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Groebner basis over large finite fields.

1) I would like to compute groebner basis of an ideal in GF(p)[x] where p has roughly 100bits. However, Singular doesn't support such large finite fields so it always falls back on the slow toy implementation of Buchberger. Are there any better options than the toy implementation?

2) Ideally (pun intended), I would like to use the .elimination_ideal method but this uses only Singular, so again no use for me, or is there any workaround?

It seems to me that because of the limitations of Singular, Sage practically doesn't support groebner basis algorithms over large finite fields which seems like a huge limitation for me. Am I missing something? I do not have magma but I suppose I could install macaulay2 and use it with sage but is it better than the toy implementation?