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 $a x^2 + b xy + c y^2 + d x +e y + f=0$ for 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), $x^2-1=0$, or $x^2=0$. (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

$$a_1 x^3 + a_2 x^2 y + a_3 x y^2 + a_4 y^3 + a_5 x^2 + a_6 x y + a_7 y^2 + a_8 x + a_9 y + a_{10} = 0,$$

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

$$y^2 = f(x)$$

or equivalently

$$y^2 - f(x)=0,$$

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

$$y^2= x^3 -3 x+ 2$$

does not define an elliptic curve, because

$$x^3-3x+2 = (x-1)^2 (x+2)$$

has $1$ as a multiple root. Similarly $y^2 = x^3$ is not an elliptic curve, but $y^2=x^3+1$ is an elliptic curve.

By scaling the coordinates and translating, one can convert any elliptic curve into one of the form $y^2 = x^3 + Ax + B$ where $A$ and $B$ are numbers. It turns out that a general curve of the form $y^2 = x^3 + Ax + B$ is an elliptic curve (i.e., $x^3+Ax+B$ is squarefree) if and only if $-(4A^3+27B^2)=0$. (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.