Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 127 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 43 + 13\cdot 127 + 7\cdot 127^{2} + 69\cdot 127^{3} + 38\cdot 127^{4} +O\left(127^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 55 + 25\cdot 127 + 98\cdot 127^{2} + 107\cdot 127^{3} + 81\cdot 127^{4} +O\left(127^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 80 + 105\cdot 127 + 83\cdot 127^{2} + 34\cdot 127^{3} + 14\cdot 127^{4} +O\left(127^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 102 + 85\cdot 127 + 3\cdot 127^{2} + 74\cdot 127^{3} + 97\cdot 127^{4} +O\left(127^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 119 + 20\cdot 127 + 13\cdot 127^{2} + 119\cdot 127^{3} + 91\cdot 127^{4} +O\left(127^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 120 + 74\cdot 127 + 32\cdot 127^{2} + 21\cdot 127^{3} + 11\cdot 127^{4} +O\left(127^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 121 + 7\cdot 127 + 56\cdot 127^{2} + 93\cdot 127^{3} + 16\cdot 127^{4} +O\left(127^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 123 + 46\cdot 127 + 86\cdot 127^{2} + 115\cdot 127^{3} + 28\cdot 127^{4} +O\left(127^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,3)(2,8)(4,7)(5,6)$ |
| $(1,2)(3,8)(4,6)(5,7)$ |
| $(1,4,3,7)(2,5,8,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,3)(2,8)(4,7)(5,6)$ | $-2$ |
| $2$ | $2$ | $(1,2)(3,8)(4,6)(5,7)$ | $0$ |
| $2$ | $2$ | $(1,5)(2,4)(3,6)(7,8)$ | $0$ |
| $2$ | $4$ | $(1,4,3,7)(2,5,8,6)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
