  We give two constructions of families of elliptic curves over cubic or
quartic fields with three, respectively four, `integral' elements in the kernel
of the tame symbol on the curves. The fields are in general non-Abelian, and
the elements linearly independent. For their integrality, we prove a new
criterion that does not ignore any torsion. We also verify Beilinson's
conjecture numerically for just over 90 of the curves.
