Best known conductors for elliptic curves of given rank

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

Sources:

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