Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 661 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 40 + 644\cdot 661 + 285\cdot 661^{2} + 110\cdot 661^{3} + 344\cdot 661^{4} +O\left(661^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 314 + 480\cdot 661 + 24\cdot 661^{2} + 289\cdot 661^{3} + 617\cdot 661^{4} +O\left(661^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 400 + 436\cdot 661 + 262\cdot 661^{2} + 446\cdot 661^{3} + 214\cdot 661^{4} +O\left(661^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 428 + 602\cdot 661 + 640\cdot 661^{2} + 390\cdot 661^{3} + 452\cdot 661^{4} +O\left(661^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 447 + 61\cdot 661 + 226\cdot 661^{2} + 630\cdot 661^{3} + 221\cdot 661^{4} +O\left(661^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 522 + 135\cdot 661 + 124\cdot 661^{2} + 292\cdot 661^{3} + 138\cdot 661^{4} +O\left(661^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 538 + 591\cdot 661 + 95\cdot 661^{2} + 262\cdot 661^{3} + 278\cdot 661^{4} +O\left(661^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 618 + 351\cdot 661 + 322\cdot 661^{2} + 222\cdot 661^{3} + 376\cdot 661^{4} +O\left(661^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,3,6,7)(2,4,5,8)$ |
| $(1,2)(5,6)$ |
| $(1,7)(2,8)(3,5)(4,6)$ |
| $(1,2)(3,4)(5,6)(7,8)$ |
| $(1,5)(2,6)(3,7)(4,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$4$ |
| $1$ |
$2$ |
$(1,2)(3,4)(5,6)(7,8)$ |
$-4$ |
| $2$ |
$2$ |
$(1,5)(2,6)(3,7)(4,8)$ |
$0$ |
| $2$ |
$2$ |
$(1,6)(2,5)(3,7)(4,8)$ |
$0$ |
| $2$ |
$2$ |
$(1,2)(5,6)$ |
$0$ |
| $4$ |
$2$ |
$(1,7)(2,8)(3,5)(4,6)$ |
$0$ |
| $4$ |
$4$ |
$(1,7,6,3)(2,8,5,4)$ |
$0$ |
| $4$ |
$4$ |
$(1,3,6,7)(2,4,5,8)$ |
$0$ |
| $4$ |
$4$ |
$(1,7,2,8)(3,6,4,5)$ |
$0$ |
| $4$ |
$4$ |
$(1,5,2,6)(3,4)$ |
$0$ |
| $4$ |
$4$ |
$(1,6,2,5)(3,4)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
