$ | $(1,4,7,6,8,5,2,3)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
