Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 127 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 17 + 53\cdot 127 + 68\cdot 127^{2} + 27\cdot 127^{3} + 71\cdot 127^{4} +O\left(127^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 31 + 91\cdot 127 + 35\cdot 127^{2} + 16\cdot 127^{3} + 65\cdot 127^{4} +O\left(127^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 46 + 105\cdot 127 + 9\cdot 127^{2} + 43\cdot 127^{3} + 122\cdot 127^{4} +O\left(127^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 58 + 105\cdot 127 + 99\cdot 127^{2} + 90\cdot 127^{3} + 123\cdot 127^{4} +O\left(127^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 69 + 21\cdot 127 + 27\cdot 127^{2} + 36\cdot 127^{3} + 3\cdot 127^{4} +O\left(127^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 81 + 21\cdot 127 + 117\cdot 127^{2} + 83\cdot 127^{3} + 4\cdot 127^{4} +O\left(127^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 96 + 35\cdot 127 + 91\cdot 127^{2} + 110\cdot 127^{3} + 61\cdot 127^{4} +O\left(127^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 110 + 73\cdot 127 + 58\cdot 127^{2} + 99\cdot 127^{3} + 55\cdot 127^{4} +O\left(127^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,2)(3,4)(5,6)(7,8)$ |
| $(1,3,8,6)(2,5,7,4)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$2$ |
| $1$ |
$2$ |
$(1,8)(2,7)(3,6)(4,5)$ |
$-2$ |
| $2$ |
$2$ |
$(1,2)(3,4)(5,6)(7,8)$ |
$0$ |
| $2$ |
$2$ |
$(1,5)(2,3)(4,8)(6,7)$ |
$0$ |
| $2$ |
$4$ |
$(1,3,8,6)(2,5,7,4)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
