Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 127 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 7 + 80\cdot 127 + 61\cdot 127^{2} + 118\cdot 127^{3} + 27\cdot 127^{4} +O\left(127^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 8 + 92\cdot 127 + 58\cdot 127^{2} + 65\cdot 127^{3} + 12\cdot 127^{4} +O\left(127^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 32 + 108\cdot 127 + 78\cdot 127^{2} + 48\cdot 127^{3} + 5\cdot 127^{4} +O\left(127^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 36 + 60\cdot 127 + 31\cdot 127^{2} + 50\cdot 127^{3} + 28\cdot 127^{4} +O\left(127^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 91 + 66\cdot 127 + 95\cdot 127^{2} + 76\cdot 127^{3} + 98\cdot 127^{4} +O\left(127^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 95 + 18\cdot 127 + 48\cdot 127^{2} + 78\cdot 127^{3} + 121\cdot 127^{4} +O\left(127^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 119 + 34\cdot 127 + 68\cdot 127^{2} + 61\cdot 127^{3} + 114\cdot 127^{4} +O\left(127^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 120 + 46\cdot 127 + 65\cdot 127^{2} + 8\cdot 127^{3} + 99\cdot 127^{4} +O\left(127^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,3)(2,5)(4,7)(6,8)$ |
| $(1,2,8,7)(3,4,6,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$2$ |
| $1$ |
$2$ |
$(1,8)(2,7)(3,6)(4,5)$ |
$-2$ |
| $2$ |
$2$ |
$(1,3)(2,5)(4,7)(6,8)$ |
$0$ |
| $2$ |
$2$ |
$(1,5)(2,6)(3,7)(4,8)$ |
$0$ |
| $2$ |
$4$ |
$(1,2,8,7)(3,4,6,5)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
