Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 127 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 6 + 33\cdot 127 + 30\cdot 127^{2} + 97\cdot 127^{3} + 99\cdot 127^{4} +O\left(127^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 8 + 65\cdot 127 + 51\cdot 127^{2} + 61\cdot 127^{3} + 36\cdot 127^{4} +O\left(127^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 10 + 25\cdot 127 + 72\cdot 127^{2} + 89\cdot 127^{3} +O\left(127^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 42 + 113\cdot 127 + 39\cdot 127^{2} + 83\cdot 127^{3} + 102\cdot 127^{4} +O\left(127^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 47 + 106\cdot 127 + 113\cdot 127^{2} + 89\cdot 127^{3} + 67\cdot 127^{4} +O\left(127^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 68 + 50\cdot 127 + 90\cdot 127^{2} + 19\cdot 127^{3} + 114\cdot 127^{4} +O\left(127^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 87 + 26\cdot 127 + 85\cdot 127^{2} + 113\cdot 127^{3} + 22\cdot 127^{4} +O\left(127^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 115 + 87\cdot 127 + 24\cdot 127^{2} + 80\cdot 127^{3} + 63\cdot 127^{4} +O\left(127^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,6,5,3)(2,8,4,7)$ |
| $(1,6,7,4,5,3,8,2)$ |
| $(1,7,5,8)(2,6,4,3)$ |
| $(1,5)(2,4)(3,6)(7,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,5)(2,4)(3,6)(7,8)$ | $-2$ |
| $4$ | $2$ | $(2,3)(4,6)(7,8)$ | $0$ |
| $2$ | $4$ | $(1,7,5,8)(2,6,4,3)$ | $0$ |
| $4$ | $4$ | $(1,6,5,3)(2,8,4,7)$ | $0$ |
| $2$ | $8$ | $(1,2,8,3,5,4,7,6)$ | $\zeta_{8}^{3} + \zeta_{8}$ |
| $2$ | $8$ | $(1,4,8,6,5,2,7,3)$ | $-\zeta_{8}^{3} - \zeta_{8}$ |
The blue line marks the conjugacy class containing complex conjugation.
