Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 131 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 6 + 98\cdot 131 + 2\cdot 131^{2} + 79\cdot 131^{3} + 94\cdot 131^{4} +O\left(131^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 22 + 99\cdot 131 + 97\cdot 131^{2} + 23\cdot 131^{3} + 25\cdot 131^{4} +O\left(131^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 50 + 90\cdot 131 + 47\cdot 131^{2} + 104\cdot 131^{3} + 5\cdot 131^{4} +O\left(131^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 55 + 8\cdot 131 + 106\cdot 131^{2} + 58\cdot 131^{3} + 88\cdot 131^{4} +O\left(131^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 76 + 122\cdot 131 + 24\cdot 131^{2} + 72\cdot 131^{3} + 42\cdot 131^{4} +O\left(131^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 81 + 40\cdot 131 + 83\cdot 131^{2} + 26\cdot 131^{3} + 125\cdot 131^{4} +O\left(131^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 109 + 31\cdot 131 + 33\cdot 131^{2} + 107\cdot 131^{3} + 105\cdot 131^{4} +O\left(131^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 125 + 32\cdot 131 + 128\cdot 131^{2} + 51\cdot 131^{3} + 36\cdot 131^{4} +O\left(131^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,2)(3,5)(4,6)(7,8)$ |
| $(1,4,2,6)(3,8,5,7)$ |
| $(1,3)(2,5)(4,7)(6,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$2$ |
| $1$ |
$2$ |
$(1,2)(3,5)(4,6)(7,8)$ |
$-2$ |
| $2$ |
$2$ |
$(1,3)(2,5)(4,7)(6,8)$ |
$0$ |
| $2$ |
$2$ |
$(1,8)(2,7)(3,4)(5,6)$ |
$0$ |
| $2$ |
$4$ |
$(1,4,2,6)(3,8,5,7)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
