The Mordell curve for a number *k* is just y^{2}=x^{3}+*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 2^{20},
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*| < 2^{26} such
that some k' = u^{6}k has at least 7 integer points with *x*
coordinate <2^{20}, all |k|<2^{32} with 16
or more, and all |k| < 2^{40} with 20 or more.

Noam Elkies
points out that the curves for *k* and -27*k* 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] |

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.