///////////////////////////////////////////////////////////////////////////
// "Growth of torsion groups of elliptic curves upon base changes"
// Enrique González-Jiménez & Filip Najman
///////////////////////////////////////////////////////////////////////////

// 13/10/2016 - Magma 2.21

// Magma script related to Lemma 8.16

// If C2xC2 subset E(Q) then G_E(2) = 2Cs
// Suppose G_E(3) = 3Ns 

A2<t,s>:=AffineSpace(Rationals(),2);
X:=Curve(A2,s^3*t^2*(t+1)^2-256*(t^2+t+1)^3);
C:=ProjectiveClosure(X); 
/* C is a singular curve and the point (0:1:0) is the unique singular point
IsSingular(C);
>true
> SingularPoints(C);
{@ (0 : 1 : 0) @}
*/
Genus(C); // 1
Pts:=Points(C:Bound:=1000); //{@ (1 : 12 : 1), (-1/2 : 12 : 1), (-2 : 12 : 1), (0 : 1 : 0), (1 : 0 : 0) @}

EE,map1:=EllipticCurve(C,C![1,12]);             
E,map2:=MinimalModel(EE); // Elliptic Curve defined by y^2 = x^3 + 1 over Rational Field
AbelianInvariants(MordellWeilGroup(E)); // Abelian Group isomorphic to Z/6



// If E has a 4-isogeny over Q 
// and suppose G_E(3) = 3Ns 

E:=EllipticCurve([0,0,16,0,0]);
Invariants(MordellWeilGroup(E));
g,m:=TorsionSubgroup(E);
/* 
>[m(a):a in g];
[ (0 : 1 : 0), (8 : -32 : 1), (0 : -16 : 1), (-4 : -8 : 1), (0 : 0 : 1), (8 : 16
*/

for a in g do 
    Pt:=m(a); 
   if Order(Pt) ne 1 then
     h:=Pt[2]/Pt[3];
     if not h in {0,-16} then
       j:=(h^2+16*h+16)^3/(h*(h+16));     
       print h,j,HasComplexMultiplication(EllipticCurveFromjInvariant(j));
     end if;
   end if;
end for;

/* OUTPUT:
-32 287496 true -16
-8 1728 true -4
16 287496 true -16
*/

//computation that the torsion group cannot be C_48 in a dihedral quartic field.


/*
// Let E/Q an elliptic curve, K a dihedral quartic field and F the quadratic field of K.
// Suppose:
//       (1)  E(Q)[2^infty]=Z/2Z 
//       (2)  Z/16Z < E(K)[2^infty] 
// Then Z/8Z < E(K)[2^infty]

load "2primary_Ss.txt";
> {r[3][1,3] : r in RZB | r[3][1,1] eq 1 and r[3][1,2] ne 1 and r[3][1,3] ne 1 and r[3][1,4] eq 4};
{ 2 }

*/