\\ \\ x3m2y3m3.gp \\ \\ \\ find x,y \n Q \\ such that \\ C: x^3-2y^3=3. \\ \\ C(Z)={[1,-1],[-5,-4]} \\ \\ C(Q) is isomorphic to E(Q) \\ E: y^2=x^3-108x^2+3888x-62208 \\ \\ E(Q)_{tors}={O} \\ E(Q)={n[64,80]:n \in Z} \\ \\ \\ g: E ---> C \\ [x,y] ---> [u,v] \\ \\ \\ u=(x^3 - 108*x^2 + (12*y + 1296)*x + 77760)/(x^3 - 36*x^2 - 1296*x - 15552) \\ v=(-x^3 + 108*x^2 + (6*y - 2592)*x + (216*y - 31104))/(x^3 - 36*x^2 - 1296*x - 15552) \\ g(p)= { local(x,y,z,u,v,d); if(p==[0],return([1,-1])); z=1; if(length(p)==3,z=p[3]); x=p[1]/z; y=p[2]/z; d=x^3 - 36*x^2 - 1296*x - 15552; if(d==0,return([0])); u=(x^3 - 108*x^2 + (12*y + 1296)*x + 77760)/d; v=(-x^3 + 108*x^2 + (6*y - 2592)*x + (216*y - 31104))/d; [u,v] } \\ \\ \\ f1(u,v)=75*u-96*v; f2(u,v)=-15*u+24*v; f3(u,v)=u^3-2*v^3; \\ \\ f: C ---> E \\ [u,v]--->[x,y] \\ \\ x=36*(u+5)/(2*v-u+3) \\ y=216*(5*u^2-8*v^2+3)/(2*v-u+3)^2 \\ f(p)= { local(x,y,d,u,v,w); if(p==[0],return([0])); w=1; if(length(p)==3,w=p[3]); u=p[1]/w; v=p[2]/w; d=2*v-u+3; if(d==0,return([64,-80])); x=36*(u+5)/d; y=216*(5*u^2-8*v^2+3)/d^2; [x,y] } f0(p)= { local(x,y,u,v,u1,v1,tt,r,d,d2); if(p==[0],return([0])); u=p[1]; v=p[2]; u1=u+5; v1=v+4; if(u1==0,return([0])); tt=v1/u1; if(tt==1/2,return([0])); d=f2(1,tt)^2-4*f1(1,tt)*f3(1,tt); d2=sqrtrational(d); if(d2<0, return([0])); x=18/(tt-1/2); y=18*d2/(tt-1/2); [x,y] } sqrtrational(x)= { local(p,q,p1,q1); p=numerator(x); q=denominator(x); if(p<0,return(-1)); p1=sqrtint(p); q1=sqrtint(q); if(p1^2==p && q1^2==q, return(p1/q1), return(-1) ) } \\ \\ C(Q) \\ rpC(n)= { local(e,p,q); e=ellinit([0, -108, 0, 3888, -62208]); p=[64,80]; for(i=0,n, q=ellpow(e,p,i); print(g(q)); if(i>0, print(g(ellpow(e,q,-1))) ) ) } \\ \\ E(Q)=Z \\ rpE(n)= { local(e,p,q); e=ellinit([0, -108, 0, 3888, -62208]); p=[64,80]; for(i=0,n, q=ellpow(e,p,i); print(q); if(i>0, print(ellpow(e,q,-1)) ) ) } \\ \\ CC: u^3-2v^3-3w^3=0 \\ [u,v,w]!=[0,0,0] \\ rpCC(n)= { local(e,p,q); e=ellinit([0, -108, 0, 3888, -62208]); p=[64,80]; for(i=0,n, q=ellpow(e,p,i); print(uvw(g(q))); if(i>0, print(uvw(g(ellpow(e,q,-1)))) ) ) } uvw(p)= { local(x,y,u,v,w); if(p==[0],return([0])); x=p[1]; y=p[2]; w=lcm(denominator(x),denominator(y)); u=x*w; v=y*w; [u,v,w] } \\ \\ find x,y,z \in Z \\ x^3-2y^3-3z^3=0 , z >= 0 \\ max(|x|,|y|)<=n find(n)= { local(z,z3); for(x=-n,n, for(y=-n,n, z3=x^3-2*y^3; if(z3 >= 0 && z3%3==0, z3=z3/3; z=sign(z3)*floor(abs(z3)^(1/3)+0.0001); if(z^3==z3 && gcd(x,gcd(y,z))==1, print([x,y,z]) ) ) ) ) }