load "load.txt";
GL18:=GL(2,Integers(18));
min:=MinimalOvergroups(GL18,reduce(Overgroup1,18));
GL2Genus(min[1]);
A:=min[1];
IsConjugateSubgroup(GL18,A,reduce(Overgroup2,18));
IsConjugateSubgroup(GL18,A,reduce(Overgroup3,18));

\\so, we conclude that the modular curves corrsponding to O1,O2,O3 all map to the modular curve corresponding to A

A3:=reduce(A,3);
_<t>:=FunctionField(Rationals());
b3:=729/(t^3-27);
j3:=(b3+3)^3*(b3+27)/b3;
E:=EllipticCurveFromjInvariant(j3);
d:=Discriminant(E);
Factorization(Numerator(d)*Denominator(d));

//the square free part is: f(x)=(x-3)*(x^2+3*x+9);
_<x>:=PolynomialRing(Rationals());
E1:=EllipticCurve((x-3)*(x^2+3*x+9));
K<w>:=QuadraticField(-3);
E:=ChangeRing(E1,K);
DescentInformation(E);
g,m:=TorsionSubgroup(E);
for i:= 1 to 6 do 
z:=(i*m(g.2))[1];
if Evaluate(Denominator(j3),z) ne 0 and (i*m(g.2))[3] ne 0 then z,Evaluate(j3,z); else print z,"cusp"; end if;  
z:=(m(g.1)+i*m(g.2))[1];
if Evaluate(Denominator(j3),z) ne 0 and (m(g.1)+i*m(g.2))[3] ne 0then z,Evaluate(j3,z); else print z,"cusp"; end if;
end for;