I'm interested in producing explicit bases for the sections of a line bundle on an embedded genus 1 curve. Let me restrict to the first case that I don't know how to do, so that I can be as concrete as possible. Take a smooth genus 1 curve E defined over QQ by an explicit cubic equation C_{0} in QQ[x,y,z]. Let D be a divisor of degree 6 on E, and I_D be defined by four cubic equations (C_{0},C_{1},C_{2},C_{3}). Note that D does not lie on any conic. Riemann-Roch says that h^0(E,O_{E}(D))=6, and I'd like to find an explicit basis of rational functions for this vector space. How do I find such a basis?

Note that if D sat on a conic defined by Q, then finding a basis is relatively easy: we could simply choose the functions x^2/Q, xy/Q, ..., z^2/Q as our rational functions.