Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 509 }$ to precision 9.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 109 + 462\cdot 509 + 499\cdot 509^{2} + 407\cdot 509^{3} + 348\cdot 509^{4} + 104\cdot 509^{5} + 320\cdot 509^{6} + 485\cdot 509^{7} + 321\cdot 509^{8} +O\left(509^{ 9 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 142 + 58\cdot 509 + 500\cdot 509^{2} + 294\cdot 509^{3} + 411\cdot 509^{4} + 28\cdot 509^{5} + 181\cdot 509^{6} + 162\cdot 509^{7} + 30\cdot 509^{8} +O\left(509^{ 9 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 200 + 451\cdot 509 + 260\cdot 509^{2} + 20\cdot 509^{3} + 391\cdot 509^{4} + 52\cdot 509^{5} + 306\cdot 509^{6} + 423\cdot 509^{7} + 265\cdot 509^{8} +O\left(509^{ 9 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 225 + 165\cdot 509 + 289\cdot 509^{2} + 445\cdot 509^{3} + 228\cdot 509^{4} + 414\cdot 509^{5} + 140\cdot 509^{6} + 197\cdot 509^{7} + 355\cdot 509^{8} +O\left(509^{ 9 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 284 + 343\cdot 509 + 219\cdot 509^{2} + 63\cdot 509^{3} + 280\cdot 509^{4} + 94\cdot 509^{5} + 368\cdot 509^{6} + 311\cdot 509^{7} + 153\cdot 509^{8} +O\left(509^{ 9 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 309 + 57\cdot 509 + 248\cdot 509^{2} + 488\cdot 509^{3} + 117\cdot 509^{4} + 456\cdot 509^{5} + 202\cdot 509^{6} + 85\cdot 509^{7} + 243\cdot 509^{8} +O\left(509^{ 9 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 367 + 450\cdot 509 + 8\cdot 509^{2} + 214\cdot 509^{3} + 97\cdot 509^{4} + 480\cdot 509^{5} + 327\cdot 509^{6} + 346\cdot 509^{7} + 478\cdot 509^{8} +O\left(509^{ 9 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 400 + 46\cdot 509 + 9\cdot 509^{2} + 101\cdot 509^{3} + 160\cdot 509^{4} + 404\cdot 509^{5} + 188\cdot 509^{6} + 23\cdot 509^{7} + 187\cdot 509^{8} +O\left(509^{ 9 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,2,8,7)(4,5)$ |
| $(1,4,8,5)(2,3,7,6)$ |
| $(1,8)(4,5)$ |
| $(2,7)(3,6)$ |
| $(1,8)(2,7)$ |
| $(1,3,8,6)(2,5,7,4)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $4$ |
| $1$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $-4$ |
| $2$ | $2$ | $(1,8)(2,7)$ | $0$ |
| $2$ | $2$ | $(1,7)(2,8)(3,4)(5,6)$ | $0$ |
| $2$ | $2$ | $(1,2)(3,4)(5,6)(7,8)$ | $0$ |
| $4$ | $2$ | $(2,7)(3,6)$ | $0$ |
| $4$ | $2$ | $(1,7)(2,8)(3,6)(4,5)$ | $2$ |
| $4$ | $2$ | $(3,5)(4,6)$ | $-2$ |
| $4$ | $2$ | $(1,5)(2,6)(3,7)(4,8)$ | $0$ |
| $4$ | $2$ | $(1,6)(2,4)(3,8)(5,7)$ | $0$ |
| $4$ | $4$ | $(1,4,8,5)(2,3,7,6)$ | $0$ |
| $4$ | $4$ | $(1,3,8,6)(2,5,7,4)$ | $0$ |
| $4$ | $4$ | $(1,2,8,7)(3,4,6,5)$ | $0$ |
| $8$ | $4$ | $(1,2,8,7)(4,5)$ | $0$ |
| $8$ | $4$ | $(1,3,7,4)(2,5,8,6)$ | $0$ |
| $8$ | $4$ | $(1,5,2,6)(3,8,4,7)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
