Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 571 }$ to precision 9.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 161 + 312\cdot 571 + 483\cdot 571^{2} + 540\cdot 571^{3} + 251\cdot 571^{4} + 556\cdot 571^{5} + 415\cdot 571^{6} + 368\cdot 571^{7} + 268\cdot 571^{8} +O\left(571^{ 9 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 190 + 458\cdot 571 + 233\cdot 571^{2} + 484\cdot 571^{3} + 556\cdot 571^{4} + 323\cdot 571^{5} + 469\cdot 571^{6} + 55\cdot 571^{7} + 185\cdot 571^{8} +O\left(571^{ 9 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 207 + 297\cdot 571 + 543\cdot 571^{2} + 267\cdot 571^{3} + 49\cdot 571^{4} + 170\cdot 571^{5} + 409\cdot 571^{6} + 69\cdot 571^{7} + 146\cdot 571^{8} +O\left(571^{ 9 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 235 + 351\cdot 571 + 396\cdot 571^{2} + 324\cdot 571^{3} + 449\cdot 571^{4} + 58\cdot 571^{5} + 82\cdot 571^{6} + 219\cdot 571^{7} + 286\cdot 571^{8} +O\left(571^{ 9 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 337 + 219\cdot 571 + 174\cdot 571^{2} + 246\cdot 571^{3} + 121\cdot 571^{4} + 512\cdot 571^{5} + 488\cdot 571^{6} + 351\cdot 571^{7} + 284\cdot 571^{8} +O\left(571^{ 9 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 365 + 273\cdot 571 + 27\cdot 571^{2} + 303\cdot 571^{3} + 521\cdot 571^{4} + 400\cdot 571^{5} + 161\cdot 571^{6} + 501\cdot 571^{7} + 424\cdot 571^{8} +O\left(571^{ 9 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 382 + 112\cdot 571 + 337\cdot 571^{2} + 86\cdot 571^{3} + 14\cdot 571^{4} + 247\cdot 571^{5} + 101\cdot 571^{6} + 515\cdot 571^{7} + 385\cdot 571^{8} +O\left(571^{ 9 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 411 + 258\cdot 571 + 87\cdot 571^{2} + 30\cdot 571^{3} + 319\cdot 571^{4} + 14\cdot 571^{5} + 155\cdot 571^{6} + 202\cdot 571^{7} + 302\cdot 571^{8} +O\left(571^{ 9 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,5)(3,6)(4,8)$ |
| $(1,5)(2,3)(4,8)(6,7)$ |
| $(2,7)(4,5)$ |
| $(3,6)(4,5)$ |
| $(1,8)(3,6)$ |
| $(1,6,8,3)(2,5,7,4)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $4$ |
| $1$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $-4$ |
| $2$ | $2$ | $(1,8)(4,5)$ | $0$ |
| $4$ | $2$ | $(1,8)(3,6)$ | $0$ |
| $4$ | $2$ | $(1,5)(2,3)(4,8)(6,7)$ | $0$ |
| $4$ | $2$ | $(1,7)(2,8)(3,4)(5,6)$ | $0$ |
| $4$ | $2$ | $(1,6)(2,4)(3,8)(5,7)$ | $0$ |
| $8$ | $2$ | $(1,5)(3,6)(4,8)$ | $0$ |
| $2$ | $4$ | $(1,4,8,5)(2,6,7,3)$ | $0$ |
| $2$ | $4$ | $(1,5,8,4)(2,6,7,3)$ | $0$ |
| $4$ | $4$ | $(1,6,8,3)(2,5,7,4)$ | $0$ |
| $4$ | $4$ | $(1,4,8,5)(2,7)(3,6)$ | $2$ |
| $4$ | $4$ | $(1,6,8,3)(2,4,7,5)$ | $0$ |
| $4$ | $4$ | $(2,6,7,3)$ | $-2$ |
| $8$ | $8$ | $(1,7,4,3,8,2,5,6)$ | $0$ |
| $8$ | $8$ | $(1,7,5,3,8,2,4,6)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
