Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 541 }$ to precision 8.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 115 + 243\cdot 541 + 267\cdot 541^{2} + 354\cdot 541^{3} + 454\cdot 541^{4} + 303\cdot 541^{5} + 55\cdot 541^{6} + 290\cdot 541^{7} +O\left(541^{ 8 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 127 + 314\cdot 541 + 440\cdot 541^{2} + 443\cdot 541^{3} + 64\cdot 541^{4} + 300\cdot 541^{5} + 368\cdot 541^{6} + 329\cdot 541^{7} +O\left(541^{ 8 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 131 + 399\cdot 541 + 202\cdot 541^{2} + 94\cdot 541^{3} + 501\cdot 541^{4} + 522\cdot 541^{5} + 529\cdot 541^{6} + 437\cdot 541^{7} +O\left(541^{ 8 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 263 + 41\cdot 541 + 387\cdot 541^{2} + 143\cdot 541^{3} + 386\cdot 541^{4} + 294\cdot 541^{5} + 88\cdot 541^{6} + 431\cdot 541^{7} +O\left(541^{ 8 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 312 + 328\cdot 541 + 46\cdot 541^{2} + 387\cdot 541^{3} + 138\cdot 541^{4} + 86\cdot 541^{5} + 268\cdot 541^{6} + 165\cdot 541^{7} +O\left(541^{ 8 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 365 + 311\cdot 541 + 24\cdot 541^{2} + 438\cdot 541^{3} + 328\cdot 541^{4} + 135\cdot 541^{5} + 476\cdot 541^{6} + 485\cdot 541^{7} +O\left(541^{ 8 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 412 + 387\cdot 541 + 261\cdot 541^{2} + 413\cdot 541^{3} + 44\cdot 541^{4} + 470\cdot 541^{5} + 529\cdot 541^{6} + 289\cdot 541^{7} +O\left(541^{ 8 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 439 + 137\cdot 541 + 533\cdot 541^{2} + 429\cdot 541^{3} + 244\cdot 541^{4} + 50\cdot 541^{5} + 388\cdot 541^{6} + 274\cdot 541^{7} +O\left(541^{ 8 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,3,8,2)(4,7,6,5)$ |
| $(1,5)(2,6)(3,4)(7,8)$ |
| $(1,5)(3,4)$ |
| $(1,5)(2,3)(4,6)$ |
| $(1,5)(7,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$4$ |
| $1$ |
$2$ |
$(1,5)(2,6)(3,4)(7,8)$ |
$-4$ |
| $2$ |
$2$ |
$(2,6)(3,4)$ |
$0$ |
| $4$ |
$2$ |
$(1,8)(2,3)(4,6)(5,7)$ |
$0$ |
| $4$ |
$2$ |
$(1,5)(3,4)$ |
$0$ |
| $8$ |
$2$ |
$(1,5)(2,3)(4,6)$ |
$0$ |
| $4$ |
$4$ |
$(2,3,6,4)$ |
$2$ |
| $4$ |
$4$ |
$(1,5)(2,4,6,3)(7,8)$ |
$-2$ |
| $4$ |
$4$ |
$(1,8,5,7)(2,3,6,4)$ |
$0$ |
| $8$ |
$4$ |
$(1,3,8,2)(4,7,6,5)$ |
$0$ |
| $8$ |
$4$ |
$(1,2,8,3)(4,5,6,7)$ |
$0$ |
| $8$ |
$4$ |
$(1,2)(3,8,4,7)(5,6)$ |
$0$ |
| $8$ |
$4$ |
$(1,2)(3,7,4,8)(5,6)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
