Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 127 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 6 + 100\cdot 127 + 38\cdot 127^{2} + 88\cdot 127^{3} + 66\cdot 127^{4} +O\left(127^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 21 + 52\cdot 127 + 119\cdot 127^{2} + 101\cdot 127^{3} + 125\cdot 127^{4} +O\left(127^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 24 + 84\cdot 127 + 5\cdot 127^{2} + 26\cdot 127^{3} + 37\cdot 127^{4} +O\left(127^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 37 + 66\cdot 127 + 109\cdot 127^{2} + 17\cdot 127^{3} + 82\cdot 127^{4} +O\left(127^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 90 + 60\cdot 127 + 17\cdot 127^{2} + 109\cdot 127^{3} + 44\cdot 127^{4} +O\left(127^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 103 + 42\cdot 127 + 121\cdot 127^{2} + 100\cdot 127^{3} + 89\cdot 127^{4} +O\left(127^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 106 + 74\cdot 127 + 7\cdot 127^{2} + 25\cdot 127^{3} + 127^{4} +O\left(127^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 121 + 26\cdot 127 + 88\cdot 127^{2} + 38\cdot 127^{3} + 60\cdot 127^{4} +O\left(127^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,3,2,4)(5,8,6,7)$ |
| $(1,2)(3,4)(5,6)(7,8)$ |
| $(1,5)(2,6)(3,7)(4,8)$ |
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.
