# Records for low-height points on elliptic curves over number fields

Elkies maintains a list of nontorsion points of low height on elliptic curves over Q; does anyone know of anything similar for curves over number fields?

Everest and Ward give examples of points of height 0.01032... and 0.009721... on curves over Q(w) for w a cube root of unity or the golden ratio respectively. I have made a modest improvement in the latter case, recovering a point of height 0.009128... .

In the context of the elliptic Lehmer problem the aim is to minimise dh(P) for d the degree of the number field, so working over quadratic extensions a point would have to have height less than 0.005 to be competitive with the examples in Elkies' table. Are there any examples?

Using PARI, I recently extended Taylor's computations based on Elkies' parameterizations and found four examples in quadratic fields of non-torsion points whose heights are smaller than those on Taylor's website.

As in Elkies' parameterization, $P=(0,0)$ for each example.
a, c and u are also as in his parameterization.

${\mathbb K}={\mathbb Q}(\sqrt{d})$ and hgtRatio=2hgt(P)/$\log \left( {\mathcal N}_{{\mathbb K}/{\mathbb Q}} \left( \Delta_{E} \right) \right)$.

w=quadgen(d), if d is congruent to 1 mod 4

hgt=0.000681, hgtRatio=0.00001794, d=10, a=3 - w, u=1 + 1/2*w, c=8/9,
a1=-118 - 53*w, a2=2520 + 1044*w, a3=-755640 - 239040*w

hgt=0.001466, hgtRatio=0.00003851, d=7, a=2 + w, u=1/7*w, c=343,
a1=-16422 - 6207*w, a2=43423359 + 16412487*w, a3=-1276637821704 - 482523741504*w

hgt=0.001481, hgtRatio=0.00003735, d=33, a=1, u=-3 + w, c=1/3,
a1=-123 + 35*w, a2=-471018 + 139674*w, a3=435016512 - 128997696*w

hgt=0.001597, hgtRatio=0.00005050, d=7, a=1/3 + 1/3*w, u=-1/2, c=288,
a1=443 + 167*w, a2=-178080 - 67308*w, a3=6681612 + 2525412*w

The example above with the smallest height also has the smallest hgtRatio, 0.00001794.
The second smallest value of hgtRatio occurs for the following example:
hgt=0.001822, hgtRatio=0.00002038, d=7, a=1/7*w, u=-5/6 + 1/6*w, c=13608, a1=4081 + 1423*w, a2=-33595908 - 12371892*w, a3=63747980088 + 23913434976*w

Since no answers have been given here or via the NMBRTHRY mailing list (and as this question is now the top hit on google for 'low height points on elliptic curves'), perhaps you'll allow me the luxury of answering my own question...

I have constructed a page detailing some points on curves over quadratic fields with height at most 0.01; two of the examples have height less than 0.005, so (scaling for degree) are competitive with some of those listed by Elkies. The table can be found here, and additional contributions would be happily accepted!

• Have you tried using (instead of elliptic divisibility sequences) the parametrizations exhibited at the end of the page you cite <math.harvard.edu/~elkies/low_height.html> to search for low-height points over quadratic fields? – Noam D. Elkies Nov 11 '11 at 15:10