# Why is this “the first elliptic curve in nature”?

The LMFDB describes the elliptic curve 11a3 (or 11.a3) as "The first elliptic curve in nature". It has minimal Weierstraß equation $$y^2 + y = x^3 - x^2.$$ My guess is that there is some problem in Diophantus' Arithmetica, or perhaps some other ancient geometry problem, that is equivalent to finding a rational point on this curve. What might it be?

Edit: Here's some extra info that I dug up and only mentioned in the comments. Alexandre Eremenko also mentions this in an answer below. The earliest-known example of an elliptic curve is one implicitly considered by Diophantus, in book IV of Arithemtica, problem 24 (Heath's translation): "To divide a given number into two numbers such that their product is cube minus its side". Actually this is a family of curves over the affine line, namely $$y(a-y)= x^3-x$$, though Diophantus, in his usual way, only provides a single rational point for the single curve corresponding to $$a=6$$. This curve is 8732.b1 in the L-functions and modular forms database (the Cremona label is 8732a1). So presumably the comment about 11a3 is not meant to mean "historically first".

• h/t to Anton Hilado for bringing this to my attention on Twitter! – David Roberts May 20 at 4:43
• Surely it's because it has smallest conductor. I'm guessing the congruent number curve (for 1, i.e. $y^2=x^3-x$) is the first elliptic curve in history. – Dror Speiser May 20 at 6:13
• product of two consecutive numbers equals product of three consecutive numbers? – liuyao May 20 at 6:20
• As far as I know, this is an informal name was coined by John Coates (and, possibly, never in writing). It is one of the three curves of smallest conductor, and has the simplest equation among those. – Tim Dokchitser May 20 at 6:25
• @liuyao : what you describe seems to be rather the curve lmfdb.org/EllipticCurve/Q/37/a/1, in my opinion – Watson May 23 at 8:44

I actually only wrote the part that says that this curve is a model for $$X_1(11)$$, not the first part, which I think was written by John Cremona.

It is standard to order elliptic curves by conductor (e.g. for statistics), and 11 is the smallest possible conductor. However, there are 3 curves with conductor 11, and no canonical way to order them as far as I know (though @François Brunault has an interesting point); for instance LMFDB labels do not order these 3 curves in the same way as Cremona labels.

This curve being the first one could maybe also be understood in terms of modular degree, although this is also ambiguous: if we order them by degree of parametrisation by $$X_1(N)$$, then this curve, being a model of $$X_1(11)$$, comes first, but if we order in terms of degree of parametrisation by $$X_0(N)$$, then 11.a2 comes first since it is a model for $$X_0(11)$$.

• From (nearly) the horse's mouth! Unless Cremona lets us know why, this might be as good as we get (if you can get him to explain, I would be curious to know). – David Roberts May 20 at 13:00
• For each isogeny class of elliptic curves over $\mathbb{Q}$, one can define a graph (which I believe is canonical) where the vertices are the elliptic curves and the arrows are the étale isogenies of prime degree. Étale means, for example, that the pull-back preserves the Néron differentials. It turns out that this graph is a tree, and the root is the $X_1(N)$-minimal curve (the smallest degree of $X_1(N) \to E$). So there is a unique curve with smallest Faltings height. This last property can fail over number fields. – François Brunault May 20 at 14:06
• PS. I think what I wrote is conditional on Stevens's conjecture on the Manin constant of $X_1(N)$-parametrisations. – François Brunault May 20 at 14:08

I can only echo Tim D's explanation: from Coates via Vlad to me. I did not know about it having minimal Faltings height.

• But the lmfdb ordering is now deterministic, so there is an answer to why $X_1(11)$ is now the first. I believe to remember that you explained me your criteria for ordering and that it implies that the minimal curve (Falting height, or lattice or ... ) is always the first one. Or am I wrong here? – Chris Wuthrich May 20 at 15:56
• Hmm, the chain of whispers stretches back... I wonder if Coates could be prevailed on to explain? :-) – David Roberts May 20 at 23:17

The closest thing I found in Diophantus is problem IV(24) which is solving the system $$X_1+X_2=a,\quad X_1X_2=Y^3-Y.$$ Diophantus sets $$X_1=x$$ and eliminates $$X_2$$ obtaining $$x(a-x)=Y^3-Y.$$ This seems to be the first elliptic curve encountered in the book of Diophantus; before that he only considers rational curves and surfaces.

Diophantus choses $$a=6$$ and obtains a solution $$x=26/27,\; Y=17/19$$.

(This little research is based on a Russian translation of Diophantus with comprehensive comments by I. G. Bashmakova, published in Moscow in 1974.)

• Yes, I mentioned this curve in the comments to the question; I was going to edit them into the main body tomorrow. I guess (I'm not an algebraic geometer) that your comment is showing that after base change to $\mathbb{Q}[i]$ one can get the Arithmetica IV(24) curve and 11a3 as fibres of the same family of curves over $\mathbb{A}^1$. – David Roberts May 20 at 13:54