Sextic CurvesΒΆ

Here you can find a list of 64 equations that corresponds to 64 rigid isotopy types of plane sextic curves. Before the equation of each type you see the topological type and then D or ND. Here D stands for dividing type and ND for nondividing type. Note that n means n ovals with no nesting, (n,1) is n ovals sitting outside eachother and all nested in another oval,(n,1),m means there is an oval with n ovals inside it and outside each other and m ovals outside it and outside each other and finally (1,1,1) is the hyperbolic type where we have an oval nested in another oval and together, they are nested in the third oval.

SixtyFour:=
[
[`0`,`ND`,x^6+y^6+z^6],
[`1`,`ND`,x^6+y^6-z^6],
[`(1,1)`,`ND`,6*(x^4+y^4-z^4)*(x^2+y^2-2*z^2)+x^5*y],
[`2`,`ND`,(x^4+y^4-z^4)*((x+4*z)^2+(y+4*z)^2-z^2)+z^6],
[`(2,1)`,`ND`,17*((x+z)^2+(y+z)^2-z^2)*(x^2+y^2-7*z^2)*((x-z)^2+(y-z)^2-z^2)+x^3*y^3],
[`(1,1),1`,`ND`,((x+2*z)^2+(y+2*z)^2-z^2)*(x^2+y^2-3*z^2)*(x^2+y^2-z^2)+x^5*y],
[`3`,`ND`,(x^2+y^2-z^2)*(x^2+y^2-2*z^2)*(x^2+y^2-3*z^2)+x^6],
[`(1,1,1)`,`D`,6*(x^2+y^2-z^2)*(x^2+y^2-2*z^2)*(x^2+y^2-3*z^2)+x^3*y^3],
[`(3,1)`,`ND`,(10*(x^4+2*x^2*y^2+y^4-x^3*z+3*x*y^2*z)+z^4)*(x^2+y^2-z^2)+x^5*y],
[`(2,1),1`,`ND`,(10*(x^4+2*x^2*y^2+y^4-x^3*z+3*x*y^2*z)+z^4)*((x+z)^2+y^2-2*z^2)+x^5*y],
[`(1,1),2`,`ND`,(10*(x^4+2*x^2*y^2+y^4-x^3*z+3*x*y^2*z)+z^4)*(x^2+(y-z)^2-z^2)+x^5*y],
[`4`,`ND`,x^6+y^6+z^6-4*x^2*y^2*z^2],
[`(4,1)`,`ND`,((x^2+3*y^2-20*z^2)*(4*x^2+y^2-16*z^2)+18*z^2*x^2)*(x^2+y^2-10*z^2)-2*z^6],
[`(4,1)`,`D`,10*(((x^2+2*y^2-16*z^2)*(2*x^2+y^2-16*z^2)+x^2*y^2)*(10*x+y+5*z)+x*z^4)*
(10*x-y-8*z)-x*z^5],
[`(3,1),1`,`ND`,((x^2+3*y^2-17*z^2)*(3*x^2+y^2-10*z^2)+15*z^2*x^2)*(x^2+4*(y+z)^2-25*z^2)
+x^3*y^3],
[`(2,1),2`,`ND`,((x^2+3*y^2-20*z^2)*(4*x^2+y^2-16*z^2)+18*z^2*x^2)*
((x+y)^2+20*(x-y-3*z)^2-24*z^2)+(y-x)*z^5],
[`(2,1),2`,`D`,((x^2+3*y^2-20*z^2)*(4*x^2+y^2-16*z^2)+18*z^2*x^2)*(x^2+8*y^2-16*z^2)-4*z^6],
[`(1,1),3`,`ND`,((x^2+2*y^2-30*z^2)*(3*x^2+y^2-20*z^2)+15*z^2*x^2)*(x^2+(4*y+16*z)^2-15*z^2)
+x^3*y^3],
[`5`,`ND`,4*((x^2+2*y^2-4*z^2)*(2*x^2+y^2-4*z^2)+z^4)*(x^2+y^2-z^2)+x^3*y^3],
[`(5,1)`,`ND`,(-4*z^2+3*x^2+4*x*y+2*y^2)*(x^2+2*(y-z)^2-8*z^2)*(2*x^2+y^2-3*z^2)-z^6],
[`(4,1),1`,`ND`,(-14*z^2+4*x^2+6*x*(y-z)+3*(y-z)^2)*(x^2+5*(y-2*z)^2-9*z^2)*
(2*x^2+(y-z)^2-15*z^2)-y*z^5],
[`(3,1),2`,`ND`,((x+z)^2+4*y^2-4*z^2)*(y^2+7*(x+z)^2-10*z^2)*(-8*z^2+(x+z)^2+4*
(2*(x+z)+2*y+z)^2)+x*z^5],
[`(2,1),3`,`ND`,((x+z)^2+3*y^2-4*z^2)*(y^2+7*(x+z)^2-12*z^2)*(-5*z^2+(x+z)^2+3*(2*(x+z)
+2*y+z)^2)
+x*z^5],
[`(1,1),4`,`ND`,((x^2+3*y^2-20*z^2)*(4*x^2+y^2-16*z^2)+18*z^2*x^2)*(y^2+8*x^2-16*z^2)
+(y+x)*z^5],
[`6`,`ND`,(-7*z^2+3*x^2+5*x*y+2*y^2)*(x^2+2*(y-z)^2-8*z^2)*(2*x^2+y^2-5*z^2)-z^6],
[`(6,1)`,`ND`,(4*x^2+4*x*y+3*y^2-4*z^2)*(x^2+3*y^2-4*z^2)*(4*x^2+y^2-4*z^2)-z^6],
[`(5,1),1`,`ND`,30*((3*y^2+(x-z)^2-5*z^2)*(3*(x-z)^2+y^2-5*z^2)+x*z^3)*((x-z)^2+y^2-2*z^2)
+(x-2*z)*z^5],
[`(5,1),1`,`D`,7*((3*(y+z)^2+x^2-48*z^2)*(3*(x+z)^2+y^2-48*z^2)-z^4)*(x^2+y^2-26*z^2)
+x*z^5+y*z^5],
[`(4,1),2`,`ND`,15*(4*x^2+y^2-3*z^2)*(x^2+3*y^2-3*z^2)*(16*y^2+(4*x-z)^2-22*z^2)+
((y-z)^3*z^3+5*x*z^5)],
[`(3,1),3`,`ND`, 34*((3*x^2+y^2-3*z^2)*(x^2+8*y^2-3*z^2)+x^2*y^2)*(2*x^2-y*z-2*z^2)
+y*(x-4*z)*z^4],
[`(3,1),3`,`D`,((x^2+3*y^2-28*z^2)*(4*x^2+y^2-20*z^2)-z^4)*((x+z)^2+y^2-12*z^2)-x*z^5],
[`(2,1),4`,`ND`,27*(-3*x^2-2*x*z-2*y*x+6*y^2)*(-2*x^2+4*y^2+(z+y)^2)*(-4*x^2+y^2+5*(z+y)^2)-
z*x^5],
[`(1,1),5`,`ND`,((x^2+3*y^2-20*z^2)*(4*x^2+y^2-16*z^2)+18*z^2*x^2)*(y^2+16*x^2-20*z^2)-
(y+x)*z^5],
[`(1,1),5`,`D`,((x^2+3*y^2-20*z^2)*(4*x^2+y^2-16*z^2)+18*z^2*x^2)*((x+y)^2+20*(x-y-3*z)^2-
24*z^2)+
(y+x)*z^5],
[`7`,`ND`,2*(4*x^2+y^2-4*z^2)*(x^2+4*y^2-5*z^2)*(x^2+y^2-4*z^2)+3*x^4*y^2+x*y^5],
[`(7,1)`,`ND`,2*(x^2+y^2-26*z^2)*(3*(y+z)^2+x^2-48*z^2)*(3*(x+z)^2+y^2-48*z^2)-z^6],
[`(6,1),1`,`ND`,(160075*(-x^2+5*y*z)*(8*(x*z+15*z^2)-(y-12*z)^2)+109*(72*z+17*x+5*y)*
(42*z+13*x+5*y)*(20*z+9*x+5*y)*(2*x+5*y))*(-x^2+5*y*z)-(x+3*z)*z^5],
[`(5,1),2`,`ND`,(5435525*((y+z)*z-x^2)*((x+2*z)*z-2*(y-x)^2)+5*(31*z-25*x+25*y)*
(49*z-5*x+50*y)*(27*z+15*x+25*y)*(37*z+35*x+25*y))*((y+z)*z-x^2)+x^5*y],
[`(4,1),3`,`ND`,(14460138*((y+z)*z-x^2)*((x+2*z)*z-2*(y-x)^2)+5*(31*z-25*x+25*y)*
(49*z-5*x+50*y)*(27*z+15*x+25*y)*(37*z+35*x+25*y))*((y+z)*z-x^2)+x^5*y],
[`(3,1),4`,`ND`,(27867506*((y+z)*z-x^2)*((x+2*z)*z-2*(y-2*x)^2)+61*(9*z+6*x+8*y)*(63*z+64*y)*
(27*z-15*x+25*y)*(37*z-35*x+25*y))*((y+z)*z- x^2)+x^5*y],
[`(2,1),5`,`ND`,40*(3*z^2+y^2-3*x^2)*(z^2+8*(y-x)^2-3*x^2)*(2*z^2-y*x-2*x^2)-
y^3*x^3-2*z*x^5+2*x^6],
[`(1,1),6`,`ND`,19*(4*z^2+y^2-4*x^2)*(z^2+8*(y-x)^2-3*x^2)*(2*z^2-y*x-2*x^2)-(2*y-3*x)*x^5],
[`8`,`ND`,12*(x^4+2*x^2*y^2+y^4-x^3*z+3*x*y^2*z)*(7*(8*x+3*z)^2+8*y^2-10*z^2)+x^5*y+2*z^6],
[`(8,1)`,`ND`,(2440488*(x^2-y*z)*(57*(-x-z)*z+(6*x+6*z-y)^2)+54*(3*z+10*x+7*y)*(z+11*x+25*y)*
(z-11*x+23*y)*(3*z-10*x+8*y))*(x^2-y*z)-61*y^6+x^2*y^4],
[`(8,1)`,`D`,((x^2+3*y^2-28*z^2)*(4*x^2+y^2-20*z^2)-z^4)*(2*x^2+y^2-12*z^2)-z^6],
[`(7,1),1`,`ND`,(529321083*(x^2-y*z)*(53*(-x-z)*z+(6*x+6*z-y)^2)+25*(3*z+10*x+8*y)*
(z+12*x+30*y)*(z-12*x+32*y)*(3*z-10*x+8*y))*(x^2-y*z)-y^6],
[`(6,1),2`,`D`,(19157935*(-x^2+5*y*z)*(8*(x*z+15*z^2)-(y-12*z)^2)+1185*(72*z+17*x+5*y)*
(42*z+13*x+5*y)*(20*z+9*x+5*y)*(2*x+5*y))*(-x^2+5*y*z)-(x+3*z)*z^5],
[`(5,1),3`,`ND`, (28920269*((y+z)*z-x^2)*((x+2*z)*z-2*(y-x)^2)+10*(31*z-25*x+25*y)*
(49*z-5*x+50*y)*(27*z+15*x+25*y)*(37*z+35*x+25*y))*((y+z)*z-x^2)+x^5*y],
[`(4,1),4`,`ND`,6761249083262*(68794627464*(1095368*(118*(x^2+y^2-3*z^2)*y+(x-2*z)*(x-12*z)*
(x-13*z))*y+(x-4*z)*(x-9*z)*(x-10*z)*(x-11*z))*y+(x-3*z)*(x-5*z)*(x-6*z)*(x-7*z)*(x-8*z))*y-
z^6],
[`(4,1),4`,`D`, 13278270242890*(52982089012*(1610519*(149*(x^2+y^2-4*z^2)*y+(x-3*z)*(x-13*z)*
(x-14*z))*y+(x-5*z)*(x-10*z)*(x-11*z)*(x-12*z))*y+(x-4*z)*(x-6*z)*(x-7*z)*(x-8*z)*(x-9*z))*y-
(x-5*z)*z^5],
[`(3,1),5`,`ND`,(26894836459*((y+z)*z-x^2)*((x+2*z)*z-2*(y-2*x)^2)+1880*(9*z+6*x+8*y)*
(63*z+64*y)*(27*z-15*x+25*y)*(37*z-35*x+25*y))*((y+z)*z-x^2)+x^5*y],
[`(2,1),6`,`D`,(93678589978*((y+z)*z-x^2)*((x+2*z)*z-2*(y-2*x)^2)+50949*(9*z+6*x+8*y)*
(73*z-18*x+72*y)*(7*z-5*x+6*y)*(28*z-27*x+18*y))*((y+z)*z-x^2)+x^5*y],
[`(1,1),7`,`ND`,23*(3*z^2+y^2-3*x^2)*(z^2+8*(y-x)^2-3*x^2)*(2*z^2-y*x-2*x^2)-(2*y-3*x)*x^5],
[`9`,`ND`,((x^2+3*y^2-20*z^2)*(4*x^2+y^2-16*z^2)+18*z^2*x^2)*((x+y)^2+20*(x-y-3*z)^2-24*z^2)+
y^2*z^4],
[`9`,`D`,((x^2+3*y^2-20*z^2)*(4*x^2+y^2-16*z^2)+18*z^2*x^2)*(y^2+16*x^2-20*z^2)+z^6],
[`(9,1)`,`ND`,(40008*(x^2-y*z)*(57*(-x-z)*z+(6*x+6*z-y)^2)+(3*z+10*x+7*y)*(z+11*x+25*y)*
(z-11*x+23*y)*(3*z-10*x+8*y))*(x^2-y*z)-y^6],
[`(8,1),1`,`ND`,(622771068*(x^2-y*z)*(57*(-x-z)*z+(6*x+6*z-y)^2)+35*(3*z+10*x+8*y)*
(z+12*x+30*y)*(z-12*x+32*y)*(3*z-10*x+8*y))*(x^2-y*z)-y^6],
[`(5,1),4`,`ND`,-3401397120*x^6-3195251840*x^5*y-2164525440*x^4*y^2-869728640*x^3*y^3+
332217600*x^2*y^4+316096000*x*y^5+53760001*y^6+1597625920*x^5*z+36848468800000000000*x^4*y*z+
7988129600000000000*x^3*y^2*z-3373286400000000000*x^2*y^3*z+3824761600000000000*x*y^4*z+
1618496000000000000*y^5*z-1199390720*x^4*z^2-7988129600000000000*x^3*y*z^2-
127552392000000000000000000000*x^2*y^2*z^2+23425600000000000000000000000*y^4*z^2+
764952320*x^3*z^3+3654393600000000000*x^2*y*z^3+
141724880000000000000000000000000000000*y^3*z^3-130099200*x^2*z^4-
3824761600000000000*x*y*z^4+11712800000000000000000000000*y^2*z^4+650496000000000000*y*z^5-
2*z^6],
[`(4,1),5`,`ND`,-3401397120*x^6-3195251840*x^5*y-2164525440*x^4*y^2-869728640*x^3*y^3+
332217600*x^2*y^4+316096000*x*y^5+53760002*y^6+1597625920*x^5*z+36848468800000000000*x^4*y*z+
7988129600000000000*x^3*y^2*z-3373286400000000000*x^2*y^3*z+3824761600000000000*x*y^4*z+
1618496000000000000*y^5*z-1199390720*x^4*z^2-7988129600000000000*x^3*y*z^2-
127552392000000000000000000000*x^2*y^2*z^2+23425600000000000000000000000*y^4*z^2+
764952320*x^3*z^3+3654393600000000000*x^2*y*z^3+
141724880000000000000000000000000000000*y^3*z^3-130099200*x^2*z^4-3824761600000000000*x*y*z^4+
11712800000000000000000000000*y^2*z^4+650496000000000000*y*z^5-z^6],
[`(1,1),8`,`ND`,(227693*(x^2-y*z)*((-x-2*z)*z+2*(y-2*z)^2)+(3*z-10*x+8*y)*(z-10*x+23*y)*
(z+11*x+22*y)*(3*z+10*x+7*y))*(x^2-y*z)+y^6],
[`10`,`ND`,19*x^6-20*x^4*y^2-20*x^2*y^4+19*y^6-20*x^4*z^2+60*x^2*y^2*z^2-20*y^4*z^2-
20*x^2*z^4-20*y^2*z^4+19*z^6],
[`(9,1),1`,`D`, (1941536164*(x^2-y*z)*(60*(-x-z)*z+(6*x+6*z-y)^2)+118*(3*z+10*x+8*y)*
(z+12*x+32*y)*(z-12*x+32*y)*(3*z-10*x+8*y))*(x^2-y*z)-y^6],
[`(5,1),5`,`D`,-3401397120*x^6-3195251840*x^5*y-2164525440*x^4*y^2-869728640*x^3*y^3+
332217600*x^2*y^4+316096000*x*y^5+53760001*y^6+1597625920*x^5*z+36848468800000000000*x^4*y*z+
7988129600000000000*x^3*y^2*z-3373286400000000000*x^2*y^3*z+3824761600000000000*x*y^4*z+
1618496000000000000*y^5*z-1199390720*x^4*z^2-7988129600000000000*x^3*y*z^2-
127552392000000000000000000000*x^2*y^2*z^2+23425600000000000000000000000*y^4*z^2+
764952320*x^3*z^3+3654393600000000000*x^2*y*z^3+
141724880000000000000000000000000000000*y^3*z^3-130099200*x^2*z^4-3824761600000000000*x*y*z^4+
11712800000000000000000000000*y^2*z^4+650496000000000000*y*z^5-z^6],
[`(1,1),9`,`D`,(340291*(x^2-y*z)*((-x-2*z)*z+2*(y-2*z)^2)+(3*z-10*x+8*y)*(z-12*x+27*y)*
(z+12*x+28*y)*(3*z+10*x+7*y))*(x^2-y*z)+y^6]
]:
nops(%);