// We compute the fiber product of j = 1728+t^2 and
// X_{9J^{0}-9a}.

F<t,j9> := FunctionField(Rationals(),2);

j1 := 1728+t^2;

c9 := (j9^3 - 3*j9 + 1)/(j9^2 - j9);
b3 := c9^3;
j2 := (b3+3)^3*(b3+27)/b3;

pol := Numerator(j1-j2);
C := Curve(AffineSpace(Rationals(),2),pol);
// C2 is the fiber product.
C2 := ProjectiveClosure(C);
FF := FunctionField(C2);
jmap := 1728+FF.1^2;

Genus(C2);

// So C2 is geometrically hyperelliptic 

chk, H, mp := IsHyperelliptic(C);
H2, mp2 := SimplifiedModel(H);

J := Jacobian(H2);
TorsionSubgroup(J);
RankBounds(J);
// The rank of J(C2) is zero.

ratpoints := Chabauty0(J);
// ratpoints is the provably complete list of rational points on H2.
// So H2 has precisely 3 rational points.

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[4];
Evaluate(jmap,pt1);
Evaluate(jmap,pt2);
Evaluate(jmap,pt3);

// All the rational points are cusps.