Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 131 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 15 + 98\cdot 131 + 106\cdot 131^{2} + 40\cdot 131^{3} + 95\cdot 131^{4} +O\left(131^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 17 + 4\cdot 131 + 113\cdot 131^{2} + 82\cdot 131^{3} + 105\cdot 131^{4} +O\left(131^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 41 + 22\cdot 131 + 45\cdot 131^{2} + 62\cdot 131^{3} + 75\cdot 131^{4} +O\left(131^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 60 + 90\cdot 131 + 2\cdot 131^{2} + 63\cdot 131^{3} + 98\cdot 131^{4} +O\left(131^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 67 + 106\cdot 131 + 6\cdot 131^{2} + 64\cdot 131^{3} + 15\cdot 131^{4} +O\left(131^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 94 + 42\cdot 131 + 76\cdot 131^{2} + 72\cdot 131^{3} + 72\cdot 131^{4} +O\left(131^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 103 + 53\cdot 131 + 56\cdot 131^{2} + 41\cdot 131^{3} + 102\cdot 131^{4} +O\left(131^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 127 + 105\cdot 131 + 116\cdot 131^{2} + 96\cdot 131^{3} + 89\cdot 131^{4} +O\left(131^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,2,8,7)(3,6,5,4)$ |
| $(1,3)(2,4)(5,8)(6,7)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$2$ |
| $1$ |
$2$ |
$(1,8)(2,7)(3,5)(4,6)$ |
$-2$ |
| $2$ |
$2$ |
$(1,3)(2,4)(5,8)(6,7)$ |
$0$ |
| $2$ |
$2$ |
$(1,4)(2,5)(3,7)(6,8)$ |
$0$ |
| $2$ |
$4$ |
$(1,2,8,7)(3,6,5,4)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
