Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 661 }$ to precision 6.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 22 + 324\cdot 661 + 280\cdot 661^{2} + 368\cdot 661^{3} + 57\cdot 661^{4} + 37\cdot 661^{5} +O\left(661^{ 6 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 237 + 365\cdot 661 + 18\cdot 661^{2} + 437\cdot 661^{3} + 104\cdot 661^{4} + 247\cdot 661^{5} +O\left(661^{ 6 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 274 + 636\cdot 661 + 659\cdot 661^{2} + 21\cdot 661^{3} + 66\cdot 661^{4} + 133\cdot 661^{5} +O\left(661^{ 6 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 380 + 252\cdot 661 + 623\cdot 661^{2} + 636\cdot 661^{3} + 629\cdot 661^{4} + 584\cdot 661^{5} +O\left(661^{ 6 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 423 + 313\cdot 661 + 379\cdot 661^{2} + 407\cdot 661^{3} + 548\cdot 661^{4} + 398\cdot 661^{5} +O\left(661^{ 6 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 431 + 467\cdot 661 + 477\cdot 661^{2} + 398\cdot 661^{3} + 281\cdot 661^{4} + 582\cdot 661^{5} +O\left(661^{ 6 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 432 + 67\cdot 661 + 20\cdot 661^{2} + 226\cdot 661^{3} + 521\cdot 661^{4} + 356\cdot 661^{5} +O\left(661^{ 6 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 447 + 216\cdot 661 + 184\cdot 661^{2} + 147\cdot 661^{3} + 434\cdot 661^{4} + 303\cdot 661^{5} +O\left(661^{ 6 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(2,7)(6,8)$ |
| $(1,2,8,3,5,7,6,4)$ |
| $(3,4)(6,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $4$ |
| $1$ | $2$ | $(1,5)(2,7)(3,4)(6,8)$ | $-4$ |
| $2$ | $2$ | $(2,7)(3,4)$ | $0$ |
| $4$ | $2$ | $(2,7)(6,8)$ | $0$ |
| $4$ | $2$ | $(1,8)(2,4)(3,7)(5,6)$ | $0$ |
| $2$ | $4$ | $(1,8,5,6)(2,3,7,4)$ | $0$ |
| $2$ | $4$ | $(1,6,5,8)(2,3,7,4)$ | $0$ |
| $4$ | $8$ | $(1,2,8,3,5,7,6,4)$ | $0$ |
| $4$ | $8$ | $(1,3,6,2,5,4,8,7)$ | $0$ |
| $4$ | $8$ | $(1,2,6,3,5,7,8,4)$ | $0$ |
| $4$ | $8$ | $(1,3,8,2,5,4,6,7)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
