Type51-60.html

Type51-60.mws

> read `/Documents and Settings/ibuki/My Documents/ESC/ESC.mpl`;
ESC();

(1)

> with(algcurves):

TypeNo.51

> qc[51]:=x^4+x^3*z-y^2*z^2;

(1.1)

> with(plots):

with(algcurves):
plot_real_curve(subs(z=1,qc[51]),x,y);

Plot_2d

> singularities(subs(z=1,qc[51]),x,y);

(1.2)

> subs(y=t*(x+z),qc[51]);

(1.3)

> Q51:=mapfactor(subs({z=1,x=U},%),U);

(1.4)

> subs(U=-1,Q51);
simplify(%);

(1.5)

> Quartic_to_Weierstrass(Q51,[-1,0]);

$`+`(`*`(`^`(Y, 2), `*`(Z)), `-`(`*`(`^`(X, 3))), `-`(`*`(`+`(3, `-`(`*`(`^`(t, 2)))), `*`(`^`(X, 2), `*`(Z)))), `-`(`*`(3, `*`(X, `*`(`^`(Z, 2))))), `-`(`*`(`^`(Z, 3)))), {X = `+`(`-`(U), `-`(1)), Y =...$
$`+`(`*`(`^`(Y, 2), `*`(Z)), `-`(`*`(`^`(X, 3))), `-`(`*`(`+`(3, `-`(`*`(`^`(t, 2)))), `*`(`^`(X, 2), `*`(Z)))), `-`(`*`(3, `*`(X, `*`(`^`(Z, 2))))), `-`(`*`(`^`(Z, 3)))), {X = `+`(`-`(U), `-`(1)), Y =...$ (1.6)

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

$`+`(`-`(`*`(3, `*`(X, `*`(`^`(Z, 2))))), `*`(`^`(Y, 2), `*`(Z)), `-`(`*`(`^`(X, 3))), `-`(`*`(`^`(Z, 3))), `*`(`+`(`-`(3), `*`(`^`(t, 2))), `*`(`^`(X, 2), `*`(Z)))), {x = X, y = Y, z = Z}, [x, y, z]$ (1.7)

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

(1.8)

> Elliptic_surface(%);

(1.9)

> Show_data();

(1.10)

Typesetting:-mrow(Typesetting:-mrow(Typesetting:-mi( (1.10)

(1.10)

TypeNo.52

> qc[52]:=x^4+y^4-x^2*y*z-x*y^2*z;

(2.1)

> with(plots):

with(algcurves):
plot_real_curve(subs(z=1,qc[52]),x,y);

Plot_2d

> singularities(subs(z=1,qc[52]),x,y);

(2.2)

> subs(z=t*(x+y),qc[52]);

(2.3)

> Q52:=mapfactor(subs({y=1,x=U},%),U);

(2.4)

> subs(U=0,Q52);
simplify(%);

(2.5)

> Quartic_to_Weierstrass(Q52,[0,1]);

(2.6)

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

(2.7)

> mapfactor(subs({X=4*X,Y=8*Y},%[1]/64),[X,Y]);

(2.8)

> Elliptic_surface(%);

(2.9)

> Show_data();

(2.10)

`+`(jay, `-`(invariant)) = `+`(`-`(`/`(`*`(64, `*`(`^`(`+`(`*`(`^`(t, 2)), 12), 3))), `*`(`+`(`*`(2, `*`(t)), `-`(1)), `*`(`^`(`+`(`*`(`^`(t, 2)), `*`(8, `*`(t)), 8), 2)))))) (2.10)

(2.10)

Typesetting:-mrow(Typesetting:-mrow(Typesetting:-mi( (2.10)

(2.10)

> latex(y^2 = x^3-2*t*x^2+(t-2)*(t+2)*x+2*t*(t+4));

{y}^{2}={x}^{3}-2\,t{x}^{2}+ \left( t-2 \right) \left( t+2 \right) x+

2\,t \left( t+4 \right)

Type No. 53

;

> qc[53]:=(x^2+y^2-2*x*z)^2-x^2*z^2-y^2*z^2;

(3.1)

> with(plots):

with(algcurves):
plot_real_curve(subs(z=1,qc[53]),x,y);

Plot_2d

> singularities(subs(z=1,qc[53]),x,y);

${[[RootOf(`+`(`*`(`^`(_Z, 2)), 1)), 1, 0], 2, 1, 1], [[0, 0, 1], 2, 1, 2]}$ (3.2)

> subs(z=t*x,qc[53]);

(3.3)

> Q53:=mapfactor(subs({y=1,x=U},%),U);

(3.4)

> subs(U=0,Q53);
simplify(%);

(3.5)

> Quartic_to_Weierstrass(Q53,[0,1]);

(3.6)

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

(3.7)

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

(3.8)

> Elliptic_surface(%);

(3.9)

> Show_data();

(3.10)

Typesetting:-mrow(Typesetting:-mrow(Typesetting:-mi( (3.10)

(3.10)

TypeNo.54

> qc[54]:=(x^2-y*z)*(x^2+y*z);

(4.1)

> with(plots):

with(algcurves):
plot_real_curve(subs(z=1,qc[54]),x,y);

Plot_2d

> singularities(subs(z=1,qc[54]),x,y);

(4.2)

> subs(y=t*(x-z),qc[54]);

(4.3)

> Q54:=mapfactor(subs({z=1,x=U},%),U);

(4.4)

> subs(U=1,Q54);
simplify(%);

(4.5)

> Quartic_to_Weierstrass(Q54,[1,1]);

(4.6)

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

(4.7)

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

(4.8)

> Elliptic_surface(%);

(4.9)

> Show_data();

(4.10)

Typesetting:-mrow(Typesetting:-mrow(Typesetting:-mi( (4.10)

(4.10)

TypeNo.55

> qc[55]:=(x^2-y*z)^2-x^3*y;

(5.1)

> with(plots):

with(algcurves):
plot_real_curve(subs(z=1,qc[55]),x,y);

Plot_2d

> singularities(subs(z=1,qc[55]),x,y);

(5.2)

> subs(y=t*(x+4*z)-4*z,qc[55]);

(5.3)

> Q55:=mapfactor(subs({z=1,x=U},%),U);

(5.4)

> subs(U=-4,Q55);
simplify(%);

(5.5)

> Quartic_to_Weierstrass(Q55,[-4,12]);

(5.6)

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

(5.7)

> mapfactor(subs({X=4*9*X,Y=27*8*Y},%[1]/(27^2*8^2)),[X,Y]);

(5.8)

> mapfactor(subs({X=4*X,Y=8*Y},%/64),[X,Y]);

(5.9)

> Elliptic_surface(%);

(5.10)

> Show_data();

(5.11)

Typesetting:-mrow(Typesetting:-mrow(Typesetting:-mi( (5.11)

(5.11)

TypeNo.56

> qc[56]:=(x^2-y*z)^2-x^3*y;

(6.1)

> with(plots):

with(algcurves):
plot_real_curve(subs(z=1,qc[56]),x,y);

Plot_2d

> singularities(subs(z=1,qc[56]),x,y);

(6.2)

> subs(z=t*(x-y),qc[56]);

(6.3)

> Q56:=mapfactor(subs({y=1,x=U},%),U);

(6.4)

> subs(U=1,Q56);
simplify(%);

(6.5)

> Quartic_to_Weierstrass(Q56,[1,0]);

(6.6)

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

(6.7)

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

(6.8)

> Elliptic_surface(%);

(6.9)

> Show_data();

(6.10)

Typesetting:-mrow(Typesetting:-mrow(Typesetting:-mi( (6.10)

(6.10)

TypeNo.57

> qc[57]:=(x^2+y^2-3*x*z)^2-4*x^2*(2*z^2-x*z);

(7.1)

> with(plots):

with(algcurves):
plot_real_curve(subs(z=1,qc[57]),x,y);

Plot_2d

> singularities(subs(z=1,qc[57]),x,y);

(7.2)

> subs(z=t*x,qc[57]);

(7.3)

> Q57:=mapfactor(subs({y=1,x=U},%),U);

(7.4)

> subs(U=0,Q57);
simplify(%);

(7.5)

> Quartic_to_Weierstrass(Q57,[0,1]);

(7.6)

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

(7.7)

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

(7.8)

> Elliptic_surface(%);

(7.9)

> Show_data();

(7.10)

Typesetting:-mrow(Typesetting:-mrow(Typesetting:-mi( (7.10)

(7.10)

TypeNo.58

> qc[58]:=(x^2+y^2-z^2)*(4*x^2+y^2-4*z^2);

(8.1)

> with(plots):

with(algcurves):
plot_real_curve(subs(z=1,qc[58]),x,y);

Plot_2d

> singularities(subs(z=1,qc[58]),x,y);

(8.2)

> subs(y=t*x+z,qc[58]);

(8.3)

> Q58:=mapfactor(subs({z=1,x=U},%),U);

(8.4)

> subs(U=0,Q58);
simplify(%);

(8.5)

> Quartic_to_Weierstrass(Q58,[0,0]);

(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)

> Show_data();

(8.10)

Typesetting:-mrow(Typesetting:-mrow(Typesetting:-mi( (8.10)

(8.10)

TypeNo.59

> qc[59]:=(x^2+4*y^2-4*z^2)*(x^2+(y-z)^2-4*z^2);

(9.1)

> with(plots):

with(algcurves):
plot_real_curve(subs(z=1,qc[59]),x,y);

Plot_2d

> singularities(subs(z=1,qc[59]),x,y);

${[[RootOf(`+`(`*`(9, `*`(`^`(_Z, 2))), `-`(32))), `/`(1, 3), 1], 2, 1, 2], [[0, -1, 1], 2, 2, 2]}$ (9.2)

> subs(y=t*(x-2*z)+z,qc[59]);

(9.3)

> Q59:=mapfactor(subs({z=1,x=U},%),U);

(9.4)

> subs(U=2,Q59);
simplify(%);

(9.5)

> Quartic_to_Weierstrass(Q59,[2,0]);

(9.6)

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

(9.7)

> mapfactor(subs({X=4*X,Y=8*Y},%[1]/64),[X,Y]);

(9.8)

> Elliptic_surface(%);

(9.9)

> Show_data();

(9.10)

`+`(jay, `-`(invariant)) = `+`(`/`(`*`(4, `*`(`^`(`+`(`*`(`^`(t, 4)), `-`(`*`(8, `*`(`^`(t, 3)))), `*`(14, `*`(`^`(t, 2))), `*`(56, `*`(t)), 1), 3))), `*`(t, `*`(`^`(`+`(`*`(`^`(t, 2)), `-`(`*`(6, `*`... (9.10)

(9.10)

Typesetting:-mrow(Typesetting:-mrow(Typesetting:-mi( (9.10)

(9.10)

TypeNo.60

> qc[60]:=x^4+y^4-x^2*z^2;

(10.1)

> with(plots):

with(algcurves):
plot_real_curve(subs(z=1,qc[60]),x,y);

Plot_2d

> singularities(subs(z=1,qc[60]),x,y);

(10.2)

> subs(z=t*y,qc[60]);

(10.3)

> Q60:=mapfactor(subs({x=1,y=U},%),U);

(10.4)

> subs(U=0,Q60);
simplify(%);

(10.5)

> Quartic_to_Weierstrass(Q60,[0,1]);

$`+`(`*`(`^`(Y, 2), `*`(Z)), `-`(`*`(`^`(X, 3))), `*`(`^`(t, 2), `*`(`^`(X, 2), `*`(Z))), `*`(4, `*`(X, `*`(`^`(Z, 2)))), `-`(`*`(4, `*`(`^`(t, 2), `*`(`^`(Z, 3)))))), {Z = `*`(`^`(U, 3)), X = `+`(`*`(...$
$`+`(`*`(`^`(Y, 2), `*`(Z)), `-`(`*`(`^`(X, 3))), `*`(`^`(t, 2), `*`(`^`(X, 2), `*`(Z))), `*`(4, `*`(X, `*`(`^`(Z, 2)))), `-`(`*`(4, `*`(`^`(t, 2), `*`(`^`(Z, 3)))))), {Z = `*`(`^`(U, 3)), X = `+`(`*`(...$ (10.6)

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

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

(10.8)

> Elliptic_surface(%);

(10.9)

> Show_data();

(10.10)

`+`(jay, `-`(invariant)) = `+`(`/`(`*`(16, `*`(`^`(`+`(`*`(`^`(t, 4)), 12), 3))), `*`(`^`(`+`(`*`(`^`(t, 2)), `-`(2)), 2), `*`(`^`(`+`(`*`(`^`(t, 2)), 2), 2))))) (10.10)

(10.10)

Typesetting:-mrow(Typesetting:-mrow(Typesetting:-mi( (10.10)

(10.10)