Moduli space for elliptic curves

I would like to find a proof of the theorem which states that the family of elliptic curves over is dimensional. I'd be particularily happy to see the computation which reduces an arbitrary non-degenerate projective algebraic curve of third degree to Legendre's form , where is the elliptic modulus. I have a book in which this is done, but the machinery used overwhelms me and I'd like to find another reference...

I understand how non-degenerate conics can be reduced to a single case, and the computation is relatively easy, because quadratic forms furnish us with an associated bilinear form for which we can find an "orthonormal" basis. For third degree curves, the computation is completely different!

Thanks!