//////////////////////////////////////////////////////////////////////////////////////////////////
//  Computation of jinvariant for the two vertices for Type V and the single vertex for Type IV //
//////////////////////////////////////////////////////////////////////////////////////////////////

LIB "all.lib";
LIB "poly.lib";
LIB "all.lib";
LIB "tropical.lib";

ring r = (0,a2,a3,a4,a5), (x,y),dp;
poly ff=y^2-x*(x-a2)*(x-a3)*(x-a4)*(x-a5);
// ff;
// -x^5+(a2+a3+a4+a5)*x^4+(-a2*a3-a2*a4-a2*a5-a3*a4-a3*a5-a4*a5)*x^3+(a2*a3*a4+a2*a3*a5+a2*a4*a5+a3*a4*a5)*x^2+y^2+(-a2*a3*a4*a5)*x

matrix m = coef(ff,xy);
// m;
// m[1,1]=x^5
// m[1,2]=x^4
// m[1,3]=x^3
// m[1,4]=x^2
// m[1,5]=y^2
// m[1,6]=x
// m[2,1]=-1
// m[2,2]=(a2+a3+a4+a5)
// m[2,3]=(-a2*a3-a2*a4-a2*a5-a3*a4-a3*a5-a4*a5)
// m[2,4]=(a2*a3*a4+a2*a3*a5+a2*a4*a5+a3*a4*a5)
// m[2,5]=1
// m[2,6]=(-a2*a3*a4*a5)

// Computation of j-invariant for the curve dual to the vertex v1 (involves the terms y^2, x, x^2, and x^3) and the vertex v2 (involves the terms y^2, x^3,x^4,x^5):
// We take the terms with x, x^2, x^3, x^4, x^5, y^2 from the matrix m above:
poly B1 =  m[2,6];
poly B2 =  m[2,4];
poly B3 =  m[2,3];
poly By2 = m[2,5];
poly B4 = m[2,2];
poly B5 = m[2,1];
poly C1 = By2*y^2 + B1*x + B2*x^2 + B3*x^3;

// C1;
// (-a2*a3-a2*a4-a2*a5-a3*a4-a3*a5-a4*a5)*x^3+(a2*a3*a4+a2*a3*a5+a2*a4*a5+a3*a4*a5)*x^2+y^2+(-a2*a3*a4*a5)*x

// We replace y by y/x to get a cubic equation and disregard issues with characteristic 3 on the residue field:
poly C2 = By2*y^2 + B3*x + B4*x^2 + B5*x^3;

// C2;
// -x^3+(a2+a3+a4+a5)*x^2+y^2+(-a2*a3-a2*a4-a2*a5-a3*a4-a3*a5-a4*a5)*x

// we use the j-invariant function from "tropical.lib" and write the two j-invariants as a plain text output.

LIB "tropical.lib";
poly j1= jInvariant(C1);
write(":w jC1Output.txt", "jC1 =", j1, "");


poly j2= jInvariant(C2);
write(":w jC2Output.txt", "jC2 =", j2, "");