Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 127 }$ to precision 6.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 9 + 92\cdot 127 + 80\cdot 127^{2} + 96\cdot 127^{3} + 94\cdot 127^{4} + 90\cdot 127^{5} +O\left(127^{ 6 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 23 + 19\cdot 127 + 58\cdot 127^{2} + 20\cdot 127^{3} + 9\cdot 127^{4} + 76\cdot 127^{5} +O\left(127^{ 6 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 36 + 53\cdot 127 + 114\cdot 127^{2} + 21\cdot 127^{3} + 113\cdot 127^{4} + 126\cdot 127^{5} +O\left(127^{ 6 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 59 + 89\cdot 127 + 115\cdot 127^{3} + 36\cdot 127^{4} + 87\cdot 127^{5} +O\left(127^{ 6 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 68 + 37\cdot 127 + 126\cdot 127^{2} + 11\cdot 127^{3} + 90\cdot 127^{4} + 39\cdot 127^{5} +O\left(127^{ 6 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 91 + 73\cdot 127 + 12\cdot 127^{2} + 105\cdot 127^{3} + 13\cdot 127^{4} +O\left(127^{ 6 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 104 + 107\cdot 127 + 68\cdot 127^{2} + 106\cdot 127^{3} + 117\cdot 127^{4} + 50\cdot 127^{5} +O\left(127^{ 6 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 118 + 34\cdot 127 + 46\cdot 127^{2} + 30\cdot 127^{3} + 32\cdot 127^{4} + 36\cdot 127^{5} +O\left(127^{ 6 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,3,2,4)(5,8,6,7)$ |
| $(1,2)(3,4)(5,6)(7,8)$ |
| $(1,5)(2,6)(3,7)(4,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,2)(3,4)(5,6)(7,8)$ | $-2$ |
| $2$ | $2$ | $(1,5)(2,6)(3,7)(4,8)$ | $0$ |
| $2$ | $2$ | $(1,7)(2,8)(3,6)(4,5)$ | $0$ |
| $2$ | $4$ | $(1,3,2,4)(5,8,6,7)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
