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 |