Plane curves of low degree

Plane curves of degree 1 are called lines. They are defined by equations of the form ax+by+c=0, where a,b are not both zero.

Plane curves of degree 2 are called conic sections or simply conics2. These have the form a x2 + b xy + c y2 + d x +e y + f=0for some numbers a,b,c,d,e,f. The conics include ellipses (including the special case of circles), parabolas, hyperbolas, as well as some ``degenerate'' cases such as xy=0 (two lines), x2-1=0, or x2=0. (Many people would exclude some or all of the last three examples from the definition of a conic.)

Plane curves of degree 3 are called cubic curves. The general form of such a curve is

a1 x3 + a2 x2 y + a3 x y2 + a4 y3 + a5 x2 + a6 x y + a7 y2 + a8 x + a9 y + a10 = 0,

where $a_1,\dots,a_{10}$ are numbers. Elliptic curves are certain cubic curves; namely they are the curves defined by equations of the form

y2 = f(x)

or equivalently

y2 - f(x)=0,

where f(x) is a squarefree polynomial of degree 3. ``Squarefree'' means that f(x) has no multiple roots. For instance

y2= x3 -3 x+ 2

does not define an elliptic curve, because

x3-3x+2 = (x-1)2 (x+2)

has 1 as a multiple root. Similarly y2 = x3 is not an elliptic curve, but y2=x3+1 is an elliptic curve.

By scaling the coordinates and translating, one can convert any elliptic curve into one of the form y2 = x3 + Ax + Bwhere A and B are numbers. It turns out that a general curve of the form y2=x3+Ax+Bis an elliptic curve (i.e., x3+Ax+B is squarefree) if and only if -(4A3+27B2)=0. (See the problems at the end.) From now on, we will always assume that our elliptic curves are defined over ${\mathbb Q}$; this means that the coefficients of the polynomial defining an elliptic curve are rational numbers.