An Elliptic Curve Question

Question

An elliptic curve is an equation of the form $y^2 = x^3 +ax +b$. There are elliptic curves that are one continuous curve, and there are some where there is a loop and a curve. What value of $a$ for a given $b$ gives a loop and a curve that touch each other? There should be only 2 roots for one of this form, rather than the expected 1 or 3.

Answer 1

The solution to the question has already been given by Somos, but it might be interesting to illustrate the appearance of the three different patterns of elliptic curves graphically by plotting the function

$$f(x,y) = -y^2 + x^3 + a\; x + b$$

as a function of $x$ and $y$ and looking at the cross section $f=0$.

The first plot shows the function $f$, which we call here "generator", without the cross section.

Now the three cases with the plane $f=0$

The parameters are shown in the graph legends.

Technical remark: I have found a simple way to rotate the 3d graphs in Mathematica: just exchange x- and y- axes.

Answer 2

In the cases you are interested in, the roots of the cubic is a single root and a double root and so the discriminant of it is zero. That is, $\,4a^3+27 b^2=0\,$ since the Wikipedia article discriminant states in the case of a cubic

The discriminant is zero if and only if at least two roots are equal.

If the curve crosses itself, then two roots approach each other and coincide. In other words, the equation has a double root.