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the following answer explains how to set the givaro implementation to benefit from the fetch_int method. Unfortunately, in this case, the number 4091^2 is too large for givaro implementation, so that the default implementation (pari) can not be replaced with givaro, which can only handle fields of size less than 2^16:

sage: type(GF(4091^2))
<class 'sage.rings.finite_rings.finite_field_pari_ffelt.FiniteField_pari_ffelt_with_category'>

sage: GF(4091^2, impl='givaro')
ValueError: q must be < 2^16