//9J0-9a
F<t,s> := FunctionField(Rationals(),2);
j1 := 1728+t^2;
j9:= (s^3 - 3*s + 1) / (s^2 - s);
b3 := j9^3;
j2 := (b3+3)^3*(b3+27)/b3;
pol := Numerator(j1-j2);
C := Curve(AffineSpace(Rationals(),2),pol);
C2 := ProjectiveClosure(C);
Genus(C2);
f,C1,m:=IsHyperelliptic(C2);
J:=Jacobian(C1);
RankBound(J);
pts:=Chabauty0(J);
pts;
for i:=1 to #pts do
Points(pts[i] @@ m);
end for;

pt := C2![1,0,0];
plac := Places(pt);
printf "The singular point %o has %o places above it.\n",pt,#plac;
pt1 := plac[1];
pt2 := plac[2];
pt3 := plac[3];
pt4 := plac[4];
FF := FunctionField(C2);
jmap := 1728+FF.1^2;
Evaluate(jmap,pt1);
Evaluate(jmap,pt2);
Evaluate(jmap,pt3);
Evaluate(jmap,pt4);

//9J0-9b
F<t,s> := FunctionField(Rationals(),2);
j1 := 1728+t^2;
j9:= −18*(s^2 - 1) / (s^3 - 3*s^2 - 9*s + 3);
b3 := j9^3;
j2 := (b3+3)^3*(b3+27)/b3;
pol := Numerator(j1-j2);
C := Curve(AffineSpace(Rationals(),2),pol);
C2 := ProjectiveClosure(C);
Genus(C2);
f,C1,m:=IsHyperelliptic(C2);
J:=Jacobian(C1);
RankBound(J);
pts:=Chabauty0(J);

for i:=1 to #pts do
Points(pts[i] @@ m);
end for;

pt := C2![1,0,0];
plac := Places(pt);
printf "The singular point %o has %o places above it.\n",pt,#plac;
pt1 := plac[1];
pt2 := plac[2];
pt3 := plac[3];
pt4 := plac[4];
FF := FunctionField(C2);
jmap := 1728+FF.1^2;
Evaluate(jmap,pt1);
Evaluate(jmap,pt2);
Evaluate(jmap,pt3);
Evaluate(jmap,pt4);



//9J0-9c
F<t,s> := FunctionField(Rationals(),2);
j1 := 1728+t^2;
j9:= 3*(s^3 + 3*s^2 - 9*s - 3) / (s^3 - 3*s^2 - 9*s + 3);
b3 := j9^3;
j2 := (b3+3)^3*(b3+27)/b3;
pol := Numerator(j1-j2);
C := Curve(AffineSpace(Rationals(),2),pol);
C2 := ProjectiveClosure(C);
Genus(C2);
f,C1,m:=IsHyperelliptic(C2);
J:=Jacobian(C1);
RankBound(J);
pts:=Chabauty0(J);
pts;