Here's an illustration of the power of the ideas involved in iterated descents. I have been interested for some time in the curves y^2 = x ( x^2 - x + (1-T^10)/5 ) for rational T. (This family is interesting because the ranks tend to be too high). I investigated about 100 of these curves in December and January. Giving them, on average, about 20-30 CPU _days_ of study with MWRANK, I found lower bounds on their ranks. (These lower bounds could often be increased a bit on the strength of parity conjectures.) With APECS I could lower the upper bound on the rank in just a few cases using Mestre's bound, but there was still considerable disparity. Here's the way the matter stood in mid-January. There was just one curve (T=0) provably of rank 0, 5 provably of rank 1, 15 or rank either 1 or 3, and so on: n rank=n or n+2 or n+4 or n+6 0 1 7 7 2 1 5 15 12 3 2 13 13 2 3 7 3 1 4 2 1 5 2 There are certain to be many more points to be found on these curves. For example, on none of the curves of rank 1 do I have any nontorsion points (the rank is presumed to be 1, not zero, as there is a unique homogeneous space). On the other hand, I was discovering fewer points as time went on. It was in regard to this question that I asked Siksek for information about computing the order of Sha; his response explaining a second 2-descent is what led to the creation of this program. Now it is a trivial matter to run all 96 curves through these maple procedures. Some are completed in seconds; the hardest (T=53/5) takes at most 10 minutes. (That curve allows 64 locally solvable homogeneous spaces, as well as 8 for the 2-isogenous curve). Less than 3 hours of real time passed for all the curves together. Very many of the upper bounds on the ranks were reduced by 2, and in ten cases by 4. Here is the situation now: (Tables also show increased rank for 6 curves assuming parity on both E/phi(E') and E'/phi(E) separately) n rk=n rk=n or n+2 0 1 14 1 19 13 2 25 5 3 13 0 4 3 0 5 3 0 TOTAL 64 32 Thus we now know the ranks of two-thirds of the curves; the average rank is between 1.6 and 2.3 . (It is true in particular that we know precisely the number of generators arising from each of the 2-isogenous cures, that is, there is no ambiguity left in Sha there after this extra descent.) The amibugity in each case is at most 2. My money is on the 14 curves of rank "0..2" to have rank 2; I don't know about the 13 curves of rank "1..3" and the 5 "2..4"'s are a good candidate for 4-descent? (T=23/9, 20/7, 26/3, 19/5, 29/5). Timings for the 96 curves: 10395 sec (just under 3 hours) elapsed time included 2811 sec (27%) finding divisors allowing solvable conics 5100 sec (49%) finding points on those conics 1776 sec (17%) deciding local solvability of all quartics 1068 sec (10%) spent on factorization (overlaps other categories) (of the 174 minutes, more than 20 were spent just on T=43/5!) Recently I have been able as well to find points on the descendant homogeneous spaces, using Cremona's program ratpoint to examine the search space as efficiently as possible. In particular I now have generators for several of the curves of rank 1; here are their x coordinates, after reascending to the original curve: T=6/1: 26192858636992454273130/7487594695430649864 T=9/1: 12774102279604761815247993795/447760174248763296454071 T=9/5: 3053993845269838705383061/10990657529368484765625 T=16/5: 401015126179795298852620608161/1432793792266856544100000000 T=9/7: 23744292777631919404889461/368402949338412391614225 T=13/7: 28676817062255975209261445/125055373686507001163881 T=15/7: 301576625016155130230572202101/1919156665733645162745650625 T=3/8: 34934519546542039345/46507934043846475776