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Integral points on elliptic curves over number fields have been implemented and the (quite large and complicated) patch is on trac, being reviewed. See http://trac.sagemath.org/ticket/10973. Note that the trac ticket you created was closed as a duplicate.

For L-functions, nothing has yet been implemented, but the underlying functionality would be provided by Tim Dokchitser's ComputeL pari/gp package. Someone needs to write that interface!