Some Curves With Small Regulator

William Stein

November 29, 2001

I examined Cremona’s newest data for elliptic curves of conductor ≤ 12000. The four smallest regulator curves in his tables are as follows:

Curve  Regulator = Min height of a point
3990V1  0.0089143244556863533937
3630Y1  0.0090386818898339096889
1430K1  0.0097398661429109179437
1470L1  0.0099641079999126058232

These curves all have rank 1. Here is some more information about each:

\begin{matrix}N & [a_{1}, a_{2}, a_{3}, a_{4}, a_{6}] & Point & N & Δ \\ 3990 & [1, 1, 1, −125615, 61201397] & (7107, −602054) & 2 · 3 · 5 · 7 · 19 & 2^{11} · 3^{4} · 5^{7} · 7^{5} · 19^{3} \\ 3630 & [1, 0, 0, −141875, 18393057] & (−386, 4153) & 2 · 3 · 5 · 11^{2} & 2^{17} · 3^{5} · 5^{7} · 11^{4} \\ 1430 & [0, 0, 1, −1089, 14308] & (11, 60) & 2 · 5 · 11 · 13 & 2^{13} · 5^{5} · 11^{3} · 13^{2} \\ 1470 & [1, 1, 1, −2990, 71147] & (27, −119) & 2 · 3 · 5 · 7^{2} & 2^{13} · 3^{2} · 5^{5} · 7^{4} \end{matrix}

The point (7107, −602054) on 3990V1 is entertaining because its naive and canonical heights differ by almost 9. Watch 2^{n}P get small then become large.

> E := EC("3990V1");
> P := E![7107,-602054];
> 2*P;
(1787 : -75374 : 1)
> 4*P;
(457 : -10204 : 1)
> 8*P;
(-2537/9 : -228958/27 : 1)
> 16*P;
(5443/9 : 380182/27 : 1)
> 32*P;
(-3185573/47961 : 87754320614/10503459 : 1)
> 64*P;
(7496507080486748587/42278403738202641 :
      -58793256118747632513363890206/8693163995352094662637689 : 1)