Minimal-known positive and negative k for Mordell curves of given rank.

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
0 1 -1 Trivial
1 2 -2 GPZ [1998]
2 15 -11 GPZ [1998]
3 113 -174 GPZ [1998]
4 2089 -2351 GPZ [1998]
5 66265 -28279 GPZ [1998]
6 1358556 -975379 Found by sieving by Womack (2000);
exhaustive-search completed in November 2001
7 47550317 -56877643 Womack (2000)
8 1632201497 -2520963512 Elkies (1999); Womack (2000)
9 185418133372 -463066403167 Womack (2000)
10 68513487607153 -56736325657288 Elkies (2001), Llorente-Quer [1988]
11 35470887868736225 -46111487743732324 Elkies (1999); Quer [1987]
12 176415071705787247056 -6533891544658786928 Quer [1987]

Sources:

Quer [1987] 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 [1988] is On the 3-Sylow subgroup of the class group of quadratic fields, Math. Comp. 50 (1988) pp 321-333.

GPZ [1998] 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.