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The reliability of the completeness claims for the elliptic curves over each number field represented in the database depends on the field and the current status of modularity of elliptic curves over the field in question. Over all totally real fields of degree \(2\) and \(3\), and most of degree $4$, it is known that all elliptic curves are modular (see [10.1007/s00222-014-0550-z] for degree $2$, [arXiv:1901.03436] for degree $3$ and Box (in preparation) for degree $4$) 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. Over totally real fields of degrees $4$, $5$ and $6$ all the curves in the database have been proved to be modular, using the criteria of [10.1007/s00222-014-0550-z], and over imaginary quadratic fields of class number $1$, all the curves in the database have also been proved to be modular using the $2$-adic Faltings-Serre method or its $3$-adic analogue.

All the data for each individual elliptic curve has been computed rigorously with no unproved conjectures or assumptions being used, except for some of the BSD invariants: