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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).