Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 131 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 26 + 51\cdot 131 + 125\cdot 131^{2} + 35\cdot 131^{3} + 8\cdot 131^{4} +O\left(131^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 32 + 91\cdot 131 + 30\cdot 131^{2} + 127\cdot 131^{3} + 16\cdot 131^{4} +O\left(131^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 55 + 30\cdot 131 + 74\cdot 131^{2} + 52\cdot 131^{3} + 98\cdot 131^{4} +O\left(131^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 58 + 5\cdot 131 + 117\cdot 131^{2} + 118\cdot 131^{3} + 53\cdot 131^{4} +O\left(131^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 61 + 53\cdot 131 + 21\cdot 131^{2} + 19\cdot 131^{3} + 118\cdot 131^{4} +O\left(131^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 81 + 60\cdot 131 + 79\cdot 131^{2} + 48\cdot 131^{3} + 79\cdot 131^{4} +O\left(131^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 86 + 70\cdot 131 + 115\cdot 131^{2} + 49\cdot 131^{3} + 36\cdot 131^{4} +O\left(131^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 126 + 29\cdot 131 + 91\cdot 131^{2} + 71\cdot 131^{3} + 112\cdot 131^{4} +O\left(131^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,2,6,5)(3,4,8,7)$ |
| $(1,3)(2,7)(4,5)(6,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$2$ |
| $1$ |
$2$ |
$(1,6)(2,5)(3,8)(4,7)$ |
$-2$ |
| $2$ |
$2$ |
$(1,3)(2,7)(4,5)(6,8)$ |
$0$ |
| $2$ |
$2$ |
$(1,7)(2,8)(3,5)(4,6)$ |
$0$ |
| $2$ |
$4$ |
$(1,2,6,5)(3,4,8,7)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
