Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 509 }$ to precision 7.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 99 + 463\cdot 509 + 352\cdot 509^{2} + 93\cdot 509^{3} + 381\cdot 509^{4} + 21\cdot 509^{5} + 340\cdot 509^{6} +O\left(509^{ 7 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 160 + 140\cdot 509 + 481\cdot 509^{2} + 362\cdot 509^{3} + 100\cdot 509^{4} + 252\cdot 509^{5} + 46\cdot 509^{6} +O\left(509^{ 7 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 311 + 257\cdot 509 + 193\cdot 509^{2} + 99\cdot 509^{3} + 301\cdot 509^{4} + 340\cdot 509^{5} + 464\cdot 509^{6} +O\left(509^{ 7 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 373 + 90\cdot 509 + 37\cdot 509^{2} + 91\cdot 509^{3} + 265\cdot 509^{4} + 508\cdot 509^{5} + 412\cdot 509^{6} +O\left(509^{ 7 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 374 + 245\cdot 509 + 157\cdot 509^{2} + 231\cdot 509^{3} + 197\cdot 509^{4} + 489\cdot 509^{5} + 287\cdot 509^{6} +O\left(509^{ 7 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 382 + 52\cdot 509 + 290\cdot 509^{2} + 213\cdot 509^{3} + 345\cdot 509^{4} + 284\cdot 509^{5} + 417\cdot 509^{6} +O\left(509^{ 7 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 417 + 335\cdot 509 + 498\cdot 509^{2} + 301\cdot 509^{3} + 279\cdot 509^{4} + 181\cdot 509^{5} + 293\cdot 509^{6} +O\left(509^{ 7 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 431 + 449\cdot 509 + 24\cdot 509^{2} + 133\cdot 509^{3} + 165\cdot 509^{4} + 466\cdot 509^{5} + 281\cdot 509^{6} +O\left(509^{ 7 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(2,6)(4,7)$ |
| $(1,3,8,5)(2,4,6,7)$ |
| $(1,2,4,3)(5,8,6,7)$ |
| $(1,8)(4,7)$ |
| $(2,6)(3,5)$ |
| $(1,3,7,2)(4,6,8,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$4$ |
| $1$ |
$2$ |
$(1,8)(2,6)(3,5)(4,7)$ |
$-4$ |
| $2$ |
$2$ |
$(1,4)(2,3)(5,6)(7,8)$ |
$0$ |
| $2$ |
$2$ |
$(2,6)(3,5)$ |
$0$ |
| $2$ |
$2$ |
$(1,7)(2,3)(4,8)(5,6)$ |
$0$ |
| $4$ |
$2$ |
$(1,5)(2,4)(3,8)(6,7)$ |
$0$ |
| $4$ |
$2$ |
$(1,6)(2,8)(3,7)(4,5)$ |
$0$ |
| $4$ |
$2$ |
$(2,6)(4,7)$ |
$0$ |
| $4$ |
$2$ |
$(2,5)(3,6)$ |
$2$ |
| $4$ |
$2$ |
$(1,8)(2,5)(3,6)(4,7)$ |
$-2$ |
| $4$ |
$4$ |
$(1,3,8,5)(2,4,6,7)$ |
$0$ |
| $4$ |
$4$ |
$(1,2,8,6)(3,7,5,4)$ |
$0$ |
| $4$ |
$4$ |
$(1,4,8,7)(2,5,6,3)$ |
$0$ |
| $8$ |
$4$ |
$(1,2,4,3)(5,8,6,7)$ |
$0$ |
| $8$ |
$4$ |
$(1,3,7,2)(4,6,8,5)$ |
$0$ |
| $8$ |
$4$ |
$(2,3,6,5)(4,7)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
