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|<=20000, |a6|<2^24) |
| 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 | [0 0 1 -7077 235516] | 5258110041 | 22.383 | Womack (2000) |
| 7 | [0 0 0 -12979 405826] | 1074680679376 | 27.703 | Womack (2000) |
| 8 | [0 1 1 -16440 1394010] | 561715239383323 | 33.962 | Womack (2000) |
| 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 vari�t�s alg�briques, 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]) |
| Womack (2000) | Not documented other than in this table: Womack* denotes curves found by Mestre-style approach |