Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 127 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 29 + 36\cdot 127 + 40\cdot 127^{2} + 53\cdot 127^{3} + 19\cdot 127^{4} +O\left(127^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 32 + 65\cdot 127 + 92\cdot 127^{2} + 61\cdot 127^{3} + 73\cdot 127^{4} +O\left(127^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 55 + 49\cdot 127 + 103\cdot 127^{2} + 101\cdot 127^{3} + 64\cdot 127^{4} +O\left(127^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 73 + 90\cdot 127 + 60\cdot 127^{2} + 4\cdot 127^{3} + 105\cdot 127^{4} +O\left(127^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 96 + 68\cdot 127 + 92\cdot 127^{2} + 30\cdot 127^{3} + 34\cdot 127^{4} +O\left(127^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 112 + 87\cdot 127 + 110\cdot 127^{2} + 75\cdot 127^{3} + 122\cdot 127^{4} +O\left(127^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 117 + 122\cdot 127 + 18\cdot 127^{2} + 118\cdot 127^{3} + 97\cdot 127^{4} +O\left(127^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 122 + 113\cdot 127 + 115\cdot 127^{2} + 61\cdot 127^{3} + 117\cdot 127^{4} +O\left(127^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,2)(3,8)(4,6)(5,7)$ |
| $(1,3)(2,4)(5,6)(7,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,5)(2,7)(3,6)(4,8)$ | $-2$ |
| $2$ | $2$ | $(1,2)(3,8)(4,6)(5,7)$ | $0$ |
| $2$ | $2$ | $(1,3)(2,4)(5,6)(7,8)$ | $0$ |
| $2$ | $4$ | $(1,4,5,8)(2,3,7,6)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
