"Source: Text/Geometry/RieSrf.text";
"Line: 1005";
"Date: Thu Sep 25 14:47:45 2025";
"Main: Fri Sep 26 14:29:19 2025";
// original file: Text/Geometry/RieSrf.text, line: 1005
// Example: H127E12 ()
print "Example: H127E12";
ei := GetEchoInput();
SetEchoInput(true);
Qxy<x,y> := PolynomialRing(Rationals(),2);
f := -4*x^4 - 5*x^3*y + x^3 + 2*x^2*y^2 - 5*x^2*y + 3*x^2
       + 3*x*y^3 + x*y - 5*x - 8*y^3 - 3;
X := RiemannSurface(f);
L := X`Fiber(0);
Pts := [ X![0,L[i]]: i in [1..#L] ];
Pts;
Print(Pts[1] : Precision := Precision(X));
Representation(Pts[1]);
InfinitePoints(X);
< Representation(P): P in InfinitePoints(X) >;
X!<Infinity(),1> eq InfinitePoints(X)[1];
X!<Infinity(),2> eq X!<Infinity(),3>;
X!<8/3,2> eq InfinitePoints(X)[3];
[ Coordinates(Pt): Pt in InfinitePoints(X) ];
X![3/4,1,0] eq InfinitePoints(X)[1];
X![-1,1,0] eq InfinitePoints(X)[2];
X![0,1,0] eq InfinitePoints(X)[3];
RamificationPoints(X);
DiscriminantPoints(X)[1];
PointsOverDiscriminantPoint(X,1);
PointsOverDiscriminantPoint(X,0);
P1 := RandomPoint(X: Finite);
P1;
P2 := RandomPoint(X: Finite := false);
P2;
SetEchoInput(ei);
