High Rank and Torsion  - Examples found using PARI

           The number indicated is a lower bound for the algebraic rank. The generators were checked to be independent ( either with PARI or mwrank ). If the rank says 7(8) this means that I found 7 indenpendent points, but the root number is 1 in the functional equation, so the rank is expected to be even, and it would be at least 8. The generators are expressed [P1,P2,...] + [Q1, Q2,...], the first set generates the free part while the second set generates the torsion part.
            These are just examples, they are not "best possible" though. For the best results known see Dujella's page .

WITH Z/3Z TORSION (notice that all of them are Mordell Curves ):

5	[0, 0, 0, 0, 109858299531561]	[[735532, 630902573], [49704, 15252915], [-4578, 10476753], [-15260, 10310419], [197379, 88314450]] + [[0, 10481331]]	11864696349408588	7/23/2003
5	[0, 0, 0, 0, 4870812295352025]	[[149286, 90541959], [141780, 87868155], [45870, 70479255], [11815, 69803020], [76791, 72963264]] + [[0, 69791205]]	526047727898018700	7/23/2003
4	[0, 0, 0, 0, 4294836225]	[[-510, 64515], [-1020, 56865], [-3855/4, 466455/8], [-1190, 51085]] + [[0, 65535]]	463842312300	7/22/2003
3	[0, 0, 0, 0, 9492561]	[[-78, 3003], [-3081/25, 345072/125], [-2054/49, 1052675/343]] + [[0, 3081]]	1025196588	7/22/2003