  We investigate the behaviour of elliptic Feynman integrals under modular
transformations. This has a practical motivation: Through a suitable modular
transformation we can achieve that the nome squared is a small quantity,
leading to fast numerical evaluations. Contrary to the case of multiple
polylogarithms, where it is sufficient to consider just variable
transformations for the numerical evaluations of multiple polylogarithms, it is
more natural in the elliptic case to consider a combination of a variable
transformation (i.e. a modular transformation) together with a redefinition of
the master integrals. Thus we combine a coordinate transformation on the base
manifold with a basis transformation in the fibre. Only in the combination of
the two transformations we stay within the same class of functions.
