//
// in this portion of the code we check if the family of elliptic curves given in
// D. Jeon, A. Schweizer "Torsion of rational elliptic curves over different types of cubic fields"
// contains all elliptic curves with torsion Z/2Z + Z/14Z
//

K<u>:=FunctionField(Rationals());
_<x>:=PolynomialRing(K);
F<v>:=ext<K|((u^3+u^2-2*u-1)*x*(x+1) +(x^3+x^2-2*x-1)*u*(u+1))>;			// the field over which curve E_u is defined, given by P. Bruin and F. Najman in "Fields of definition of elliptic curves with prescribed torsion"
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)>;		// the field over which curve E_u is defined, given by D. Jeon and A. Schweizer in "Torsion of rational elliptic curves over different types of cubic fields"

tr,f:=IsIsomorphic(F,L);								// here we check whether the fields are isomorphic; tr=true and f is an isomorphism between F and L

// here is the model E_u presented in the paper of D. Jeon and A. Schweizer defined over the field L
A:=-(u^12+4*u^11-10*u^10-68*u^9+3*u^8+552*u^7+4*u^6-2568*u^5+2103*u^4+1684*u^3+1958*u^2+396*u+37)/(48*(u^2+3)^3*(u^6+4*u^5+13*u^4-40*u^3+19*u^2+36*u+31));
B:=(u^24+8*u^23+12*u^22-120*u^21-518*u^20+504*u^19+5068*u^18+568*u^17-24009*u^16-15024*u^15+62936*u^14+183120*u^13-550452*u^12-851984*u^11+4384056*u^10-3808912*u^9+1467519*u^8-4083672*u^7+3590300*u^6+5512360*u^5+6945498*u^4+2943128*u^3+893052*u^2+120024*u+3753)/(864*(u^2+3)^6*(u^6+4*u^5+13*u^4-40*u^3+19*u^2+36*u+31)^2);
E1:=EllipticCurve([A, B]);
E1:=BaseChange(E1, L);

// this part is from https://math.mit.edu/~drew/X1/X1_2_14.txt
q := (u+v)/(v-u);
t := (u-v)*(u+v)*(u+v+2)/(u^3+u^2*v+2*u^2+u*v^2+2*u*v+v^3+2*v^2);
E2 := EllipticCurve([f(0),f(t^2-2*q*t-2),f(0),f(-(t^2-1)*(q*t+1)^2),f(0)]); 		// we create an elliptic curve with coefficients after they are mapped by the isomorphism f

IsIsomorphic(E1,E2);									// we chech whether the curves are isomorphic, this returns "true"