Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 131 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 22 + 89\cdot 131 + 20\cdot 131^{2} + 123\cdot 131^{3} + 122\cdot 131^{4} +O\left(131^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 27 + 31\cdot 131 + 37\cdot 131^{2} + 7\cdot 131^{3} + 63\cdot 131^{4} +O\left(131^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 28 + 101\cdot 131 + 124\cdot 131^{2} + 124\cdot 131^{3} + 61\cdot 131^{4} +O\left(131^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 60 + 36\cdot 131 + 35\cdot 131^{2} + 90\cdot 131^{3} + 77\cdot 131^{4} +O\left(131^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 80 + 8\cdot 131 + 83\cdot 131^{2} + 60\cdot 131^{3} + 84\cdot 131^{4} +O\left(131^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 96 + 54\cdot 131 + 106\cdot 131^{2} + 103\cdot 131^{3} + 36\cdot 131^{4} +O\left(131^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 99 + 53\cdot 131 + 92\cdot 131^{2} + 116\cdot 131^{3} + 111\cdot 131^{4} +O\left(131^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 114 + 17\cdot 131 + 24\cdot 131^{2} + 28\cdot 131^{3} + 96\cdot 131^{4} +O\left(131^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,2)(3,5)(4,7)(6,8)$ |
| $(1,3,7,8)(2,6,4,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$2$ |
| $1$ |
$2$ |
$(1,7)(2,4)(3,8)(5,6)$ |
$-2$ |
| $2$ |
$2$ |
$(1,2)(3,5)(4,7)(6,8)$ |
$0$ |
| $2$ |
$2$ |
$(1,6)(2,3)(4,8)(5,7)$ |
$0$ |
| $2$ |
$4$ |
$(1,3,7,8)(2,6,4,5)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
