Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 131 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 11 + 92\cdot 131 + 48\cdot 131^{2} + 112\cdot 131^{3} + 59\cdot 131^{4} +O\left(131^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 34 + 109\cdot 131 + 90\cdot 131^{2} + 129\cdot 131^{3} + 96\cdot 131^{4} +O\left(131^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 52 + 5\cdot 131 + 19\cdot 131^{2} + 35\cdot 131^{3} + 85\cdot 131^{4} +O\left(131^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 57 + 108\cdot 131 + 69\cdot 131^{2} + 78\cdot 131^{3} + 8\cdot 131^{4} +O\left(131^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 75 + 22\cdot 131 + 61\cdot 131^{2} + 52\cdot 131^{3} + 122\cdot 131^{4} +O\left(131^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 80 + 125\cdot 131 + 111\cdot 131^{2} + 95\cdot 131^{3} + 45\cdot 131^{4} +O\left(131^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 98 + 21\cdot 131 + 40\cdot 131^{2} + 131^{3} + 34\cdot 131^{4} +O\left(131^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 121 + 38\cdot 131 + 82\cdot 131^{2} + 18\cdot 131^{3} + 71\cdot 131^{4} +O\left(131^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,2)(3,6)(4,5)(7,8)$ |
| $(1,3)(2,5)(4,7)(6,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$2$ |
| $1$ |
$2$ |
$(1,7)(2,8)(3,4)(5,6)$ |
$-2$ |
| $2$ |
$2$ |
$(1,2)(3,6)(4,5)(7,8)$ |
$0$ |
| $2$ |
$2$ |
$(1,3)(2,5)(4,7)(6,8)$ |
$0$ |
| $2$ |
$4$ |
$(1,5,7,6)(2,3,8,4)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
