The [LMFDB](https://www.lmfdb.org/) describes the elliptic curve [11a3](https://www.lmfdb.org/EllipticCurve/Q/11/a/3) (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?




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**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](https://mathoverflow.net/a/360867/). The earliest-known example of an elliptic curve is one implicitly considered by Diophantus, in book IV of *Arithmetica*, problem 24 ([Heath's translation](https://archive.org/details/diophantusofalex00heatiala/page/184/mode/1up)): "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](https://www.lmfdb.org/EllipticCurve/Q/8732/b/1) 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".



Why is this "the first elliptic curve in nature"?
I asked Kevin Buzzard to ask John Coates directly, and it's basically as people have surmised: the moniker is due to the fact the curve appears first in Cremona's book as it has the smallest possible conductor, and it has the smallest coefficients. It is *not* due to historical priority, as Coates knows of 8th/9th century Arabic manuscripts that discuss $y^2 = x^3 - x$, whereas the first occurrence of the "first curve in nature" is apparently a book of Fricke on elliptic functions (I think from 1922, but I'm not sure).


