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It seems when people talk about modular forms they tend to forget that they are very related to families of elliptic curves. Here I want to explain some simple way to understand the connection. We will consider modular forms for the full modular group SL(2, \mathbf Z).

So consider the simplest family of elliptic curves, the Weierstrass family:

y^2=x^3+ax+b.

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