Type No. 1 

>	QC[1]:=x^4+y^4-z^4;

 

(1.1)

 

>	plot_real_curve(subs(z=1,QC[1]),x,y);

 

>	mapfactor(subs({z=1,x=U},subs(y=t*(x-2*z),QC[1])),U);

 

(1.2)

 

>	factor(subs(U=1,%));

 

(1.3)

 

>	Quartic_to_Weierstrass(%%,[1,t^2]);

 

(1.4)

 

>	step5(%[1],{x = X, z = Z, y = Y}, [x, y, z]);

 

(1.5)

 

>	mapfactor(subs({X=t^4*X,Y=t^6*Y},%[1])/t^12,[X,Y]);

 

(1.6)

 

>	Elliptic_surface(%): Show_data();

 

(1.7)

 

(1.7)

 

(1.7)

 

(1.7)

 

(1.7)

 

(1.7)

 

(1.7)

 

(1.7)

 

(1.7)

 

(1.7)

 

(1.7)

 

(1.7)

 

Type No. 2 

>	QC[2]:=x^4+y^4-2*z^4;

 

(2.1)

 

>	mapfactor(subs({z=1,x=U},subs(y=t*(x-z)+z,QC[2])),U);

 

(2.2)

 

>	Quartic_to_Weierstrass(%/2,[1,0]);

 

(2.3)

 

>	Elliptic_surface(%): Show_data();

 

(2.4)

 

(2.4)

 

(2.4)

 

(2.4)

 

(2.4)

 

(2.4)

 

(2.4)

 

(2.4)

 

(2.4)

 

(2.4)

 

(2.4)

 

(2.4)

 

>	ifactor(discrim(Show(Disc)/(t+1)^2,t));

 

(2.5)

 

>	factor(Show(Disc) mod 3);

 

(2.6)

 

Type No. 3 

>	mapfactor(subs({z=1,x=U},subs(y=t*(x-z),QC[1])),U);

 

(3.1)

 

>	Quartic_to_Weierstrass(%/2,[1,0]);

 

(3.2)

 

>	step5(%[1],{x = X, z = Z, y = Y}, [x, y, z]);

 

(3.3)

 

>	Elliptic_surface(%): Show_data();

 

(3.4)

 

(3.4)

 

(3.4)

 

(3.4)

 

(3.4)

 

(3.4)

 

(3.4)

 

(3.4)

 

(3.4)

 

(3.4)

 

(3.4)

 

(3.4)

 

(3.5)

 

Type No. 4 

>	QC[3]:=x^4+y^4+x^2*z^2-y^2*z^2;

 

(4.1)

 

>	plot_real_curve(subs(z=1,QC[3]),x,y);

 

>	mapfactor(subs({x=U,z=1},subs(y=t*x+z,QC[3])),U);

 

(4.2)

 

>	Quartic_to_Weierstrass(%,[0,0]);

 

(4.3)

 

>	Elliptic_surface(%): Show_data();

 

(4.4)

 

(4.4)

 

(4.4)

 

(4.4)

 

(4.4)

 

(4.4)

 

(4.4)

 

(4.4)

 

(4.4)

 

(4.4)

 

(4.4)

 

(4.4)

 

Type No. 5 

>	QC[4]:=x^4+y^4-x^3*z-x^2*z^2;

 

(5.1)

 

>	plot_real_curve(subs(z=1,QC[4]),x,y);

 

>	singularities(subs(z=1,x^4+y^4-x^3*z-x^2*z^2),x,y);

 

(5.2)

 

>	subs({x=1,y=U},subs(z=y*t*2,x^4+y^4-x^3*z-x^2*z^2));

 

(5.3)

 

>	Quartic_to_Weierstrass(%,[0,1]);

 

(5.4)

 

>	step5(%[1],{x = X, z = Z, y = Y}, [x, y, z]);

 

(5.5)

 

>	Elliptic_surface(%): Show_data();

 

(5.6)

 

(5.6)

 

(5.6)

 

(5.6)

 

(5.6)

 

(5.6)

 

(5.6)

 

(5.6)

 

(5.6)

 

(5.6)

 

(5.6)

 

(5.6)

 

Type No. 6 

>	QC[5]:=x^4+y^4-y^2*z^2-x^3*z;

 

(6.1)

 

>	plot_real_curve(subs(z=1,QC[5]),x,y);

 

>	singularities(subs(z=1,QC[5]),x,y);

 

(6.2)

 

>	mapfactor(subs({x=U,z=1},subs(y=t*(x-z),QC[5])),U);

 

(6.3)

 

>	Quartic_to_Weierstrass(%,[1,0]);

 

(6.4)

 

>	Elliptic_surface(%); Show_data();

 

(6.5)

 

(6.5)

 

(6.5)

 

(6.5)

 

(6.5)

 

(6.5)

 

(6.5)

 

(6.5)

 

(6.5)

 

(6.5)

 

(6.5)

 

(6.5)

 

Type No. 7 

>	QC[6]:=(x^2+y^2)^2-x^2*z^2+y^2*z^2;

 

(7.1)

 

>	plot_real_curve(subs(z=1,QC[6]),x,y);

 

>	mapfactor(subs({z=1,x=U},subs(y=t*x+z,QC[6])),U);

 

(7.2)

 

>	Quartic_to_Weierstrass(%/2,[0,1]);

 

(7.3)

 

>	step5(%[1],{x = X, z = Z, y = Y}, [x, y, z]);

 

(7.4)

 

>	Elliptic_surface(%); Show_data();

 

(7.5)

 

(7.5)

 

(7.5)

 

(7.5)

 

(7.5)

 

(7.5)

 

(7.5)

 

(7.5)

 

(7.5)

 

(7.5)

 

(7.5)

 

(7.5)

 

Type No.8 

>

 

>	qc[8]:=x^3*y-x^2*z^2+y^2*z^2;

 

(8.1)

 

>	with(plots): with(algcurves): plot_real_curve(subs(z=1,qc[8]),x,y);

 

>	singularities(subs(z=1,qc[8]),x,y);

 

(8.2)

 

>	subs(y=t*x+z,qc[8]);

 

(8.3)

 

>	Q8:=mapfactor(subs({z=1,x=U},%),U);

 

(8.4)

 

>	subs(U=0,Q8); simplify(%);

 

(8.5)

 

(8.5)

 

>	Quartic_to_Weierstrass(Q8,[0,1]);

 

(8.6)

 

>	step5(%[1],{x=X,z=Z,y=Y},{x,y,z});

 

(8.7)

 

>	mapfactor(subs({X=X,Y=Y},%[1]),[X,Y]);

 

(8.8)

 

>	Elliptic_surface(%);

 

(8.9)

 

(8.9)

 

>	Show_data();

 

(8.10)

 

(8.10)

 

(8.10)

 

(8.10)

 

(8.10)

 

(8.10)

 

(8.10)

 

(8.10)

 

(8.10)

 

(8.10)

 

Type No.9 

>	qc[9]:=x^4+y^4-x^2*z^2;

 

(9.1)

 

>	with(plots): with(algcurves): plot_real_curve(subs(z=1,qc[9]),x,y);

 

>	singularities(subs(z=1,qc[9]),x,y);

 

(9.2)

 

>	subs(y=t*x+z,qc[9]);

 

(9.3)

 

>	Q9:=mapfactor(subs({z=1,x=U},%),U);

 

(9.4)

 

>	subs(U=0,Q9); simplify(%);

 

(9.5)

 

(9.5)

 

>	Quartic_to_Weierstrass(Q9,[0,1]);

 

(9.6)

 

>	step5(%[1],{x=X,z=Z,y=Y},{x,y,z});Elliptic_surface(%);

 

(9.7)

 

(9.7)

 

(9.7)

 

>	Show_data();

 

(9.8)

 

(9.8)

 

(9.8)

 

(9.8)

 

(9.8)

 

(9.8)

 

(9.8)

 

(9.8)

 

(9.8)

 

(9.8)

 

Type No.10 

>	qc[10]:=x^4+y^4+x^3*y-x^2*z^2;

 

(10.1)

 

>	with(plots): with(algcurves): plot_real_curve(subs(z=1,qc[10]),x,y);

 

>	singularities(subs(z=1,qc[10]),x,y);

 

(10.2)

 

>	subs(y=t*(x-z),qc[10]);

 

(10.3)

 

>	Q10:=mapfactor(subs({z=1,x=U},%),U);

 

(10.4)

 

>	subs(U=1,Q10); simplify(%);

 

(10.5)

 

(10.5)

 

>	Quartic_to_Weierstrass(Q10,[1,0]);

 

(10.6)

 

(10.7)

 

>	step5(%[1],{x=X,z=Z,y=Y},{x,y,z});

 

(10.8)

 

>	Elliptic_surface(%);

 

(10.9)

 

(10.9)

 

>	Show_data();

 

(10.10)

 

(10.10)

 

(10.10)

 

(10.10)

 

(10.10)

 

(10.10)

 

(10.10)

 

(10.10)

 

(10.10)

 

(10.10)