An algorithm for computing the Weierstrass normal form
Let f in C[x,y] be a polynomial defining an algebraic
curve of genus 1. Then the algebraic function field
C(x)[y]/(f) is isomorphic to C(X)[Y]/(F) for some
polynomial F=Y^2 + (polynomial in X of degree 3).
In this paper we compute such F, and also compute
this isomorphism and its inverse by computing the
images of x, y, X and Y. This can be used to reduce
the problem of integration in C(x)[y]/(f) to the
field C(X)[Y]/(F), for which special purpose
integration algorithms exist. Furthermore the
j-invariant can be obtained.
Note that when the problem is non-trivial, i.e.
when the degree of f is > 3, then f must have
singularities in order to have genus 1. A convenient
and efficient way to deal with these singularities
is to use an integral basis.
For the implementation see the file genus1 in
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See also the paper on the hyperelliptic case.