The reliability of the completeness claims for the elliptic curves over each field represented in the database depends on the field and the current status of modularity of elliptic curves over the field in question. Over totally real fields of degree $$2$$ and $$3$$ it is known that all elliptic curves are modular and hence the database contains all curves of conductor norm up to the bound for that field. Over other fields we can only say that the database contains all modular elliptic curves within this range.