**Villegas:** If you say that the curve has rank and look at regulator
in terms of -function, then look at BSD, then big imply
small regulator hence small height. If you believe that the
-function is essentially constant....

**Mazur:** In rank 2 case, what is ratio between shortest and second shortest.

**Weissman:** Higher dimensional? S. David. Can formulate an analogue for
abelian varieties, but I don't know much so I shouldn't summarize.
See Number Theory III, by Lang.

**Elkies:**
Thanks! The four examples you list are quite remarkable also for having
lots of integer points: sixteen pairs on each of the top two examples
(#1: 1-21 except 8,11,13,16,17; #2: 1-18 except 10,11,17 and also 21);
fourteen on #3 (1-18 except 7,11,13,14), and fifteen on #4
(1-15 except 8,13, and also 18 and 25). In each case I don't claim
that my list is complete--I looked only up to 50 and certainly didn't
attempt a proof that there aren't yet more integer multiples, though
for rank 1 it's reasonably straightforward to do in practice. Anyway,
these examples beat the best I that I could find in the tables,
where there were never more than 12 integral pairs, and the largest
multiple was 24P for **735F**. For consecutive integer points, the curve
**618F**
ties your record of nine for **3630Y**. I suspect that you might find
even more such examples by setting a more generous limit on the height
and looking specifically for many integral multiples, for integral
with large, or for many consecutive multiples (could there be a case
where is integral for each in ?).

Update: is realized by the point on the curve of conductor . The point is also integral.

The largest integral multiple I can find for now is , for , (whose first multiples, and , , , are also integral).

**Acknowledgement:** It is a pleasure to thank Matt Baker for giving
a really cool talk, the people who made comments above, and John Cremona for
computing a table of regulators of elliptic curves.