I've recently started to look at elliptic curves and have three basic questions:

Is it correct to say that elliptic curves $E$ in the projective plane are in

**bijective**correspondence with lattices $L$ in the complex plane via $E$ <--> $C/L$.If so, is there an explicit expression of the lattice generators in terms of the equation defining the curve? Or, at least, is there a simple example of a curve and its corresponding lattice?

Since every elliptic curve is a Lie group, it must have a corresponding Lie algebra. Is there an explicit expression of the Lie algebra in terms of the equation or lattice? Or, again, a simple example of a curve and its Lie algebra (or, even better, an example of a curve, its lattice, and its Lie algebra).