Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 131 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 6 + 71\cdot 131 + 93\cdot 131^{2} + 48\cdot 131^{3} + 75\cdot 131^{4} +O\left(131^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 7 + 45\cdot 131 + 64\cdot 131^{2} + 106\cdot 131^{3} + 96\cdot 131^{4} +O\left(131^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 18 + 77\cdot 131 + 79\cdot 131^{2} + 99\cdot 131^{3} + 92\cdot 131^{4} +O\left(131^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 30 + 62\cdot 131 + 106\cdot 131^{2} + 123\cdot 131^{3} + 2\cdot 131^{4} +O\left(131^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 102 + 68\cdot 131 + 24\cdot 131^{2} + 7\cdot 131^{3} + 128\cdot 131^{4} +O\left(131^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 114 + 53\cdot 131 + 51\cdot 131^{2} + 31\cdot 131^{3} + 38\cdot 131^{4} +O\left(131^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 125 + 85\cdot 131 + 66\cdot 131^{2} + 24\cdot 131^{3} + 34\cdot 131^{4} +O\left(131^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 126 + 59\cdot 131 + 37\cdot 131^{2} + 82\cdot 131^{3} + 55\cdot 131^{4} +O\left(131^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,2,3,5)(4,8,7,6)$ |
| $(1,4)(2,6)(3,7)(5,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$2$ |
| $1$ |
$2$ |
$(1,3)(2,5)(4,7)(6,8)$ |
$-2$ |
| $2$ |
$2$ |
$(1,4)(2,6)(3,7)(5,8)$ |
$0$ |
| $2$ |
$2$ |
$(1,6)(2,7)(3,8)(4,5)$ |
$0$ |
| $2$ |
$4$ |
$(1,2,3,5)(4,8,7,6)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
