reason to change elliptic curves to normal form I would like to know about the concept of why we change the curve into Weierstrass normal form.
I mean for a specific example, if we have $x^3 +7y^3 + 64z^3=0$ and I went through all the steps and I changed it to $y^2= x^3 +ax^2 +bx + c$ .
I did this and I got very huge number like in trillions for $a, b, c$.
so we have $P$($2,2,-1$) belong to our first curve, what is this point considered w.r.t. the normal form?
I understood the process but I don't know why we do that and how can we find the rational points.
Thanks for your help in advance.
 A: First of all, ultimately you need to deal with such curves without using explicit equations. The higher-dimensional analogues of elliptic curves (called abelian varieties) do not easily submit to equation-theoretic reasoning.
Even for elliptic curves, the group law is best thought of geometrically, not in terms of equations for the curve.
If we do use equations, a reason for working with Weierstrass equations is related to the definition of an elliptic curve over a field: it is not just given by a smooth cubic, but also must have a point defined over the field. A famous example of your type of cubic that has rational coefficients but no point with rational coordinates is $3x^3 + 4y^3 + 5z^3 = 0$ (viewed in $\mathbf P^2$, so $[0:0:0]$ does not count). The projective curve defined by this equation (in algebraic geometry) is smooth of genus 1 but it is not an elliptic curve over $\mathbf Q$.
Every curve over a field defined by a Weierstrass equation has a rational point over that field: $[0:1:0]$. This point is also rather special geometrically among all points on a curve defined by a Weierstrass equation: the tangent line to the curve at that point is the line at infinity $Z = 0$ and this tangent line meets the curve at $[0:1:0]$ with multiplicity 3 (it is called a flex point), not just multiplicity at least 2. This can help you figure out how to find the Weierstrass equation for an elliptic curve given by a different equation: make a change of coordinates to move the flex point to $[0:1:0]$ so that the tangent line to the curve at the flex point turns into $Z = 0$.
