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## Return to Answer

2 correction, following David Speyer's comment.
1. No, for any cubic curve in the plane, there is a family of cubic plane curves (3 or 4 dimensional - I forget Edit: 8-dimensional, with a transitive action of PGL3) that are isomorphic as curves. For any lattice in C, there is a 1-dimensional family of lattices that form isomorphic curves. Over C, the curves are classified by their j-invariant (a complex number).
2. You can fix the lattice and choose generators so that one of the generators is 1 and the other lies in the upper half plane. There is then an action of SL2(Z) on the choices of generators, yielding an action on the upper half plane. If you choose a fundamental domain for this action, you get a "canonical" choice of lattice for each elliptic curve.
3. The Lie algebra is the unique 1-dimensional Lie algebra, whose bracket is zero. One can sometimes find more interesting information using the formal group law, but that mostly applies when you work in characteristic p.
1
1. No, for any cubic curve in the plane, there is a family of cubic plane curves (3 or 4 dimensional - I forget) that are isomorphic as curves. For any lattice in C, there is a 1-dimensional family of lattices that form isomorphic curves. Over C, the curves are classified by their j-invariant (a complex number).
2. You can fix the lattice and choose generators so that one of the generators is 1 and the other lies in the upper half plane. There is then an action of SL2(Z) on the choices of generators, yielding an action on the upper half plane. If you choose a fundamental domain for this action, you get a "canonical" choice of lattice for each elliptic curve.
3. The Lie algebra is the unique 1-dimensional Lie algebra, whose bracket is zero. One can sometimes find more interesting information using the formal group law, but that mostly applies when you work in characteristic p.