// ========================================
// SECTION 2 :: TRANSLATION TO GEOMETRY
// ========================================

//-----------------------------------------
// (0)
//-----------------------------------------

// Hyperelliptic equation
_<t>:=PolynomialRing(Rationals());
F0:=HyperellipticCurve(16*t^4 -144*t^3+340*t^2-252*t+49);
EF0:=EllipticCurve(F0,F0![1,3]);
// 2 quadrics intersection
Pr32<X1,X2, X3,X4> := ProjectiveSpace(Rationals(), 3);
C0:=Curve(Pr32,[4*X1^2-9*X3^2+5*X4^2,4*X2^2-7*X3^2+3*X4^2]);
EC0:=EllipticCurve(C0,C0![-1,1,1,1]);
// Weierstrass Model 
E0:=EllipticCurve(t*(t-8)*(t+27));

&and[IsIsomorphic(EC0,EF0),IsIsomorphic(EC0,E0),IsIsomorphic(EF0,E0)];

CremonaReference(E0);
MordellWeilGroup(E0);

/*
true
1680g2
Abelian Group isomorphic to Z/2 + Z/2 + Z
Defined on 3 generators
Relations:
    2*$.1 = 0
    2*$.2 = 0
*/

//-----------------------------------------
// (1)
//-----------------------------------------

// Hyperelliptic equation
_<t>:=PolynomialRing(Rationals());
F1:=HyperellipticCurve(16*t^4 -160*t^3+384*t^2-280*t+49);
EF1:=EllipticCurve(F1,F1![1,3]);
// 2 quadrics intersection
Pr32<X0, X2, X3,X4> := ProjectiveSpace(Rationals(), 3);
C1:=Curve(Pr32,[4*X0^2-10*X3^2+6*X4^2,4*X2^2-7*X3^2+3*X4^2]);
EC1:=EllipticCurve(C1,C1![-1,1,1,1]);
// Weierstrass Model 
E1:=EllipticCurve(t*(t-12)*(t+30));

&and[IsIsomorphic(EC1,EF1),IsIsomorphic(EC1,E1),IsIsomorphic(EF1,E1)];

CremonaReference(E1);
MordellWeilGroup(E1);

/*
true
20160bg2
Abelian Group isomorphic to Z/2 + Z/2 + Z + Z
Defined on 4 generators
Relations:
    2*$.1 = 0
    2*$.2 = 0
*/

//-----------------------------------------
// (2)
//-----------------------------------------

// Hyperelliptic equation
_<t>:=PolynomialRing(Rationals());
F2:=HyperellipticCurve(36*t^4 + 96*t^3 - 236*t^2 + 80*t + 25);
EF2:=EllipticCurve(F2,F2![1,1]);
// 2 quadrics intersection
Pr32<X0, X1, X3,X4> := ProjectiveSpace(Rationals(), 3);
C2:=Curve(Pr32,[5*X0^2-6*X1^2+X3^2, 9*X0^2-10*X1^2+X4^2]);
EC2:=EllipticCurve(C2,C2![-1,1,1,1]);
// Weierstrass Model 
E2<x,y,z>:=EllipticCurve(t*(t+4)*(t+54));

&and[IsIsomorphic(EC2,EF2),IsIsomorphic(EC2,E2),IsIsomorphic(EF2,E2)];


CremonaReference(E2);
MordellWeilGroup(E2);

/*
true
960h2
Abelian Group isomorphic to Z/2 + Z/2 + Z
Defined on 3 generators
Relations:
    2*$.1 = 0
    2*$.2 = 0

*/

//-----------------------------------------
// (3)
//-----------------------------------------

// Hyperelliptic equation
_<t>:=PolynomialRing(Rationals());
F3:=HyperellipticCurve(100*t^4-360*t^3+472*t^2-252*t+49);
EF3:=EllipticCurve(F3,F3![1,3]);
// 2 quadrics intersection
Pr32<X0, X1, X2,X4> := ProjectiveSpace(Rationals(), 3);
C3:=Curve(Pr32,[2*X0^2-3*X1^2+X2^2, 9*X0^2-10*X1^2+X4^2]);
EC3:=EllipticCurve(C3,C3![-1,1,1,1]);
// Weierstrass Model 
E3<x,y,z>:=EllipticCurve(t*(t-7)*(t+20));

&and[IsIsomorphic(EC3,EF3),IsIsomorphic(EC3,E3),IsIsomorphic(EF3,E3)];


CremonaReference(E3);
MordellWeilGroup(E3);

/*
true
840h2
Abelian Group isomorphic to Z/2 + Z/2 + Z
Defined on 3 generators
Relations:
    2*$.1 = 0
    2*$.2 = 0

*/

//-----------------------------------------
// (4)
//-----------------------------------------

// Hyperelliptic equation
_<t>:=PolynomialRing(Rationals());
F4:=HyperellipticCurve(36*t^4 - 72*t^3 + 72*t^2 - 60*t + 25);
EF4:=EllipticCurve(F4,F4![1,1]);
// 2 quadrics intersection
Pr32<X0, X1, X2,X3> := ProjectiveSpace(Rationals(), 3);
C4:=Curve(Pr32,[2*X0^2-3*X1^2+X2^2, 5*X0^2-6*X1^2+X3^2]);
EC4:=EllipticCurve(C4,C4![-1,1,1,1]);
// Weierstrass Model 
E4:=EllipticCurve(t*(t-12)*(t-15));

&and[IsIsomorphic(EC4,EF4),IsIsomorphic(EC4,E4),IsIsomorphic(EF4,E4)];


CremonaReference(E4);
MordellWeilGroup(E4);

/*
true
360e2
Abelian Group isomorphic to Z/2 + Z/2 + Z
Defined on 3 generators
Relations:
    2*$.1 = 0
    2*$.2 = 0

*/