/////////////////////////////////////////////////////////////////////
// "On the torsion of rational elliptic curves over sextic fields" //
//        Harris B. Daniels and Enrique González-Jiménez           //
/////////////////////////////////////////////////////////////////////

// 2/8/2018 - Magma 2.23

// Magma script related to Proposition 8 (q) 


//f1 is the j-map from X_{G3} (G3=2Cn in Sutherland's notation) to P1 
//f2 is the j-map from X_0(13) to P1 
// both taken from Zywina, On the possible images of the mod l representations associated to elliptic curves over Q.


F<t> := FunctionField(Rationals());
f1 := t^2+1728;
f2:=(t^2+5*t+13)*(t^4+7*t^3+20*t^2+19*t+)^3/t;

R<x,y>:=PolynomialRing(Rationals(),2);
g:=Numerator(Evaluate(f1,x)-Evaluate(f2,y));
assert IsIrreducible(g);
C<X,Y,Z>:=ProjectiveClosure(Curve(AffineSpace(R),g));
assert Genus(C) eq 1;
P := C![1,0,0];
E,map := EllipticCurve(C,P);
assert Rank(E) eq 0;
assert CremonaReference(E) eq "52a2";

T,mp := TorsionSubgroup(E);
PTS := [mp(t) : t in T];

//Pull back the torsion points and add the presages to Set.
Set := {};
PHI := map;
for p in PTS do
Set := Set join RationalPoints(p @@ PHI);
end for;
assert Set eq { C![1,0,0] };
assert Set eq SingularPoints(C);
/*
Thus there are only 1 singular point on C and no affine points.
*/