Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 571 }$ to precision 8.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 62 + 16\cdot 571 + 382\cdot 571^{2} + 560\cdot 571^{3} + 5\cdot 571^{4} + 395\cdot 571^{5} + 144\cdot 571^{6} + 241\cdot 571^{7} +O\left(571^{ 8 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 117 + 419\cdot 571 + 205\cdot 571^{2} + 186\cdot 571^{3} + 456\cdot 571^{4} + 240\cdot 571^{5} + 427\cdot 571^{6} + 137\cdot 571^{7} +O\left(571^{ 8 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 132 + 283\cdot 571 + 393\cdot 571^{2} + 145\cdot 571^{3} + 317\cdot 571^{4} + 404\cdot 571^{5} + 146\cdot 571^{6} + 61\cdot 571^{7} +O\left(571^{ 8 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 256 + 153\cdot 571 + 570\cdot 571^{2} + 485\cdot 571^{3} + 436\cdot 571^{4} + 55\cdot 571^{5} + 274\cdot 571^{6} + 567\cdot 571^{7} +O\left(571^{ 8 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 316 + 417\cdot 571 + 85\cdot 571^{3} + 134\cdot 571^{4} + 515\cdot 571^{5} + 296\cdot 571^{6} + 3\cdot 571^{7} +O\left(571^{ 8 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 440 + 287\cdot 571 + 177\cdot 571^{2} + 425\cdot 571^{3} + 253\cdot 571^{4} + 166\cdot 571^{5} + 424\cdot 571^{6} + 509\cdot 571^{7} +O\left(571^{ 8 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 455 + 151\cdot 571 + 365\cdot 571^{2} + 384\cdot 571^{3} + 114\cdot 571^{4} + 330\cdot 571^{5} + 143\cdot 571^{6} + 433\cdot 571^{7} +O\left(571^{ 8 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 510 + 554\cdot 571 + 188\cdot 571^{2} + 10\cdot 571^{3} + 565\cdot 571^{4} + 175\cdot 571^{5} + 426\cdot 571^{6} + 329\cdot 571^{7} +O\left(571^{ 8 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,8)(2,7)(3,6)(4,5)$ |
| $(1,4)(2,3)(5,8)(6,7)$ |
| $(2,4,7,5)$ |
| $(2,7)(4,5)$ |
| $(1,6,8,3)(2,5,7,4)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $-2$ |
| $2$ | $2$ | $(2,7)(4,5)$ | $0$ |
| $4$ | $2$ | $(1,4)(2,3)(5,8)(6,7)$ | $0$ |
| $1$ | $4$ | $(1,6,8,3)(2,4,7,5)$ | $-2 \zeta_{4}$ |
| $1$ | $4$ | $(1,3,8,6)(2,5,7,4)$ | $2 \zeta_{4}$ |
| $2$ | $4$ | $(1,6,8,3)(2,5,7,4)$ | $0$ |
| $2$ | $4$ | $(2,4,7,5)$ | $\zeta_{4} - 1$ |
| $2$ | $4$ | $(2,5,7,4)$ | $-\zeta_{4} - 1$ |
| $2$ | $4$ | $(1,8)(2,5,7,4)(3,6)$ | $-\zeta_{4} + 1$ |
| $2$ | $4$ | $(1,8)(2,4,7,5)(3,6)$ | $\zeta_{4} + 1$ |
| $4$ | $4$ | $(1,5,8,4)(2,3,7,6)$ | $0$ |
| $4$ | $8$ | $(1,7,6,5,8,2,3,4)$ | $0$ |
| $4$ | $8$ | $(1,5,3,7,8,4,6,2)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
