# From an eigenfom with $\mathbf{Q}$-coefficients to $j$-invariants

Given a cuspidal, classical eigenform $f\in S_2(\Gamma_0(N))$ of weight $2$ and with $\mathbf{Q}$-coefficients is there a way of describing the set $J_f$ of $j$-invariants of the elliptic curves lying in the isogeny class determined by $f$?

Has this question been looked at? Thanks.

• Compute the periods from a $q$-expansion by this projecteuclid.org/euclid.em/1047649997 Then compute $j(\tau)$ and recognize it rationally. Then find all isogenous curves. For the last two steps, see Cremona's book (which also gives the first, but not so refined). – GuestPoster Jun 5 '15 at 9:18
• Is there an actual question here? I'm confused. – user74597 Jun 14 '15 at 18:00
• I think the question is there. I agree that's somewhat vague. I will soon retire it. – Tommaso Centeleghe Jun 24 '15 at 9:45

Didn't John Cremona write a book about this question? (And then compute all such elliptic curves for $N$ less than a bizillion or so?)

http://homepages.warwick.ac.uk/~masgaj/book/amec.html

• Also, Cremona's tables are available in searchable form interact.sagemath.org/Tables/cremona/INDEX.html SAGE implements modular symbols doc.sagemath.org/html/en/reference/modsym/index.html , which is the main toolkit used for the computation. I haven't found a one line command in SAGE to just do this, but I expect one exists; this is dead in the center of William Stein's interests. – David E Speyer Jun 5 '15 at 3:33
• I might suggest wstein.org/books/modform/stein-modform.pdf as a more modern updating of Cremona. The specific question is answered in Section 10.6 – David E Speyer Jun 5 '15 at 3:40
• @DavidSpeyer You should browse Cremona's tables on the LMFDB website lmfdb.org/EllipticCurve/Q, the link you provide is obsolete. – Aurel Jun 5 '15 at 8:50
• Thank you all for your answers. I guess what I had in mind was a characterization of $J_f$ starting from $f$, and not necessarily a concrete procedure to calculate $J_f$. – Tommaso Centeleghe Jun 9 '15 at 8:48