additional information to
as explained in the article, the fine Tate-Shafarevich group is a certain canonically defined subgroup of the usual Tate-Shafarevich group of an elliptic curve over an number field.
the section 4 contains numerical computations. we include here further information on these computations
'the fine Tate-Shafarevich group'
there is a very badly written pari-script that performs a complete 2-descent on an elliptic curve of rank 0 with rational 2-torsion
it extracts the information on the fine subgroup.
using the following script
and the list of curves with non-trivial 2-torsion in sha obtained from cremona's tables, one obtains the
(!! there are still a few minor bugs !!)