Curves with moderate rank and interesting torsion group

There used to be a fashion among the minor princes of Germany for cabinets containing curiosities: nautilus shells, dried puffer-fish, semi-precious stones carved in peculiar shapes ...

This fashion died out with the German minor princes, but here's my curiosity cabinet: it contains elliptic curves with interesting torsion groups. The data comes from a long Pari run using the parameterisations (found in Kubert's paper in the 1976 Proceedings of the LMS, and implemented in torseg.gp) of curves with specific torsion groups; for conductors less than 5300, it matches the data found in Cremona's table of curves.

These parameterisations give curves with highly-composite discriminants, so they tend to have very small conductors in comparison to their discriminants; accordingly, analytic-rank methods work effectively on them (whilst the huge discriminants mean that mwrank-style homogenous-space-enumeration approaches don't work very well: so the ranks in this table are analytic ones, and sha=25 means only that |tors|^2 * L(1) / (Omega * tamagawa product) = 25.

I have two tables: one gives the smallest conductor observed for a given rank and torsion, the other the smallest conductor observed among curves of rank zero with a given Sha and torsion.

I couldn't find a way to write the actual curves into these tables without making them even uglier than they are already, so to see a specific curve you should load torseg.gp into Pari and issue the command given after the conductor in the appropriate cell of the table. Alternatively, click on the command and the curve will appear in the bottom frame.

Minimum conductor found at given rank and torsion

Minimum conductor found at rank zero, given sha and torsion