Type21-30.html

Type21-30.mws

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

(1)

> with(algcurves):

TypeNo.21

> qc[21]:=(2*x^2-z^2+2*y^2)^2-4*x*y*(x-y)^2;

(1.1)

> with(plots):

with(algcurves):
plot_real_curve(subs(z=1,qc[21]),x,y);

Plot_2d

> singularities(subs(z=1,qc[21]),x,y);

(1.2)

> subs(y=t*x,qc[21]);

(1.3)

> Q21:=mapfactor(subs({z=1,x=U},%),U);

(1.4)

> subs(U=0,Q21);
simplify(%);

(1.5)

> Quartic_to_Weierstrass(Q21,[0,1]);

(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)

Typesetting:-mrow(Typesetting:-mrow(Typesetting:-mi( (1.11)

(1.11)

TypeNo.22

> qc[22]:=2*x^4+y^4-3*x^2*y*z-2*y^3*z+y^2*z^2;

(2.1)

> with(plots):

with(algcurves):
plot_real_curve(subs(z=1,qc[22]),x,y);

Plot_2d

> singularities(subs(z=1,qc[22]),x,y);

(2.2)

> subs(y=t*(x-z)+2*z,qc[22]);

(2.3)

> Q22:=mapfactor(subs({z=1,x=U},%),U);

(2.4)

> subs(U=1,Q22);
simplify(%);

(2.5)

> Quartic_to_Weierstrass(Q22,[1,0]);

(2.6)

> Elliptic_surface(%);

(2.7)

> Show_data();

(2.8)

Typesetting:-mrow(Typesetting:-mrow(Typesetting:-mi( (2.8)

(2.8)

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

2\,{x}^{4}+{y}^{4}-3\,{x}^{2}yz-2\,{y}^{3}z+{y}^{2}{z}^{2}

TypeNo.23

> qc[23]:=(3*x^2+y^2)^2-6*x^2*z^2+2*y^2*z^2;

(3.1)

> with(plots):

with(algcurves):
plot_real_curve(subs(z=1,qc[23]),x,y);

Plot_2d

> singularities(subs(z=1,qc[23]),x,y);

${[[RootOf(`+`(`*`(3, `*`(`^`(_Z, 2))), 1)), 1, 0], 2, 1, 2], [[0, 0, 1], 2, 1, 2]}$ (3.2)

> subs(z=t*(x-y)+2*y,qc[23]);

(3.3)

> Q23:=mapfactor(subs({y=1,x=U},%),U);

(3.4)

> subs(U=1,Q23);
simplify(%);

(3.5)

> Quartic_to_Weierstrass(Q23,[1,0]);

(3.6)

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

(3.7)

> Elliptic_surface(%);

(3.8)

> Show_data();

(3.9)

Typesetting:-mrow(Typesetting:-mrow(Typesetting:-mi( (3.9)

(3.9)

TypeNo.24

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

(4.1)

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

(4.2)

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

(4.3)

> Elliptic_surface(%):
Show_data();

(4.4)

`+`(jay, `-`(invariant)) = `+`(`-`(`/`(`*`(16, `*`(`^`(`+`(`*`(`^`(t, 4)), 12, `-`(`*`(12, `*`(`^`(t, 2))))), 3))), `*`(`+`(t, `-`(1)), `*`(`+`(t, 1), `*`(`^`(`+`(`*`(`^`(t, 4)), `*`(4, `*`(`^`(t, 2))... (4.4)

(4.4)

Typesetting:-mrow(Typesetting:-mrow(Typesetting:-mi( (4.4)

(4.4)

TypeNo.25

> qc[25]:=x^4+x^3*z-y^2*z^2;

(5.1)

> with(plots):

with(algcurves):
plot_real_curve(subs(z=1,qc[25]),x,y);

Plot_2d

> singularities(subs(z=1,qc[25]),x,y);

(5.2)

> subs(y=t*x+z,qc[25]);

(5.3)

> Q25:=mapfactor(subs({z=1,x=U},%),U);

(5.4)

> subs(U=0,Q25);
simplify(%);

(5.5)

> Quartic_to_Weierstrass(Q25,[0,I]);

$`+`(`*`(`^`(Y, 2), `*`(Z)), `-`(`*`(2, `*`(t, `*`(X, `*`(Y, `*`(Z)))))), `*`(2, `*`(Y, `*`(`^`(Z, 2)))), `-`(`*`(`^`(X, 3))), `-`(`*`(4, `*`(X, `*`(`^`(Z, 2)))))), {Y = `+`(`*`(`*`(4, `*`(I)), `*`(V))...$
$`+`(`*`(`^`(Y, 2), `*`(Z)), `-`(`*`(2, `*`(t, `*`(X, `*`(Y, `*`(Z)))))), `*`(2, `*`(Y, `*`(`^`(Z, 2)))), `-`(`*`(`^`(X, 3))), `-`(`*`(4, `*`(X, `*`(`^`(Z, 2)))))), {Y = `+`(`*`(`*`(4, `*`(I)), `*`(V))...$ (5.6)

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

$`+`(`-`(`*`(`^`(X, 2), `*`(Z, `*`(`^`(t, 2))))), `*`(2, `*`(`+`(t, `-`(2)), `*`(X, `*`(`^`(Z, 2))))), `-`(`*`(`^`(Z, 3))), `-`(`*`(`^`(X, 3))), `*`(`^`(Y, 2), `*`(Z))), {x = X, z = Z, y = `+`(Y, `*`(t...$
$`+`(`-`(`*`(`^`(X, 2), `*`(Z, `*`(`^`(t, 2))))), `*`(2, `*`(`+`(t, `-`(2)), `*`(X, `*`(`^`(Z, 2))))), `-`(`*`(`^`(Z, 3))), `-`(`*`(`^`(X, 3))), `*`(`^`(Y, 2), `*`(Z))), {x = X, z = Z, y = `+`(Y, `*`(t...$ (5.7)

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

(5.8)

> Elliptic_surface(%);

(5.9)

> Show_data();

(5.10)

Typesetting:-mrow(Typesetting:-mrow(Typesetting:-mi( (5.10)

(5.10)

TypeNo.26

> qc[26]:=x^4+y^4-x*y^2*z;

(6.1)

> with(plots):

with(algcurves):
plot_real_curve(subs(z=1,qc[26]),x,y);

Plot_2d

> singularities(subs(z=1,qc[26]),x,y);

(6.2)

> subs(y=t*x+z,qc[26]);

(6.3)

> Q26:=mapfactor(subs({z=1,x=U},%),U);

(6.4)

> subs(U=0,Q26);
simplify(%);

(6.5)

> Quartic_to_Weierstrass(Q26,[0,1]);

(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)

TypeNo.27

> qc[27]:=x^4+y^4+x^3*y-x*y^2*z;

(7.1)

> with(plots):

with(algcurves):
plot_real_curve(subs(z=1,qc[27]),x,y);

Plot_2d

> singularities(subs(z=1,qc[27]),x,y);

(7.2)

> subs(z=t*y+2*x,qc[27]);

(7.3)

> Q27:=mapfactor(subs({x=1,y=U},%),U);

(7.4)

> subs(U=0,Q27);
simplify(%);

(7.5)

> Quartic_to_Weierstrass(Q27,[0,1]);

(7.6)

(7.7)

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

$`+`(`-`(`*`(`^`(X, 3))), `*`(`^`(Y, 2), `*`(Z)), `-`(`*`(64, `*`(`+`(9, `*`(`^`(t, 2))), `*`(`^`(Z, 3))))), `*`(8, `*`(`^`(X, 2), `*`(Z))), `*`(16, `*`(`+`(4, t), `*`(X, `*`(`^`(Z, 2)))))), {y = `+`(Y...$
$`+`(`-`(`*`(`^`(X, 3))), `*`(`^`(Y, 2), `*`(Z)), `-`(`*`(64, `*`(`+`(9, `*`(`^`(t, 2))), `*`(`^`(Z, 3))))), `*`(8, `*`(`^`(X, 2), `*`(Z))), `*`(16, `*`(`+`(4, t), `*`(X, `*`(`^`(Z, 2)))))), {y = `+`(Y...$ (7.8)

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

(7.9)

> Elliptic_surface(%);

(7.10)

> Show_data();

(7.11)

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

(7.11)

TypeNo.28

> qc[28]:=(z^2-x*z-y^2)^2-x^3*y;

(8.1)

> with(plots):

with(algcurves):
plot_real_curve(subs(z=1,qc[28]),x,y);

Plot_2d

> singularities(subs(z=1,qc[28]),x,y);

(8.2)

> subs(y=t*x,qc[28]);

(8.3)

> Q28:=mapfactor(subs({z=1,x=U},%),U);

(8.4)

> subs(U=0,Q28);
simplify(%);

(8.5)

> Quartic_to_Weierstrass(Q28,[0,1]);

(8.6)

(8.7)

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

(8.8)

> Elliptic_surface(%);

(8.9)

> Show_data();

(8.10)

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

(8.10)

TypeNo.29

> qc[29]:=(x^2+y^2-3*x*z)^2-4*x^2*(2*z^2-x*z);

(9.1)

> with(plots):

with(algcurves):
plot_real_curve(subs(z=1,qc[29]),x,y);

Plot_2d

> singularities(subs(z=1,qc[29]),x,y);

(9.2)

> subs(y=t*(x-2*z),qc[29]);

(9.3)

> Q29:=mapfactor(subs({z=1,x=U},%),U);

(9.4)

> subs(U=2,Q29);
simplify(%);

(9.5)

> Quartic_to_Weierstrass(Q29,[2,2]);

(9.6)

(9.7)

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

(9.8)

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

(9.9)

> Elliptic_surface(%);

(9.10)

> Show_data();

(9.11)

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

(9.11)

TypeNo.30

> qc[30]:=x^4+y^4-x*y^2*z;

(10.1)

> with(plots):

with(algcurves):
plot_real_curve(subs(z=1,qc[30]),x,y);

Plot_2d

> singularities(subs(x=1,qc[30]),y,z);

(10.2)

> subs(z=t*(y-2*x)+2*x,qc[30]);

(10.3)

> Q30:=mapfactor(subs({x=1,y=U},%),U);

(10.4)

> subs(U=2,Q30);
simplify(%);

(10.5)

> Quartic_to_Weierstrass(Q30,[2,3]);

(10.6)

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

(10.7)

> mapfactor(subs({X=9*X,Y=27*Y},%[1])/729,[X,Y]);

(10.8)

> Elliptic_surface(%);

(10.9)

> Show_data();

(10.10)

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

(10.10)

> latex(y^2 = x^3+(-4*t+22)*x^2+(4*t^2-56*t+156)*x+25*t^2-192*t+360);

{y}^{2}={x}^{3}+ \left( -4\,t+22 \right) {x}^{2}+ \left( 4\,{t}^{2}-56

\,t+156 \right) x+25\,{t}^{2}-192\,t+360