Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 131 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 1 + 72\cdot 131 + 101\cdot 131^{2} + 84\cdot 131^{3} + 96\cdot 131^{4} +O\left(131^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 29 + 78\cdot 131 + 113\cdot 131^{2} + 61\cdot 131^{3} + 2\cdot 131^{4} +O\left(131^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 50 + 7\cdot 131 + 31\cdot 131^{2} + 45\cdot 131^{3} + 54\cdot 131^{4} +O\left(131^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 78 + 33\cdot 131 + 91\cdot 131^{2} + 22\cdot 131^{3} + 85\cdot 131^{4} +O\left(131^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 83 + 34\cdot 131 + 77\cdot 131^{2} + 10\cdot 131^{3} + 50\cdot 131^{4} +O\left(131^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 92 + 72\cdot 131 + 81\cdot 131^{2} + 61\cdot 131^{3} + 38\cdot 131^{4} +O\left(131^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 94 + 68\cdot 131 + 70\cdot 131^{2} + 123\cdot 131^{3} + 35\cdot 131^{4} +O\left(131^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 98 + 25\cdot 131 + 88\cdot 131^{2} + 113\cdot 131^{3} + 29\cdot 131^{4} +O\left(131^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,2)(3,8)(4,6)(5,7)$ |
| $(1,3,6,7)(2,5,4,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$2$ |
| $1$ |
$2$ |
$(1,6)(2,4)(3,7)(5,8)$ |
$-2$ |
| $2$ |
$2$ |
$(1,2)(3,8)(4,6)(5,7)$ |
$0$ |
| $2$ |
$2$ |
$(1,5)(2,3)(4,7)(6,8)$ |
$0$ |
| $2$ |
$4$ |
$(1,3,6,7)(2,5,4,8)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
