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".