SetNthreads(5);
K := Integers();
R<x,y,A,B,a0,a1,a2,a3,a4,a5,t>:=PolynomialRing(K,11);
//R<x,y,A,B,a0,a1,a2,a3,a4,a5>:=PolynomialRing(K,10);
S<X,Y,Z,AA,BB,aa0,aa1,aa2,aa3,aa4,aa5,T>:=PolynomialRing(K,12);

//----------------------------(P,q,detJ)-------------------------------------
PP:=Y^2*Z-(X^3+AA*X*Z^2+BB*Z^3);
qq:=aa0*X^2 + aa1*X*Y + aa2*X*Z^2 + aa3*Y^2*Z + aa4*Y*Z + aa5*Z^2;
phi := hom<S->R | x,y,1,A,B,a0,a1,a2,a3,a4,a5,t>;
P:=phi(PP);
q:=phi(qq);

W := [P, q, a1*x+a4, 6*t-1];
GW := GroebnerBasis(W);

GW;