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\f0\fs24 \cf0 /////// Computation of the correspondence curve X_C in characteristic 2\
\
// to get the general equation, one takes\
F<a,b,c,d,e,f>:=FunctionField(GF(2),6);\
// otherwise\
F<t>:=GF(2^4);\
a:=Random(F);b:=Random(F);c:=Random(F);d:=Random(F);e:=Random(F);f:=Random(F);\
\
// We take here an ordinary curve C (we will need a second copy C2 for technical reasons)\
P<x,y,z>:=PolynomialRing(F,3);\
P2<X,Y>:=PolynomialRing(P,2);\
\
C:=(a*x^2+b*x*y+c*x*z+d*y^2+e*y*z+f*z^2)^2-x*y*z*(x+y+z);\
phi:=hom<P -> P2  | X,Y,1>;\
C2:=phi(C);\
\
// An equation of the tangent to the point (x,y) is \
\
T:=Derivative(C,x)*X+Derivative(C,y)*Y+Derivative(C,z);\
\
// We express the coordinate X of a point of C=C2 which is on the tangent at the point (x,y)\
\
R:=Resultant(T,C2,Y);\
\
// We need now to see R as a polynomial over the function field of the curve in (x,y).\
\
// we make the following morphisms\
//  x -> u, y-> v, z-> 1, X->w->W and Y-> 1\
\
F2<u>:=FunctionField(F);\
P3<v>:=PolynomialRing(F2);\
psi:=hom< P-> P3 | u,v,1>;\
P4<w>:=PolynomialRing(P3);\
psi2:=hom< P2 -> P4 | psi,w,1>;\
F3<v>:=FunctionField(psi(C));\
// For explicit computation, check the genus \
// Genus(F3);\
P4<W>:=PolynomialRing(F3);\
\
R2:=P4!psi2(R);\
\
// Now we need to get rid of the double solution X=x (W=u in the new coordinates) of R so we get an equation of degree 2 in W\
\
XC:=R2 div(W-u)^2;\
R2 mod (W-u)^2;\
A:=Coefficient(XC,2);B:=Coefficient(XC,1);\
C:=Coefficient(XC,0);\
\
// Actually A,B,C are perfect square in F(u,v) so we take their square root\
\
t,Ar:=IsSquare(A);\
 t,Br:=IsSquare(B);\
 t,Cr:=IsSquare(C);\
\
// Here is the field of XC\
\
FXC:=FunctionField(Ar*W^2+Br*W+Cr);\
// Genus(FXC);\
}