Type11-20.html

Type11-20.mws

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

(1)

> with(algcurves):

TypeNo.11

> qc[11]:=y^2*z^2-x^3*z-x^3*y;

(1.1)

> with(plots):

with(algcurves):
plot_real_curve(subs(z=1,qc[11]),x,y);

Plot_2d

> singularities(subs(z=1,qc[11]),x,y);

(1.2)

> subs(y=t*(x-z)+z,qc[11]);

(1.3)

> Q11:=mapfactor(subs({z=1,x=U},%),U);

(1.4)

> subs(U=1,Q11);
simplify(%);

(1.5)

> Quartic_to_Weierstrass(Q11,[1,I]);

(1.6)

(1.7)

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

(1.8)

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

(1.9)

> Elliptic_surface(%);

(1.10)

> Show_data();

(1.11)

TypeNo.12

> qc[12]:=(x^2+y^2)^2-x^2*z^2+y^2*z^2;

(2.1)

> with(plots):

with(algcurves):
plot_real_curve(subs(z=1,qc[12]),x,y);

Plot_2d

> singularities(subs(z=1,qc[12]),x,y);

${[[0, 0, 1], 2, 1, 2], [[RootOf(`+`(`*`(`^`(_Z, 2)), 1)), 1, 0], 2, 1, 2]}$ (2.2)

> subs(y=t*(x-z)+z,qc[12]);

(2.3)

> Q12:=mapfactor(subs({z=1,x=U},%),U);

(2.4)

> subs(U=1,Q12);
simplify(%);

(2.5)

> Quartic_to_Weierstrass(Q12,[1,2]);

(2.6)

(2.7)

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

(2.8)

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

(2.9)

> Elliptic_surface(%);

(2.10)

> Show_data();

(2.11)

Typesetting:-mrow(Typesetting:-mrow(Typesetting:-mi( (2.11)

(2.11)

TypeNo.13

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

(3.1)

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

(3.2)

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

$`+`(`*`(`^`(Y, 2), `*`(Z)), `-`(`*`(`^`(X, 3))), `-`(`*`(`+`(`*`(4, `*`(`^`(t, 4))), 4), `*`(X, `*`(`^`(Z, 2)))))), {Z = `*`(`^`(U, 3)), X = `+`(`*`(2, `*`(`+`(V, 1), `*`(U)))), Y = `+`(`*`(4, `*`(V))...$
$`+`(`*`(`^`(Y, 2), `*`(Z)), `-`(`*`(`^`(X, 3))), `-`(`*`(`+`(`*`(4, `*`(`^`(t, 4))), 4), `*`(X, `*`(`^`(Z, 2)))))), {Z = `*`(`^`(U, 3)), X = `+`(`*`(2, `*`(`+`(V, 1), `*`(U)))), Y = `+`(`*`(4, `*`(V))...$ (3.3)

> Elliptic_surface(%):
Show_data();

(3.4)

TypeNo.14

> qc[14]:=(x^2+y^2)^2-x^2*z^2+y^2*z^2;

(4.1)

> with(plots):

with(algcurves):
plot_real_curve(subs(z=1,qc[14]),x,y);

Plot_2d

> singularities(subs(z=1,qc[14]),x,y);

${[[0, 0, 1], 2, 1, 2], [[RootOf(`+`(`*`(`^`(_Z, 2)), 1)), 1, 0], 2, 1, 2]}$ (4.2)

> subs(y=t*(x-z),qc[14]);

(4.3)

> Q14:=mapfactor(subs({z=1,x=U},%),U);

(4.4)

> subs(U=1,Q14);
simplify(%);

(4.5)

> Quartic_to_Weierstrass(Q14,[1,0]);

(4.6)

(4.7)

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

(4.8)

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

(4.9)

> Elliptic_surface(%);

(4.10)

> Show_data();

(4.11)

`+`(jay, `-`(invariant)) = `+`(`-`(`/`(`*`(16, `*`(`^`(`+`(`*`(3, `*`(`^`(t, 2))), 1), 6))), `*`(`^`(t, 2), `*`(`+`(`*`(27, `*`(`^`(t, 4))), `*`(18, `*`(`^`(t, 2))), `-`(1)), `*`(`^`(`+`(`*`(`^`(t, 2)... (4.11)

(4.11)

Typesetting:-mrow(Typesetting:-mrow(Typesetting:-mi( (4.11)

(4.11)

TypeNo.15

> qc[15]:=(x^2+y^2-3*x*z)^2-4*x^2*(2*z^2-x*z);

(5.1)

> with(plots):

with(algcurves):
plot_real_curve(subs(z=1,qc[15]),x,y);

Plot_2d

> singularities(subs(z=1,qc[15]),x,y);

(5.2)

> subs(y=t*(x+z),qc[15]);

(5.3)

> Q15:=mapfactor(subs({z=1,x=U},%),U);

(5.4)

> subs(U=-1,Q15);
simplify(%);

(5.5)

> Quartic_to_Weierstrass(Q15,[-1,2]);

(5.6)

(5.7)

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

(5.8)

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

(5.9)

> Elliptic_surface(%);

(5.10)

> Show_data();

(5.11)

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

(5.11)

TypeNo.16

> qc[16]:=x^4+y^4+x^3*y-x*y^2*z;

(6.1)

> with(plots):

with(algcurves):
plot_real_curve(subs(z=1,qc[16]),x,y);

Plot_2d

> singularities(subs(z=1,qc[16]),x,y);

(6.2)

> subs(z=t*(x-y)+2*y,qc[16]);

(6.3)

> Q16:=mapfactor(subs({y=1,x=U},%),U);

(6.4)

> subs(U=1,Q16);
simplify(%);

(6.5)

> Quartic_to_Weierstrass(Q16,[1,1]);

(6.6)

(6.7)

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

(6.8)

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

(6.9)

> Elliptic_surface(%);

(6.10)

> Show_data();

(6.11)

Typesetting:-mrow(Typesetting:-mrow(Typesetting:-mi( (6.11)

(6.11)

> latex(y^2 = x^3+(-t+9)*x^2+(21-5*t)*x+14-6*t+t^2);

{y}^{2}={x}^{3}+ \left( -t+9 \right) {x}^{2}+ \left( 21-5\,t \right) x

+14-6\,t+{t}^{2}

TypeNo.17

> qc[17]:=(x^2-y*z)^2+y*(4*x^3-x*y^2-y^3);

(7.1)

> with(plots):

with(algcurves):
plot_real_curve(subs(z=1,qc[17]),x,y);

Plot_2d

> singularities(subs(z=1,qc[17]),x,y);

(7.2)

> subs(y=t*x+z,qc[17]);

(7.3)

> Q17:=mapfactor(subs({z=1,x=U},%),U);

(7.4)

> subs(U=0,Q17);
simplify(%);

(7.5)

> Quartic_to_Weierstrass(Q17,[0,0]);

(7.6)

(7.7)

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

(7.8)

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

(7.9)

> Elliptic_surface(%);

(7.10)

> Show_data();

(7.11)

Typesetting:-mrow(Typesetting:-mrow(Typesetting:-mi( (7.11)

(7.11)

TypeNo.18

> qc[18]:=x^4+y^4-x^2*y*z-x*y^2*z;

(8.1)

> with(plots):

with(algcurves):
plot_real_curve(subs(z=1,qc[18]),x,y);

Plot_2d

> singularities(subs(z=1,qc[18]),x,y);

(8.2)

> subs(y=t*(x-z)+z,qc[18]);

(8.3)

> Q18:=mapfactor(subs({z=1,x=U},%),U);

(8.4)

> subs(U=1,Q18);
simplify(%);

(8.5)

> Quartic_to_Weierstrass(Q18,[1,0]);

(8.6)

(8.7)

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

(8.8)

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

(8.9)

> Elliptic_surface(%);

(8.10)

> Show_data();

(8.11)

Typesetting:-mrow(Typesetting:-mrow(Typesetting:-mi( (8.11)

(8.11)

TypeNo.19

> qc[19]:=x^4+y^4-x^2*z^2;

(9.1)

> with(plots):

with(algcurves):
plot_real_curve(subs(z=1,qc[19]),x,y);

Plot_2d

> singularities(subs(z=1,qc[19]),x,y);

(9.2)

> subs(y=t*(x-z),qc[19]);

(9.3)

> Q19:=mapfactor(subs({z=1,x=U},%),U);

(9.4)

> subs(U=1,Q19);
simplify(%);

(9.5)

> Quartic_to_Weierstrass(Q19,[1,0]);

$`+`(`*`(`^`(Y, 2), `*`(Z)), `-`(`*`(`^`(X, 3))), `-`(`*`(5, `*`(`^`(X, 2), `*`(Z)))), `-`(`*`(8, `*`(X, `*`(`^`(Z, 2))))), `-`(`*`(`+`(4, `*`(4, `*`(`^`(t, 4)))), `*`(`^`(Z, 3))))), {X = `+`(`*`(2, `*...$
$`+`(`*`(`^`(Y, 2), `*`(Z)), `-`(`*`(`^`(X, 3))), `-`(`*`(5, `*`(`^`(X, 2), `*`(Z)))), `-`(`*`(8, `*`(X, `*`(`^`(Z, 2))))), `-`(`*`(`+`(4, `*`(4, `*`(`^`(t, 4)))), `*`(`^`(Z, 3))))), {X = `+`(`*`(2, `*...$ (9.6)

(9.7)

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

$`+`(`*`(`^`(Y, 2), `*`(Z)), `-`(`*`(`^`(X, 3))), `-`(`*`(5, `*`(`^`(X, 2), `*`(Z)))), `-`(`*`(8, `*`(X, `*`(`^`(Z, 2))))), `-`(`*`(4, `*`(`+`(1, `*`(`^`(t, 4))), `*`(`^`(Z, 3)))))), {x = X, y = Y, z =...$ (9.8)

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

(9.9)

> Elliptic_surface(%);

(9.10)

> Show_data();

(9.11)

Typesetting:-mrow(Typesetting:-mrow(Typesetting:-mi( (9.11)

(9.11)

(9.12)

TypeNo.20

> qc[20]:=(x^2+y^2-2*x*z)^2-x^2*z^2-y^2*z^2;

(10.1)

> with(plots):

with(algcurves):
plot_real_curve(subs(z=1,qc[20]),x,y);

Plot_2d

> singularities(subs(z=1,qc[20]),x,y);

${[[RootOf(`+`(`*`(`^`(_Z, 2)), 1)), 1, 0], 2, 1, 1], [[0, 0, 1], 2, 1, 2]}$ (10.2)

> subs(y=t*(x-2*z),qc[20]);

(10.3)

> Q20:=mapfactor(subs({z=1,x=U},%),U);

(10.4)

> subs(U=2,Q20);
simplify(%);

(10.5)

> Quartic_to_Weierstrass(Q20,[2,2*I]);

(10.6)

(10.7)

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

(10.8)

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

(10.9)

> Elliptic_surface(%);

(10.10)

> Show_data();

(10.11)

`+`(jay, `-`(invariant)) = `+`(`/`(`*`(4, `*`(`^`(`+`(`-`(9), `*`(54, `*`(`^`(t, 2))), `*`(47, `*`(`^`(t, 4)))), 3))), `*`(`^`(t, 2), `*`(`+`(`*`(289, `*`(`^`(t, 4))), `*`(342, `*`(`^`(t, 2))), `-`(27... (10.11)

(10.11)

Typesetting:-mrow(Typesetting:-mrow(Typesetting:-mi( (10.11)

(10.11)