In the case the modular curve $X_0(N)$ has genus one, is there a reference for the explicit map between such curve and the corresponding elliptic curve in the lmfdb database? (just having the explicit map to the j-line would be enough).
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2$\begingroup$ Do you want to know the map, or do you want a paper you can reference that writes these down? All the information you seek is implemented in Magma's "Small modular curves" database. $\endgroup$– Jeremy RouseSep 22, 2016 at 14:01
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$\begingroup$ Thank you Jeremy. Magma implementation does the job for the example I had in mind. Still, is there a good reference for the code? (I assume it is based in some article). $\endgroup$– A. PacettiSep 22, 2016 at 15:46
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The maps $j : X_{0}(N) \to \mathbb{P}^{1}$ are given in Magma's ''Small modular curves'' database. In each case, they construct functions on the modular curves $X_{0}(N)$ out of eta products, modular forms attached to elliptic curves, theta series of binary quadratic forms, or weight $2$ Eisenstein series. It appears this has been done independently of any papers on the subject. (I think you can get Magma to spit out its representation of the modular functions in question.)
It seems that one of the first to write down equations for the genus one $X_{0}(N)$ was Ligozat (see the paper here), and it seems that Jacques Velu was the one who computed the elliptic curves over $\mathbb{Q}$ with exotic isogenies (for example, those with a cyclic $17$-isogeny), according to the "Remarks on Isogenies" in "Modular Functions of One Variable, IV", the book edited by Birch and Kuyk. (This is the source Andrew Wiles references in his proof of Fermat's Last Theorem for the fact that the non-cuspidal rational points on $X_{0}(15)$ correspond to non-semistable curves.)
answered Sep 22, 2016 at 17:28
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$\begingroup$ I looked at Ligozat article, but it is not explicit and he refers to an article by Frike for this computation (see section 4). He needs the eta products you mention and only claims they do exist citing Frike's article. My problem was for N=49, and in Antwerp the reference is an article of Joly Jean-René that I cuoldn't found online. Still I do believe magma =D $\endgroup$ Sep 22, 2016 at 21:20