TITLE: How to show modularity of an elliptic curve?
QUESTION [27 upvotes]: In the days before [W, TW, BCDT], how did people show that specific elliptic curves over $\mathbb{Q}$ were modular? For instance, I was reading through a paper of Buhler, Gross and Zagier from 1985 on the curve 5077a, and they say that modularity can be checked by a finite computation in the 422-dimensional space of cuspforms of weight 2 and level 5077 (and remark at the end that Serre and Mestre have checked it). A google search brought up the name "Faltings-Serre method": was this the technique of choice? Also, are there any good references for it?

REPLY [5 votes]: The answer is outlined in Don Zagier's 1985 paper Modular points, modular curves, modular surfaces and modular forms.