Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 127 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 38 + 104\cdot 127 + 40\cdot 127^{2} + 10\cdot 127^{3} + 127^{4} +O\left(127^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 49 + 4\cdot 127 + 103\cdot 127^{2} + 65\cdot 127^{3} + 13\cdot 127^{4} +O\left(127^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 68 + 26\cdot 127 + 69\cdot 127^{2} + 121\cdot 127^{3} + 19\cdot 127^{4} +O\left(127^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 72 + 82\cdot 127 + 95\cdot 127^{2} + 96\cdot 127^{3} + 105\cdot 127^{4} +O\left(127^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 76 + 88\cdot 127 + 81\cdot 127^{2} + 17\cdot 127^{3} + 9\cdot 127^{4} +O\left(127^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 91 + 87\cdot 127 + 74\cdot 127^{2} + 23\cdot 127^{3} + 80\cdot 127^{4} +O\left(127^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 120 + 118\cdot 127 + 112\cdot 127^{2} + 71\cdot 127^{3} + 82\cdot 127^{4} +O\left(127^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 124 + 121\cdot 127 + 56\cdot 127^{2} + 100\cdot 127^{3} + 68\cdot 127^{4} +O\left(127^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,6)(2,8)(3,4)(5,7)$ |
| $(1,5,6,7)(2,3,8,4)$ |
| $(1,3,6,4)(2,7,8,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$2$ |
| $1$ |
$2$ |
$(1,6)(2,8)(3,4)(5,7)$ |
$-2$ |
| $2$ |
$4$ |
$(1,5,6,7)(2,3,8,4)$ |
$0$ |
| $2$ |
$4$ |
$(1,3,6,4)(2,7,8,5)$ |
$0$ |
| $2$ |
$4$ |
$(1,2,6,8)(3,5,4,7)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
