The Mordell curve for a number k is just y2=x3+k. As in the conductor table, cells with a green background are known to be minimal, as a result of a search in the range -1000000 < k < 1400000 using Cremona's mwrank; I performed this search between September and November 2001, and it took just over eight weeks on three Athlon/850 computers. Contact me if you're interested in the results; I have for each k in that range a collection of generators and the value of the 2-Selmer rank.
A pink background denotes a k-value which is found to have high rank after being selected by sieving for curves with many integer points with x co-ordinate less than 220, a grey background comes from Quer's approach involving looking at the 3-component of the Selmer group, and a blue background is constructed in some other way. If I give two sources separated with semi-colons, the left-hand one is for the positive k and the right-hand for the negative one.
I have looked at all all |k| < 226 such that some k' = u6k has at least 7 integer points with x coordinate <220, all |k|<232 with 16 or more, and all |k| < 240 with 20 or more.
Noam Elkies points out that the curves for k and -27k are isogenous, so it's possible to find a k of one sign given one of the other; the values in cells with a bright yellow background were found this way and are extremely unlikely to be anything approaching optimal.
For rank <=8, the rank was computed with Cremona's mwrank without using conjectures (taking up to a couple of hours on a 500MHz Pentium-III); for higher ranks I found the given number of independent points using Cremona's findinf and used Mestre's bounds, as implemented in APECS and conditional on various analytic properties of L-series, together with the parity conjecture, to show that the rank was in fact as given.
If you plot log(abs(best k)) against rank, at least for the rank <=9 for which the k in this table might be expected to be a good estimate of the actual minimum, you get a curve fitted very well by a parabola; I have no idea whether there is theoretical underpinning for this.
To perform the exhaustive search required to colour the rank-7 curves green would involve checking slightly over 100 million curves; experimental timings suggest that this would take mwrank roughly one year running on 500 1GHz computers. This is a task two orders of magnitude smaller than the largest distributed computations ever performed (SETI@home, distributed.net or mersenne.org); it would still be a substantial problem in logistics.
|Rank||Smallest positive k||Largest negative k||Source|
|6||1358556||-975379||Found by sieving by Womack (2000);
exhaustive-search completed in November 2001
|8||1632201497||-2520963512||Elkies (1999); Womack (2000)|
|10||68513487607153||-56736325657288||Elkies (2001), Llorente-Quer |
|11||35470887868736225||-46111487743732324||Elkies (1999); Quer |
Quer  is Corps quadratiques de 3-rang 6 et courbes elliptiques de rang 12, C. R. Acad. Sc. Paris. I 305 (1987) 215-218.
Llorente-Quer  is On the 3-Sylow subgroup of the class group of quadratic fields, Math. Comp. 50 (1988) pp 321-333.
GPZ  is the main paper about computational results on Mordell curves for small k: J Gebel, A. Petho, H.G. Zimmer, On Mordell's Equation, Compositio Mathematica 110: 335-367, 1998.
Professor J-F Mestre found a family of K providing curves of rank at least 6 in his paper Rang de courbes elliptiques d'invariant donné, available on-line and published in C. R. Acad. Sc. Paris 314 (1992), 919-922, and extended the construction to rank at least 7 in Rang de courbes elliptiques d'invariant nul, also on-line and published in C. R. Acad. Sc. Paris 321 (1995), 1235-1236
Jesper Petersen and Tom Womack can be contacted by email.