Computation with two algebraic integers

I have two algebraic integers g and h, and



- I know the minimal polynomial of g, stored in a variable called minpoly_for_g, which has degree 18 in g

- I know the minimal polynomial of h, stored in a variable called minpoly_for_h, which has degree 72 in h

- I also know an extra relation between g and h, in the form of a polynomial with integer coefficients called

relator, and which has degree 17 in g, 71 in h. This relation ensures that g is unique in terms of h.



My goal is to compute the (polynomial) expression of g in terms of h from those data. So far, all my attempts failed

because of integer overflow at some point. For example, the most obvious method of applying the Euclidean algorithm

overflows after a few iterations.



Any suggestions to attack this problem ?



To give an idea in terms of size, sizebyte(minpoly_for_h)=11432, sizebyte(minpoly_for_g)=824, sizebyte(relator)=6184.