Roadmap for elliptic curves (awaiting review)

Elliptic Curves

Additional pages

• Add page with list of elliptic curves over Q that have a point of canonical height at most some small'ish epsilon. (Noam Elkies has this data.)
• Add page with elliptic curves of smallest known conductor and Mordell-Weil group of the form F x Z^r for the 15 possible finite groups F and r = 0,1,2,...

Additional data (over Q)

• Currently the database contains all elliptic curves defined over $\mathbb{Q}$ of conductor at most 379998. Add the much larger Stein-Watkins database, which includes curves with larger conductor. This would not be complete. Currently we do not have as much data about all the Stein-Watkins curves as for those in the Cremona database: for example we do not have generators for all the curve of rank 1. Also Mike Bennett's 2015 data, Randall Rathbun's data on congruent number curves for parameters up to 1000000.
• More data about each curve:
• Fisher has explicit models of genus one curves representing every element of order 3 and order 5 in Sha, and one could easily do similar for elements of order 2. These could be shown on a curve's home page.
• Base change: the user could select an extension field and see the page for the base-changed curve. This would be easy for cases where the base-change curve is already in the database.

Elliptic curves over number fields

• Data available for each curve should be extended to include Mordell-Weil generators, which have in most cases not yet been computed.