Elliptical Curves

I recently read an interesting article in Quanta Magazine that involved elliptic curves. It's a fairly high level article but an excerpt read:

Their starting point was one of the central objects in number theory: elliptic curves. Just like circles and lines, elliptic curves are both numbers and shapes. They are pairs of numbers, \(x\) and \(y\), that serve as solutions to an algebraic equation like \(y^2 = x^3 − 2x\). The graph of those solutions creates a geometric shape that looks vaguely like a vertical line extruding a bubble.

This was followed by the diagram shown in Figure 1:

Figure 1

It turns out that these elliptic curves are very important in cryptography as this following YouTube video explained:

Many elliptic curves can be written as \(y^2=x^3+ax+b\) and all the curves shown in Figure 1 are of this type. What's of interest to me is that there are integer solutions to these sorts of equations, in other words there are points \( (x,y) \) such that \(x\) and \(y\) are both integers. These could be termed elliptic Diophantine equations.

Let's consider \(y^2=x^3-x+1\) as a particular example (the graph is shown in Figure 1, bottom left). When \(x=0\), \(y \pm 1\) but what about when \(y=0\)? Here we have \(x^3-x+1=0\) and there are three solutions, two complex and one real. What is the real solution? $${\left(\frac{1}{18} \, \sqrt{23} \sqrt{3} - \frac{1}{2}\right)}^{\frac{1}{3}} + \frac{1}{3 \, {\left(\frac{1}{18} \, \sqrt{23} \sqrt{3} - \frac{1}{2}\right)}^{\frac{1}{3}}} \approx -1.3247$$So let's check what integral solutions there are between -1 and 100. Figure 2 shows the points that satisfy (permalink):

Figure 2

Extending the range to one million does not throw up any new points so these seem to be the only points that are integers. I have a book that deals with this very subject that is described as follows:

This book presents in a unified and concrete way the beautiful and deep mathematics - both theoretical and computational - on which the explicit solution of an elliptic Diophantine equation is based. It collects numerous results and methods that are scattered in the literature. Some results are hidden behind a number of routines in software packages, like Magma and Maple; professional mathematicians very often use these routines just as a black-box, having little idea about the mathematical treasure behind them. Almost 20 years have passed since the first publications on the explicit solution of elliptic Diophantine equations with the use of elliptic logarithms. The "art" of solving this type of equation has now reached its full maturity. The author is one of the main persons that contributed to the development of this art. The monograph presents a well-balanced combination of a variety of theoretical tools (from Diophantine geometry, algebraic number theory, theory of linear forms in logarithms of various forms - real/complex and p-adic elliptic - and classical complex analysis), clever computational methods and techniques (LLL algorithm and de Weger's reduction technique, AGM algorithm, Zagier's technique for computing elliptic integrals), ready-to-use computer packages. A result is the solution in practice of a large general class of Diophantine equations.

This is rather too detailed for me but the mention of Magma and Maple make me wonder what tools SageMath has to offer in terms of the solution of elliptic Diophantine equations. The book was published in August of 2013. As it turns out SageMath offers two commands. For the equation \(y^2=x^3+ax+b\), the commands are:

E=EllipticCurve([\(a,b\)])

E.integral_points()

For the case of \(y^2=x^3-x+1\), this produces the output (\(a\)=-1 and \(b\)=1):

[(-1 : 1 : 1), (0 : 1 : 1), (1 : 1 : 1), (3 : 5 : 1), (5 : 11 : 1), (56 : 419 : 1)]

The output is a little confusing but it would seem that the final 1 in each triplet signifies that the middle number can be positive or negative. It is thus equivalent to:

[(-1, ±1), ((0, ±1)), (1, ±1), (3, ±5), (5, ±11), (56, ±419)]

This is what is got earlier using my longer method. Elliptic curves is a big topic and I've only scratched the surface here. Figure 3 shows the points on the curve except for (56, ±419).

Figure 3: created using GeoGebra Classic