TITLE: Elliptic Curve in Weierstrass Form
QUESTION [5 upvotes]: My book says that an Elliptic Curve is a curve of the form
$$y^2+a_1xy+a_3y=x^3+a_2x^2+a_4x+a_6\text{ over a field.}$$
My question is: Why there are some parts like $a_7xy^2$ missing?

REPLY [6 votes]: An elliptic curve over a field $F$, in fact, is a projective, smooth curve of genus $1$, with at least one point defined over $F$. It turns out (as Lord Shark discusses) there is a change of variables that brings any elliptic curve to a model of the form you write (which is called a Weierstrass equation). In fact, if the characteristic of $F$ is not $2$ or $3$, then you can bring it to a model of the form $y^2=x^3+Ax+B$, which is called a short Weierstrass form.
For instance, the curve $C$ over $\mathbb{Q}$ given by $x^3+y^3=1$ (the curve $X^3+Y^3=Z^3$ in projective space) is also an elliptic curve (there is at least one point, namely $(1,-1)$, the point $[1,-1,0]$ in projective coordinates), even though it is not given a priori by a Weierstrass form. A change of variables brings $C$ to the equation
$$y^2 - 9y = x^3 - 27$$
in Weierstrass form. 
If you are interested in how one finds such changes of variables, this is briefly explained in Silverman and Tate's "Rational Points on Elliptic Curves".