Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 523 }$ to precision 8.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 7 + 40\cdot 523 + 130\cdot 523^{2} + 522\cdot 523^{3} + 4\cdot 523^{4} + 470\cdot 523^{5} + 262\cdot 523^{6} + 389\cdot 523^{7} +O\left(523^{ 8 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 33 + 374\cdot 523 + 20\cdot 523^{2} + 370\cdot 523^{3} + 420\cdot 523^{4} + 432\cdot 523^{5} + 448\cdot 523^{6} + 77\cdot 523^{7} +O\left(523^{ 8 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 117 + 195\cdot 523 + 54\cdot 523^{2} + 358\cdot 523^{3} + 474\cdot 523^{4} + 55\cdot 523^{5} + 433\cdot 523^{6} + 286\cdot 523^{7} +O\left(523^{ 8 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 124 + 149\cdot 523 + 153\cdot 523^{2} + 408\cdot 523^{3} + 280\cdot 523^{4} + 43\cdot 523^{5} + 44\cdot 523^{6} + 71\cdot 523^{7} +O\left(523^{ 8 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 153 + 428\cdot 523 + 484\cdot 523^{2} + 415\cdot 523^{3} + 338\cdot 523^{4} + 367\cdot 523^{5} + 153\cdot 523^{6} + 304\cdot 523^{7} +O\left(523^{ 8 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 375 + 469\cdot 523 + 447\cdot 523^{2} + 343\cdot 523^{3} + 348\cdot 523^{4} + 417\cdot 523^{5} + 427\cdot 523^{6} + 195\cdot 523^{7} +O\left(523^{ 8 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 380 + 64\cdot 523 + 510\cdot 523^{2} + 253\cdot 523^{3} + 271\cdot 523^{4} + 12\cdot 523^{5} + 238\cdot 523^{6} + 141\cdot 523^{7} +O\left(523^{ 8 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 381 + 370\cdot 523 + 290\cdot 523^{2} + 465\cdot 523^{3} + 474\cdot 523^{4} + 291\cdot 523^{5} + 83\cdot 523^{6} + 102\cdot 523^{7} +O\left(523^{ 8 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,8)(2,4)$ |
| $(1,8)(2,5,4,7)$ |
| $(1,5)(2,3)(4,6)(7,8)$ |
| $(3,6)(5,7)$ |
| $(2,4)(5,7)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$4$ |
| $1$ |
$2$ |
$(1,8)(2,4)(3,6)(5,7)$ |
$-4$ |
| $2$ |
$2$ |
$(1,6)(2,5)(3,8)(4,7)$ |
$0$ |
| $2$ |
$2$ |
$(1,8)(3,6)$ |
$0$ |
| $2$ |
$2$ |
$(1,6)(2,7)(3,8)(4,5)$ |
$0$ |
| $4$ |
$2$ |
$(1,5)(2,3)(4,6)(7,8)$ |
$0$ |
| $4$ |
$2$ |
$(3,6)(5,7)$ |
$0$ |
| $4$ |
$2$ |
$(1,3)(2,4)(5,7)(6,8)$ |
$-2$ |
| $4$ |
$2$ |
$(1,5)(2,6)(3,4)(7,8)$ |
$0$ |
| $4$ |
$2$ |
$(1,3)(6,8)$ |
$2$ |
| $4$ |
$4$ |
$(1,5,8,7)(2,3,4,6)$ |
$0$ |
| $4$ |
$4$ |
$(1,5,8,7)(2,6,4,3)$ |
$0$ |
| $4$ |
$4$ |
$(1,6,8,3)(2,7,4,5)$ |
$0$ |
| $8$ |
$4$ |
$(1,4,6,7)(2,3,5,8)$ |
$0$ |
| $8$ |
$4$ |
$(1,6,8,3)(5,7)$ |
$0$ |
| $8$ |
$4$ |
$(1,2,6,7)(3,5,8,4)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
