Reliability of elliptic curve data over number fields

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 [ https//doi.org/10.1090/tran/8557] 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:

• the analytic rank and special L-value were computed using the Magma function AnalyticRank(), which assumes that the L-function satisfies the Hasse-Weil conjecture and computes successive derivatives at the critial point to a fixed precision until it finds a non-zero value. This assumption on the Hasse-Weil conjecture is satisfied by all the elliptic curves in the database which are known to be modular.
• the analytic order of Ш was in all cases computed approximately from its definition, given by the BSD formula, and rounded to the nearest integer; in all cases this is a positive integer square.
• the heights of generators, regulators, and periods have been computed to 128-bit precision in all cases (with the regulator being exactly 1 for curves of rank 0).
• the special L-value and analytic order of Ш have been computed to the same precision over fields of degree 2 and 3, and to lower precision over fields of higher degree.