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 Arithmetica, 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".
 A: 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).
A: 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 $X_0(11)$.
A: I can only echo Tim D's explanation: from Coates via Vlad to me.  I did not know about it having minimal Faltings height. 
A: 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.)
