Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 571 }$ to precision 7.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 145 + 299\cdot 571 + 418\cdot 571^{2} + 19\cdot 571^{3} + 76\cdot 571^{4} + 507\cdot 571^{5} + 322\cdot 571^{6} +O\left(571^{ 7 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 191 + 32\cdot 571 + 319\cdot 571^{2} + 262\cdot 571^{3} + 179\cdot 571^{4} + 104\cdot 571^{5} + 296\cdot 571^{6} +O\left(571^{ 7 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 226 + 186\cdot 571 + 154\cdot 571^{2} + 28\cdot 571^{3} + 217\cdot 571^{4} + 562\cdot 571^{5} + 433\cdot 571^{6} +O\left(571^{ 7 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 242 + 371\cdot 571 + 503\cdot 571^{2} + 196\cdot 571^{3} + 119\cdot 571^{4} + 478\cdot 571^{5} + 317\cdot 571^{6} +O\left(571^{ 7 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 245 + 225\cdot 571 + 49\cdot 571^{2} + 105\cdot 571^{3} + 425\cdot 571^{4} + 555\cdot 571^{5} + 426\cdot 571^{6} +O\left(571^{ 7 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 372 + 306\cdot 571 + 399\cdot 571^{2} + 384\cdot 571^{3} + 382\cdot 571^{4} + 356\cdot 571^{5} + 552\cdot 571^{6} +O\left(571^{ 7 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 382 + 321\cdot 571 + 170\cdot 571^{2} + 67\cdot 571^{3} + 157\cdot 571^{4} + 324\cdot 571^{5} + 209\cdot 571^{6} +O\left(571^{ 7 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 484 + 540\cdot 571 + 268\cdot 571^{2} + 77\cdot 571^{3} + 156\cdot 571^{4} + 537\cdot 571^{5} + 294\cdot 571^{6} +O\left(571^{ 7 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(2,6)(3,5)(4,7)$ |
| $(1,7,8,4)(2,6,5,3)$ |
| $(1,8)(2,5)$ |
| $(3,6)(4,7)$ |
| $(1,8)(4,7)$ |
| $(1,2)(3,4)(5,8)(6,7)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$4$ |
| $1$ |
$2$ |
$(1,8)(2,5)(3,6)(4,7)$ |
$-4$ |
| $2$ |
$2$ |
$(2,5)(3,6)$ |
$0$ |
| $4$ |
$2$ |
$(1,8)(2,5)$ |
$0$ |
| $4$ |
$2$ |
$(1,2)(3,4)(5,8)(6,7)$ |
$0$ |
| $4$ |
$2$ |
$(1,5)(2,8)(3,4)(6,7)$ |
$0$ |
| $4$ |
$2$ |
$(1,4)(2,3)(5,6)(7,8)$ |
$0$ |
| $8$ |
$2$ |
$(2,6)(3,5)(4,7)$ |
$0$ |
| $2$ |
$4$ |
$(1,4,8,7)(2,3,5,6)$ |
$0$ |
| $2$ |
$4$ |
$(1,7,8,4)(2,3,5,6)$ |
$0$ |
| $4$ |
$4$ |
$(1,3,8,6)(2,4,5,7)$ |
$0$ |
| $4$ |
$4$ |
$(2,6,5,3)$ |
$2$ |
| $4$ |
$4$ |
$(1,2,8,5)(3,4,6,7)$ |
$0$ |
| $4$ |
$4$ |
$(1,8)(2,6,5,3)(4,7)$ |
$-2$ |
| $8$ |
$8$ |
$(1,2,7,3,8,5,4,6)$ |
$0$ |
| $8$ |
$8$ |
$(1,2,7,6,8,5,4,3)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
