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

There are four classes of curves here, which I've colour-coded in what I hope is a meaningful way:

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|<=2^16, |a6|<2^26) |

Cream |
I don't know how this curve was found |

Grey |
Curve was found by a Mestre-style method (mine is documented here): the rank has not always been proven not to be more than I claim |

If the conductor is displayed in a saturated (rather than a pastel) colour, it is prime.

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 |

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

6 | [1 1 0 -2582 48720] | 5187563742 | 22.369 | Watkins (2002) |

7 | [0 0 0 -10012 346900] | 382623908456 | 26.670 | Watkins (2002) |

8 | [0 0 1 -23737 960366] | 457532830151317 | 33.757 | Womack (2002) |

9 | [0 1 1 -3529920 2567473020] | 484154179417645171 | 40.721 | Womack* (2000) |

10 | [0 1 0 -73169143545 8305634997295659] | 1971056874401658426264 | 49.033 | Womack* (2000) |

11 | [0 0 1 -56874727 151924164456] | 1803406168183626767102437 | 55.852 | Mestre (1986) |

APECS |
The exam(4) table in Ian Connell's elliptic-curve system. |

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 an 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.maths.nottingham.ac.uk/personal/jec/ftp/data |

Mestre (1986) |
J. F. Mestre, Formules explicites et minorations
de conducteurs de variétés algébriques,
Compositio Math. 58 (1986) pp 209-232;
contained a very good rank-8 example as well as this
rank-11 one. |

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

Watkins (2002) |
Mark J Watkins; personal communication. |

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