// We compute the fiber product of j = 1728+t^2 and
// X_{3D^{0}-3a}.

F<t,d3> := FunctionField(Rationals(),2);

j1 := 1728+t^2;

b3 := 729/(d3^3-27);
j2 := (b3+3)^3*(b3+27)/b3;

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

pt := C2![1,0,0];
IsSingular(pt);
plac := Places(pt);

E, phi := EllipticCurve(C2,plac[1]);
E2 := MinimalModel(E);
MordellWeilGroup(E2);
// So E2 is y^2 = x^3 - 27, which has two rational points.
// We've already found two degree 1 places on C2, namely plac[1] and plac[3].

Evaluate(jmap,plac[1]);
Evaluate(jmap,plac[3]);

// So both rational points on the fiber product are cusps.