//
//
// code used in Example 2.3
//
//

//
// u=2
//

_<y>:=PolynomialRing(Rationals());
K<u>:=FunctionField(Rationals());
_<x>:=PolynomialRing(K);

u:=2;
L<w>:=ext<K|((u^2-1)*x^3+(u^3+2*u^2-9*u-2)*x^2-9*(u^2-1)*x-u^3-2*u^2+9*u+2)>;

A2:=(-4*(u^6+2*u^5+15*u^4-20*u^3+15*u^2+18*u+33)*(u-1)^2*(u+1)^2)/((u^2+3)^3*(u^6+4*u^5+13*u^4-40*u^3+19*u^2+36*u+31));
A4:=(64*(u^6+2*u^5+3*u^4-20*u^3+39*u^2+18*u+21)*(u-1)^6*(u+1)^6)/((u^2+3)^6*(u^6+4*u^5+13*u^4-40*u^3+19*u^2+36*u+31)^2);
A6:=(4096*(u-1)^12*(u+1)^12)/((u^6+4*u^5+13*u^4-40*u^3+19*u^2+36*u+31)^3*(u^2+3)^9);
E:=MinimalModel(EllipticCurve([1,A2,0,A4,A6]));

E;
LocalInformation(E,2);

J<a>:=NumberField(3*y^3-4*y^2-27*y+4);
J;
TorsionSubgroup(BaseChange(E,J));





//
// u=1/2
//

_<y>:=PolynomialRing(Rationals());
K<u>:=FunctionField(Rationals());
_<x>:=PolynomialRing(K);

u:=1/2;
L<w>:=ext<K|((u^2-1)*x^3+(u^3+2*u^2-9*u-2)*x^2-9*(u^2-1)*x-u^3-2*u^2+9*u+2)>;

A2:=(-4*(u^6+2*u^5+15*u^4-20*u^3+15*u^2+18*u+33)*(u-1)^2*(u+1)^2)/((u^2+3)^3*(u^6+4*u^5+13*u^4-40*u^3+19*u^2+36*u+31));
A4:=(64*(u^6+2*u^5+3*u^4-20*u^3+39*u^2+18*u+21)*(u-1)^6*(u+1)^6)/((u^2+3)^6*(u^6+4*u^5+13*u^4-40*u^3+19*u^2+36*u+31)^2);
A6:=(4096*(u-1)^12*(u+1)^12)/((u^6+4*u^5+13*u^4-40*u^3+19*u^2+36*u+31)^3*(u^2+3)^9);
E:=MinimalModel(EllipticCurve([1,A2,0,A4,A6]));

E;
LocalInformation(E,2);

J<a>:=NumberField(-6*y^3-47*y^2+54*y+47);
J;
TorsionSubgroup(BaseChange(E,J));