Original table created by Tom Womack. Updated by John Cremona, April 2012.

The curves are given in the form [a1, a2, a3, a4, a6].

There are five classes of curves here, which are colour-coded as follows (note that Elkies & Watkins (2004) completely supercedes Womack's records):

Green |
Curve is known to have the smallest conductor for that rank, as a result of an exhaustive search over conductors |

Blue |
Curve is known to have the smallest conductor for that rank among curves with max(a4,a6) <= its value of max(a4,a6) |

Pink |
Curve was found by the sieve-driven search documented here (with bounds |a4|<=20000, |a6|<2^24) |

Grey |
Curve was found by the Mestre-style method documented here |

Orange |
Curve was found by Elkies and Watkins (see their paper in ANTS VI) |

Rank |
Curve |
Conductor |
log(N) |
Source |

0 | [0, -1, 1, 0, 0] | 11 | 2.398 | Cremona [1997] |

1 | [0, 0, 1, -1, 0] | 37 | 3.611 | Cremona [1997] |

2 | [0, 1, 1, -2, 0] | 389 | 5.964 | Cremona [1997] |

3 | [0, 0, 1, -7, 6] | 5077 | 8.532 | Cremona* |

4 | [1, -1, 0, -79, 289] | 234446 | 12.365 | APECS, Cremona* |

5 | [0, 0, 1, -79, 342] | 19047851 | 16.762 | BMcG [1990] |

6 | [1 1 0 -2582 48720] | 5187563742 | 22.370 | Elkies & Watkins 2004 |

7 | [0,0,0,-10012,346900] | 382623908456 | 26.670 | Elkies & Watkins 2004 |

8 | [1,-1,0,-106384,13075804] | 249649566346838 | 33.151 | Elkies & Watkins 2004 |

9 | [1,-1,0,-135004,97151644] | 32107342006814614 | 38.008 | Elkies & Watkins 2004 |

10 | [0,0,1,-16312387,25970162646] | 10189285026863130793 | 43.768 | Elkies & Watkins 2004 |

11 | [0,0,1,-16359067,26274178986] | 18031737725935636520843 | 51.246 | Elkies & Watkins 2004 |

APECS |
The exam(4) table in Ian Connell's elliptic-curve system (proved minimal by Cremona in 2012). |

BMcG [1990] |
A. Brumer & O. McGuinness, The Behaviour of
the Mordell-Weil Group of Elliptic Curves, Bulletin
of the AMS 23 #2 (Oct 1990) pp 375-382 |

Buddenhagen |
provided the r=9 example to Ian Connell for APECS |

Cremona[1997] |
J E Cremona, Algorithms for Modular Elliptic
Curves, 2nd Edition, pub. CUP, ISBN 0521598206 |

Cremona* |
The extended table found at http://www.warwick.ac.uk/staff/J.E.Cremona/ftp/data |

Elkies & Watkins |
Elliptic curves of large rank and small conductor, ANTS VI |

Mestre (1986) |
the paper in Math: Comp: 58 about constructing elliptic curves of large rank; contained a very good rank-8 example. |

Suess (2000) |
Nigel Suess's PhD thesis (contained the reasonable rank-7 example [0, 0, 1, -5707, 151416]) |

Womack (2000) |
Not documented other than in this table: Womack* denotes curves found by Mestre-style approach |