Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 647 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 172 + 80\cdot 647 + 556\cdot 647^{2} + 194\cdot 647^{3} + 147\cdot 647^{4} +O\left(647^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 223 + 175\cdot 647 + 376\cdot 647^{2} + 500\cdot 647^{3} + 437\cdot 647^{4} +O\left(647^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 325 + 151\cdot 647 + 31\cdot 647^{2} + 132\cdot 647^{3} + 308\cdot 647^{4} +O\left(647^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 418 + 512\cdot 647 + 516\cdot 647^{2} + 347\cdot 647^{3} + 621\cdot 647^{4} +O\left(647^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 452 + 441\cdot 647 + 630\cdot 647^{2} + 538\cdot 647^{3} + 517\cdot 647^{4} +O\left(647^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 456 + 173\cdot 647 + 521\cdot 647^{2} + 47\cdot 647^{3} + 377\cdot 647^{4} +O\left(647^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 561 + 326\cdot 647 + 71\cdot 647^{2} + 127\cdot 647^{3} + 168\cdot 647^{4} +O\left(647^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 629 + 78\cdot 647 + 531\cdot 647^{2} + 51\cdot 647^{3} + 10\cdot 647^{4} +O\left(647^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,8)(2,6)(3,4)(5,7)$ |
| $(1,4)(5,6)$ |
| $(1,4)(2,7)(3,8)(5,6)$ |
| $(1,5,4,6)(2,7)$ |
| $(1,5)(2,8)(3,7)(4,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $4$ |
| $1$ | $2$ | $(1,4)(2,7)(3,8)(5,6)$ | $-4$ |
| $2$ | $2$ | $(1,5)(2,8)(3,7)(4,6)$ | $0$ |
| $2$ | $2$ | $(1,4)(5,6)$ | $0$ |
| $2$ | $2$ | $(1,6)(2,8)(3,7)(4,5)$ | $0$ |
| $4$ | $2$ | $(1,8)(2,6)(3,4)(5,7)$ | $0$ |
| $4$ | $4$ | $(1,8,5,2)(3,6,7,4)$ | $0$ |
| $4$ | $4$ | $(1,2,5,8)(3,4,7,6)$ | $0$ |
| $4$ | $4$ | $(1,5,4,6)(2,7)$ | $0$ |
| $4$ | $4$ | $(1,6,4,5)(2,7)$ | $0$ |
| $4$ | $4$ | $(1,2,4,7)(3,6,8,5)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
