Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 631 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 45 + 309\cdot 631 + 110\cdot 631^{2} + 149\cdot 631^{3} + 11\cdot 631^{4} +O\left(631^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 49 + 570\cdot 631 + 480\cdot 631^{2} + 284\cdot 631^{3} + 480\cdot 631^{4} +O\left(631^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 156 + 72\cdot 631 + 581\cdot 631^{2} + 99\cdot 631^{3} + 310\cdot 631^{4} +O\left(631^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 207 + 25\cdot 631 + 611\cdot 631^{2} + 263\cdot 631^{3} + 395\cdot 631^{4} +O\left(631^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 219 + 586\cdot 631 + 456\cdot 631^{2} + 16\cdot 631^{3} + 187\cdot 631^{4} +O\left(631^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 250 + 430\cdot 631 + 169\cdot 631^{2} + 239\cdot 631^{3} + 57\cdot 631^{4} +O\left(631^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 397 + 508\cdot 631 + 218\cdot 631^{2} + 450\cdot 631^{3} + 188\cdot 631^{4} +O\left(631^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 571 + 21\cdot 631 + 526\cdot 631^{2} + 388\cdot 631^{3} + 262\cdot 631^{4} +O\left(631^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,2,6,7)(3,8,4,5)$ |
| $(2,7)(5,8)$ |
| $(1,4,7,8,6,3,2,5)$ |
| $(3,4)(5,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $4$ |
| $1$ | $2$ | $(1,6)(2,7)(3,4)(5,8)$ | $-4$ |
| $2$ | $2$ | $(3,4)(5,8)$ | $0$ |
| $4$ | $2$ | $(2,7)(5,8)$ | $0$ |
| $4$ | $2$ | $(1,2)(3,8)(4,5)(6,7)$ | $0$ |
| $2$ | $4$ | $(1,2,6,7)(3,8,4,5)$ | $0$ |
| $2$ | $4$ | $(1,7,6,2)(3,8,4,5)$ | $0$ |
| $4$ | $8$ | $(1,4,7,8,6,3,2,5)$ | $0$ |
| $4$ | $8$ | $(1,8,2,4,6,5,7,3)$ | $0$ |
| $4$ | $8$ | $(1,4,7,5,6,3,2,8)$ | $0$ |
| $4$ | $8$ | $(1,5,2,4,6,8,7,3)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
