Collaborative Calculation in the Cloud
Sage is an open-source computer algebra system with extensive support for computing with elliptic curves. You can access Sage via SageMathCloud using any browser; no software installation is required. Sage provides both a command-line interface and a graphical user interface. Tutorials and many examples can be found on-line (Google is your friend). You can download a copy of Sage to run on your own machine if you wish.
LMFDB
The L-functions and modular forms database contains a wealth of information about elliptic curves, modular forms, and their L-functions, including a complete list of all elliptic curves over Q for conductor up to 400,000 (and counting) that incorporates isogeny graphs, images of Galois representations, rational points, and invariants related to the conjecture of Birch and Swinnerton-Dyer.
Explicit Formulas Database
Explicit formulas implementing the group operation on elliptic curves in a wide variety of forms can be found here. These formulas have been carefully optimized and verified for correctness.