Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 127 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 16 + 101\cdot 127 + 33\cdot 127^{2} + 28\cdot 127^{3} + 49\cdot 127^{4} +O\left(127^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 41 + 37\cdot 127 + 52\cdot 127^{2} + 77\cdot 127^{3} + 123\cdot 127^{4} +O\left(127^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 58 + 59\cdot 127 + 59\cdot 127^{2} + 68\cdot 127^{3} + 88\cdot 127^{4} +O\left(127^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 61 + 20\cdot 127 + 112\cdot 127^{2} + 110\cdot 127^{3} + 116\cdot 127^{4} +O\left(127^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 66 + 106\cdot 127 + 14\cdot 127^{2} + 16\cdot 127^{3} + 10\cdot 127^{4} +O\left(127^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 69 + 67\cdot 127 + 67\cdot 127^{2} + 58\cdot 127^{3} + 38\cdot 127^{4} +O\left(127^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 86 + 89\cdot 127 + 74\cdot 127^{2} + 49\cdot 127^{3} + 3\cdot 127^{4} +O\left(127^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 111 + 25\cdot 127 + 93\cdot 127^{2} + 98\cdot 127^{3} + 77\cdot 127^{4} +O\left(127^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,3,8,6)(2,4,7,5)$ |
| $(1,2)(3,5)(4,6)(7,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $-2$ |
| $2$ | $2$ | $(1,2)(3,5)(4,6)(7,8)$ | $0$ |
| $2$ | $2$ | $(1,4)(2,3)(5,8)(6,7)$ | $0$ |
| $2$ | $4$ | $(1,3,8,6)(2,4,7,5)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
