load "load.txt";
GL18:=GL(2,Integers(18));
G4:=reduce(Overgroup4,18);
G5:=reduce(Overgroup5,18);
G6:=reduce(Overgroup6,18);
G7:=reduce(Overgroup7,18);
G8:=reduce(Overgroup8,18);
G9:=reduce(Overgroup9,18);
G10:=reduce(Overgroup10,18);
MinimalOvergroups(GL18, G4);
S:=SL(2,Integers(18));
G:=GL(2,Integers(18));
S1:=MinimalOvergroups(G,S)[1];
assert GL2DeterminantIndex(S1) eq 3;
IsConjugate(S1, S1 meet G5,S1 meet G6);
IsConjugate(S1, S1 meet G4,S1 meet G6);
// this shows that all the curves are isomorphic over Q(\zeta_9)^+


//G7 computations. We first show that such an elliptic curve has full 2-torsion over Q(zeta_9)^+

f,psi,F:=FindCanonicalModel(realize(reduce(Overgroup4,18),18),400: ReturnBasis:=true);
K:=Parent(Coefficients(F[1])[1]);
L<q>:=LaurentSeriesRing(K,400);
F:=[&+[Coefficient(L!f,3*n)*q^n : n in [1..133]] : f in F]; //q^3 -> q
L<q>:=LaurentSeriesRing(K,133);
F:=[L!f : f in F];
f1:=F[1];
f2:=F[2];
f1:=f1/Coefficient(f1,1);
f2:=f2/Coefficient(f2,3); 
x:=f1/f2;
y:=q*Derivative(x)/f2;
y:=-y/2;
x:=x+O(q^128);
dif:=y^2-x^5;
a4:=Coefficient(dif,-8);
dif:=dif-a4*x^4;
a3:=Coefficient(dif,-6);
dif:=dif-a3*x^3;
a2:=Coefficient(dif,-4);
dif:=dif-a2*x^2;
a1:=Coefficient(dif,-2);
dif:=dif-a1*x;
a0:=Coefficient(dif,0);
dif:=dif-a0;
dif;
P<z>:=PolynomialRing(K);
f:=z^5+a4*z^4+a3*z^3+a2*z^2+a1*z+a0;
C:=HyperellipticCurve(f);
K:=Subfields(CyclotomicField(9),3)[1,1];
CQ:=ChangeRing(HyperellipticCurve(f),Rationals());
C:=ChangeRing(CQ,K);
aut:=Automorphisms(C);
G:=AutomorphismGroup(C,[aut[6]]);
E,f:=CurveQuotient(G);
assert Genus(E) eq 1;
DescentInformation(E);
g,m:=TorsionSubgroup(E);
pts:=Points(C:Bound:=10);
#g;
A:={@ @};
for i:=1 to #g do 
A:= A join Points((m(i*g.2)) @@ f);
A:= A join Points(m(g.1+i*g.2) @@ f);
end for;
assert pts eq A;
assert GL2RationalCuspCount(G4) eq 3 and GL2CuspCount(G4) eq 6;