Generators [X,Y] for the curve Y^2 = X^3 + n^2*X^2 - 72*n*X - 16*(4*n^3+27) for 1 <= n <= 1000. This list is complete except for 7 curves, all of rank 1, for which a generator is not yet known: 609, 697, 748, 769, 802, 901, 907 (Note that until a generator is found in each case, we cannot deteremine whether there are points on the curves which have X < 0.) When n=3, the curve is of genus zero. For the other 999 values, the curve has rank 0 in 450 cases, rank 1 in 456 cases, rank 2 in 92 cases, rank 3 once. The statement that the ranks are not higher than the numbers of points shown is as always based on various assumed conjectures (BSD etc.) and on the accuracy of several complicated machine computations (e.g. L-series nonvanishing) Use at your own risk, etc. 450 Curves whose rank is zero: n = 1, 2, 4, 5, 7, 8, 11, 12, 22, 23, 24, 25, 27, 28, 32, 33, 34, 37, 39, 42, 43, 45, 46, 48, 49, 50, 52, 55, 56, 58, 59, 60, 61, 65, 68, 75, 78, 79, 80, 81, 82, 85, 88, 89, 90, 91, 93, 95, 97, 100, 104, 111, 114, 115, 118, 121, 125, 131, 134, 135, 137, 138, 139, 140, 141, 144, 146, 150, 152, 153, 157, 163, 165, 168, 169, 170, 171, 173, 176, 179, 180, 182, 183, 184, 188, 193, 194, 198, 199, 200, 205, 211, 213, 216, 221, 222, 223, 225, 226, 227, 228, 231, 232, 234, 238, 239, 240, 241, 247, 249, 251, 252, 254, 257, 258, 259, 260, 264, 265, 266, 271, 276, 277, 279, 282, 283, 284, 285, 286, 287, 289, 292, 295, 296, 297, 301, 307, 308, 309, 315, 319, 322, 324, 327, 328, 330, 331, 337, 338, 340, 341, 342, 343, 345, 347, 348, 352, 353, 355, 356, 359, 361, 362, 363, 365, 367, 371, 373, 374, 377, 378, 382, 385, 387, 389, 390, 392, 393, 396, 398, 399, 400, 401, 402, 403, 404, 415, 416, 419, 423, 424, 426, 428, 432, 433, 438, 440, 441, 445, 447, 448, 449, 453, 457, 459, 460, 461, 462, 463, 464, 465, 466, 472, 473, 474, 475, 477, 479, 480, 481, 482, 485, 486, 487, 488, 490, 491, 492, 494, 496, 498, 499, 500, 502, 503, 508, 509, 513, 514, 516, 519, 520, 522, 523, 525, 528, 530, 531, 533, 537, 538, 540, 541, 543, 544, 545, 548, 549, 552, 555, 557, 560, 562, 565, 566, 568, 571, 572, 573, 575, 577, 582, 583, 584, 586, 592, 601, 603, 604, 605, 608, 610, 614, 615, 620, 623, 624, 625, 626, 627, 631, 639, 640, 642, 644, 646, 647, 651, 652, 658, 660, 662, 665, 667, 669, 670, 673, 674, 678, 679, 684, 687, 688, 689, 691, 698, 702, 704, 705, 706, 709, 712, 716, 720, 722, 727, 729, 731, 736, 737, 738, 739, 740, 742, 743, 744, 749, 750, 752, 753, 756, 758, 760, 761, 763, 764, 767, 768, 770, 771, 772, 774, 775, 776, 780, 785, 786, 788, 791, 792, 793, 794, 795, 801, 804, 805, 807, 808, 812, 815, 817, 822, 823, 824, 826, 828, 829, 830, 834, 835, 837, 838, 840, 842, 848, 852, 854, 855, 856, 860, 861, 862, 863, 864, 865, 869, 871, 872, 878, 879, 881, 882, 884, 887, 890, 894, 896, 898, 904, 906, 912, 913, 914, 915, 916, 920, 921, 926, 927, 928, 930, 931, 936, 937, 939, 942, 943, 945, 947, 948, 950, 952, 954, 960, 962, 963, 964, 967, 973, 976, 977, 979, 980, 984, 988, 989, 993, 996, 999, 1000 456(-7) Curves whose rank is 1, with a generator shown. (Note: some lines are very long; not suitable for human reading.) Of these, the following are precisely the ones for which there are solutions with X < 0 (apart from some subset of the 7 unsolved curves): [6, 9, 10, 13, 14, 17, 18, 19, 21, 26, 29, 30, 38, 51, 53, 54, 57, 66, 67, 73, 74, 83, 86, 101, 102, 110, 113, 117, 122, 130, 133, 142, 145, 147, 158, 162, 166, 177, 178, 181, 197, 201, 206, 209, 210, 214, 217, 218, 229, 230, 233, 237, 242, 243, 245, 250, 269, 270, 273, 275, 278, 281, 290, 293, 294, 298, 302, 305, 306, 310, 314, 318, 325, 326, 333, 339, 346, 349, 350, 354, 357, 358, 369, 386, 394, 397, 410, 413, 414, 417, 429, 430, 434, 435, 437, 442, 451, 454, 458, 467, 470, 489, 505, 510, 518, 534, 542, 554, 558, 578, 579, 585, 593, 594, 595, 597, 598, 613, 617, 618, 621, 629, 630, 633, 637, 638, 641, 645, 649, 650, 657, 666, 677, 685, 686, 694, 707, 710, 723, 725, 730, 733, 734, 745, 746, 754, 755, 757, 762, 765, 773, 778, 787, 789, 790, 797, 813, 818, 819, 833, 841, 846, 849, 853, 857, 873, 874, 877, 889, 902, 905, 910, 918, 925, 933, 946, 949, 953, 957, 958, 961, 965, 970, 974, 982, 986, 990] 6, [-20,28] 9, [-24, 36] 10, [-68, 364] 13, [-104, 812] 14, [-36, 260] 15, [36, -288] 16, [196, 4108] 17, [-40, 364] 18, [-276, 1908] 19, [-332, 1792] 20, [36, 28] 21, [-120, 2052] 26, [-52, 812] 29, [-376, 8060] 30, [-84, 2052] 31, [100, 2912] 35, [48, -260] 36, [84, 2556] 38, [-516, 15652] 40, [2356, 148148] 44, [50036, 11406884] 47, [480, 24724] 51, [-972, 39168] 53, [-712, 32500] 54, [-164, 8036] 57, [-2280, 70956] 62, [53460/841, 12254284/24389] 63, [6564, 673632] 64, [49324/729, 27634708/19683] 66, [-2532, 108108] 67, [-399020/121, 151476224/1331] 70, [130612/729, 233048780/19683] 71, [68, -608] 72, [2257951620/1247689, 210768585826212/1393668613] 73, [-89717798504/158030041, 77265730632501572/1986595645411] 74, [-4996, 109460] 83, [-76, -1792] 84, [636, 55404] 86, [-420, -34532] 87, [100, -5824] 92, [6408640436/9579025, 1882920523965884/29647082375] 96, [6436, 805132] 98, [263636/2401, 881221748/117649] 99, [51614976/7921, 586752290676/704969] 101, [-8440, 354172] 102, [-13927451079540/106215372649, 362520187661304361548/34616333453917643] 103, [88, 3500] 107, [118811616/1311025, 5942682758636/1501123625] 108, [100, 6020] 109, [2723707216/3352561, 558871998491444/6138539191] 110, [-5428, 443324] 112, [40155138290151330664945/32505331107937801, 8087337059337646625285073313321361/5860462500472878688184549] 113, [-122776/625, 310312132/15625] 116, [780, 92492] 117, [-2600, 273644] 119, [1284, 159200] 120, [3628, 486980] 122, [-2643647942807940/5099778876361, 705127723710168111397156/11516672543340879109] 123, [112321830657456/67551961, 1195812539922182464332/555209567459] 124, [255562108/212521, 15127407257732/97972181] 127, [3460, 484096] 128, [2900724/49, 5582315404/343] 130, [-2210582868215531/138879202225, 25825080929380286738953/51755417897179625] 132, [79780, 24873436] 133, [-1335218610915546200/105974403705769, -980583897458924448846603980/1090941523841420218603] 142, [-121493651778952370463359660420/54727454720812833584888041, 120287786196999563903919614306662909447283156/404862801081772719402232087203316428661] 143, [379071876/25, 7385416617824/125] 145, [-7927161573992/806162449, -23810049066526185932/22889370414457] 147, [-20300, 734464] 148, [474709952116/30591961, 507906129346525132/169204136291] 155, [35499683, 211584388550] 156, [7406052424260/77810041, 22584970661264526348/686362371661] 158, [-7594613659/1247689, 1165384138306081/1393668613] 159, [144, -16380] 162, [-420, 65484] 164, [30591819110788868/188224982079121, 53582461790475804906816572/2582354712109303429831] 166, [-243956/9, 15522164/27] 167, [16564138200/93876721, 21783501340519436/909571549769] 172, [2039311243516/2044396225, 16035036167297770636/92437375313375] 175, [1109899012/22801, 47195777681536/3442951] 177, [-1600532606457895400719288872602056552863000/4216489864294939248323713357587084541129, 17551502920649953589434705332665169089975857243800664103187317772/273795679730722189468772898207282169644797335073220630314117] 178, [-212, 32500] 181, [-2371563109880/211208089, 5057290624209964364/3069487157437] 185, [352, 62244] 187, [340591981745053002312053920/42088384723360325817769, 14498347780378019947728495917378963933252/8634623149897546519454398931640853] 189, [22742256/121, 118313265636/1331] 190, [1163487131205442830852071332/22564302942549058507089, 51746392711936829342039973496389355274180/3389478494326077097438129088948313] 191, [48356, 14084512] 192, [1860, 365364] 197, [-35320, 2086276] 201, [-696, 136836] 202, [51687079924/970225, 15615632257238132/955671625] 203, [5197877/4, 12037000375/8] 204, [10708, 2449252] 206, [-38542180/312481, 1620552571708/174676879] 207, [79790880/5329, 1400584304052/389017] 208, [144218082455092931845113219796/671780221164330334728163801, 657204394366990296524991542736908858606421308/17411685718233026181096760754074764004451] 209, [-1557, 318682] 210, [-52056876312199091748/168799777577955001, 4147705982323730129174129587356/69351810786146534626307501] 212, [51469825940/26327161, 57090032080652236/135084663091] 214, [-10682312110334965077208616804/4754894040400359488182225, 4854455966509438814125240694468588446911556/10368388582010591764281445600539801625] 217, [-77317267716437838648675598616/33237784197159331712692225, 94188268368654267963184728524191810733686748/191623203208459131578479176779724826625] 218, [-5731, 1171363] 219, [5461008/11881, 126558209268/1295029] 224, [3097121676/18447025, 2091822821617724/79229972375] 229, [-1038289059821148341589512/29791179942100943521, 751584633747195259372990261810940812/162604124367728419593103052431] 230, [-1044, 236132] 233, [-264932235633337069/10799182736521, 150181168420826115662753602/35488393099765411931] 235, [340348701352/18003049, 393174074107425260/76386936907] 236, [262308/1681, 1565701732/68921] 237, [-51960, 3371004] 242, [-13240761774533608536580/2578913676234724609, 4913567173616396335338865982053108/4141476562308816538735313023] 243, [-204, 39168] 244, [165796, 78701924] 245, [-79568458568/2601769, 22022158721863740/4196653397] 246, [54420, 18449532] 250, [-164, 26068] 253, [202661273427275440/7345081052761, 166231384466919168303676060/19906499112652859741] 255, [730564/169, 2499924896/2197] 256, [780660070376352623810284/2072883014114186950729, 8581399124372618356633292117440159924/94376135339346600496802383994683] 263, [1090894263396/63001, 1141667613670846688/15813251] 267, [4445420265214413914736473169787340374932950269/3238630417106699078151061671643554852900, 303993262444198686972590502541952648808600334829945831864949720596797/184307108462071585360492767683043818089961007672223550583000] 268, [62310632106483000796/10890663753705649, 1810308948790548807805843400404/1136531623506881639379607] 269, [-264, 61516] 270, [-1225427104524296840857764/16976389186033477201, 135087066784727867822651552193718428/69946821529831045363482844199] 272, [7124463252209102200965978808500/2030957947465691535913547689, 89314447914114881100545543592323850809878229732/91527457118522425439422161261621260912363] 273, [-13939118102631576/411225930361, 1801811603152920639671532/263706441136668109] 275, [-44585388/841, 194420681216/24389] 278, [-42612, 7934500] 280, [16756/81, 32230612/729] 281, [-73960, 5230292] 288, [2398476/121, 8457511572/1331] 290, [-4837348/28561, 140746155116/4826809] 291, [51289691280/7295401, 41945342374614708/19704878101] 293, [-1226561248769559031864791864/8945168819106639267267001, -57359773335361852713814423192329210678068/26753635875745924417132882184384275501] 294, [-1316, -381860] 298, [-1472929736186870466251015171/41057914071206162855209, 68663614479998347019713703898676159815971/8319463528510413618354324801134677] 299, [517248, 402872428] 300, [3876, 1186812] 302, [-169918671506925329841193444/21281785653705319275625, 7149790786890003535176906258765848572204/3104646065699586744911053857203125] 303, [288, 76500] 305, [-1080409101462872/11716547049, 3334068954262470033548/1268234202224907] 306, [-74931, 10248021] 310, [-642155911878298104042773901401822070246517281538523636182979/6951748870074769226144121271434879363849653145854316561, 103359738631164596007262652097199255406773627478502078221408064370327787142794309544467357/18329098809445159087357843604481905258806417159758068701102243018624981906836696441] 312, [13921209/6400, 350589964077/512000] 313, [8515104256/9025, 825538116825604/857375] 314, [-3492, 1076012] 317, [639869850174400676385/8948988207224464, 25103279992008124319182284559463/846566158722142134586688] 318, [-66756, 12375612] 320, [112531963409424526397361417396080354002172/2145112918685837849664171012438705241, 64858736936642039609447738731054536110377766380073269091589236/3141776142907266811450525263640054718463545179071394211] 321, [40050189/4, 254759352939/8] 325, [-63937016/625, 92180698148/15625] 326, [-164, 25308] 332, [283340317328037795164271932346145360621121665500/41635734109957007207561134833186812926416721, 625299767062853506090891580959296560038221356005698750445446702574306924/268657729220755736620968283892526095409136003157558662888547108519] 333, [-25168172430099740896393998761549553391781736/240605063734431793015019346898998427561, 30950566205517809565381903788114298111126143085350674989531854172/3732133281161498533736081171262037684390158409370877651691] 334, [2777614804/841, 148840592974580/24389] 335, [176, 32732] 336, [155980, 80880668] 339, [-2636, 881920] 344, [26276, 9992044] 346, [-225700463252/2725801, 71594127782988188/4500297451] 349, [-249448685996417754560895199275871840992220591016/2102016109967189636743698743119204514605969, 20233067602578425272484918931885639754480949278353797966315579745458828/3047572599736882169177456790090037139615556259239732262620639047] 350, [-745693170243/10883401, 571573252503990019/35904339899] 351, [1979527731400477272/266592383109649, 11679161034108085632582339276/4352828990226942262393] 354, [-9476, 3224780] 357, [-3202591385402023889489793486120/281371293012584952800354449, 18300283533324608640603073236439567827979365108/4719757437475210598355658418530778598457] 358, [-32150277109771298651/990613746116289, 313046586401613271942106626271/31178593885697763962913] 360, [243148, 148449340] 364, [13615454302265970409596533404/4637390697931938000461449, 10775574264074945461811300729706428884963388/9986436154431990677809532175360143707] 368, [519394814037910468540572916/97043310206067577401, 11985948958900281428120866858527611201468/955979112281292996375791098899] 369, [-5352, 1934892] 372, [53803036746909739483497863722874392471975843251189687334882900773156/1195751969268495782929372774683360713002061777472021868704025, 395254662338033141592149778339643928247196104101977090810635010130336911696433947141504882327927710996/1307560092437545287552923374474863572937852891941954161339746741748639353557059898525519875] 375, [180036, 101948832] 379, [12925570993092952849974582257550208/59954327709177718833186620641, 1896911040000065179387791016481462543661370923692372/14680160579377723144802855676380689635739439] 380, [498820308/1442401, 203426240457700/1732323601] 384, [3924/25, 363132/125] 386, [-823643789339240410151587124348196/2533627581224464999777434984649, 442510679256856695582605998215072096726947348633476/4032869484764945644434149651710362692108043643] 388, [4599707210668438336408040199311617312347336916226371357452695065603276/1434413930158398000780111836109542855662124840350882612293488601961, 2157493835530437811204313617692458080624093594134446328801737519281493673519468921879459585668819526955708/1717954831903959971884957712276251711992689476808332153061390186537358846508372554430752729996601291] 391, [1444/9, 282464/27] 394, [-161604440344352938570393061391798362084/1615459065473245658172928364473321, 1526052467367413191795478183357910099314421100260537461156/64929780796405621439363132303370207946536492155381] 395, [6542579697432/1728979561, 108657528463490606236/71892699125941] 397, [-100558314500790104179108520/12563359823715237058369, 4358712680938248145241580588459199244012/1408181704981552203107903489410847] 407, [83770119780957975555076/23987202627900625, 24814001306775786989585106805105824/3715090568284058696265625] 408, [254479034665716/183250369, 4295977102393682837004/2480660245153] 410, [-56911918085133348/1782542744161, 28039009721103902874034324/2379906686041490159] 411, [79912288/121, 800526993308/1331] 412, [59158849/625, 760480827743/15625] 413, [-616, 244948] 414, [-8623425430852236132633019284/2805520666194842898694681, 5917644782181541318336069568263279161218676/4699159738751619730794210112680407171] 417, [-30396386430636439944/76695788524602025, -3194654928574761596394751273596/21240135125313768410591125] 420, [110513169/49, 1206350172441/343] 421, [834909913778427664/3684284463601, 1018424242522930714489301324/7071796129374475849] 422, [968005918910890329485936588308446836637733665/137476816699118171500919582875518243856, 30495848357057092500585351594144193215564032313266171430925303243951/1611922662178661248623639073298750259449810489643567810496] 425, [27666864/25, 156953044612/125] 427, [5740170215747146532326551075649918280487507555784862619387640/35501515392741731143855034013991339311676626619015078521, 20060324301632163264649259732378030445329696294395650520332081445177620576033506261517301916/211529204994676786387740771454937259942957367500760805357549446386672508224801311181] 429, 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180183849627970903227173172113495228/93556940703013467055734301] 802, [0,0] 803, [24040770686419008/104534858644441, 33198977860432047896781500/1068788333263394360989] 806, [85420870226778168729934564/3419395639931856966241, 4102507325209930947262499315696187638076/199951207138022489745684311465711] 811, [411318911751611078389504/694664397967250475241, 8118701106460985580119121501913836764/18308913055599051524033407100789] 813, [-18483637838722824/954644289481, 14464813038864468832672452/932743794836016379] 814, [78192107087236838898207880120918795355429149997766291832218115803056149130780979610455798750439261730202833297178670294586417048419969/162345575601825268109628276325955998654369321311151649656214373923274621215467673988580271931965008469484296992960887432628037142784, 22590706182077316041269381299628514268292115976443769282590359162018192247322573390568124519949448968032105158743360092448734233183970075261399139975487468147650282284266733154817752564329928483099818047/65412490694711698207759680508812002351009161712022072055200777560420213428053123165457067492215395007930140923527510122346606118081720930872216168879677852640894847308017752922118285078366944514048] 818, [-6484, 5274868] 819, [-352716, 198915840] 820, [1969843612/729, 97701526491580/19683] 821, [11702240/1369, 357597045236/50653] 831, [1284, 1050624] 832, [2666785/9, 7954351993/27] 833, [-3310936904200/2846969449, 144109387691596978300/151905748890293] 836, [9894633777492/577777369, 201234868263794900276/13888034618653] 839, [709732/9, 1883794976/27] 841, [-194548691785882376070843931233402893108014990407464/509451380204245331463136827779286444917245561, 2504896853584553620564510172609416977334242485270820840790779779648163659332/11498842216209329158712064715151368332500041637623602187940850288691] 843, [6265814572620612909/6106582322500, 20405258519337783131516986923/15090280906245875000] 844, [401151288988/398161, 331957397309098772/251239591] 846, [-3620, -3048452] 847, [7620918993254060080695155310050599797148999430633973883803199019707750756/13388252546937548169726194579548430808323898171176028174577824280989025, 681812302229940035388364461664173071257693604552199051324839442019796508729068038322150998473898835240012259104/1549122793089627103035798533583106884723430566520983193603888286288988661187568158029300129719791410507375] 849, [-918598359250808616/1742492401225, 533571921363304600297161948/2300150956851042875] 850, [2043952996/25, 92814721659044/125] 851, [154223958672/146689, 78708067800201244/56181887] 853, [-62113951201173744142404223711708854168008/353929837831162281434994570317004409, 27457475576301009527854990892389152176422566358451599892289500/210559948472029219428049680749869867602559512683060173] 857, [-719080, 89145644] 859, [2447473907657322064/37746241577209, 13472072978283051887157781612/231905245763331922573] 866, [30911108533891301687058856266325390568861818148/475145961620013847345874660103010729854841, 19235666462136753466754580711131035316189625709958107125217105998509348/327522065110106348181495274572885856033713884053734322015017139] 867, [108594600/121, 1534026839436/1331] 868, [40639673195616613180122918580/619710228504912052420969, 28952383469004939227093140784347328009242540/487846278726320925766278870858687797] 870, [413605860869193624477858424360522785471336346596/212644293311725653819798629097320849266574281, 5215181848890873907393468324549883242980781854108537239730438770877678828/3100848857278274418650762601505643454138482615089109804651515581179] 873, [-1256, 1076012] 874, [-182339/25, 792591709/125] 875, [6491734324802538279/2165544663080569, 264023660519637030260790514624/100774515266400130689997] 877, [-12979891837527953316690727323690805705052558767880/39051183633691795860198285574779834786081544441, 1577490788171686986674746852021228407909465754780477298261187248940446238236/7717049796849479808886597483676123304494593547992505782382688389489011] 880, [38582943961203769/4824274849, 7937109179185771535142101/335079658186993] 883, [39843416481617589478518374402994294577711/231217819762567465650405993866190289, 18693356510685011880939406705081083012697992485399369002110196/111181352443122298711367956462725042300809349825349913] 885, [199936, 198245348] 888, [3230721/6241, 201241675593/493039] 889, [-151396684332879029476201735043192/346928040073165713751838561481, 66367655697376371683262749265375140933125738370452/204342688936196956412998268210736634010005371] 891, [169814856/1225, 5739429695796/42875] 892, [99215860/363609, 25776330499052/219256227] 895, [34894276/361, 628157339488/6859] 899, [3866641250519982456951911771360859384/463538196384713654416057433449, 7963097038392592708273835493966363626432486490462135292/315593674444774951379864381085763378878043957] 900, [25044148/729, 621329086988/19683] 901, [0,0] 902, [-180740316715524/3962073025, 9969814054235289498076/249392686558625] 903, [1923839448/121, 86519402869500/1331] 905, [-3864, -3481892] 907, [0,0] 909, [41376/25, 186231924/125] 910, [-11123678659671176195002674493025879981934545283787508/888305323118817597073808789479639249503101862481, 9466331573413968043565108747446515064353218591075896672999495136724006615707356/837227330247070198370668262234596750098639677206056782259842012613148121] 911, [34596, 32166368] 917, [10628320361297266168109733777700333959984624131950267293275502801902370422889659682125415238019627330784497427861831296592594276273639173124542318813884904165498990720112354031949310113997459088995257309/3525716992428468524284232971923510678043294534325285672738193418949743678269795858782604524988237425772536323532680465823831178266887176036594187215094927597556010079149490255311395810539008100, 1095866166670325577668259997726054814107332536160131355830865160418614446861843185527501766999520027692366874623040365443844041561502027401083536305642436349184084717117281802961323656786030336231100956085522018725059932865870891323003944240341152953195533474146985395264274563528497621831317304736149523/6620200963632215485071126897921771963688601934274432924721208747730110845324552398198610684104737523269964023109232654658601878452960066156114653441958031980031154337954198036570865071353668269174723047152305562291545574852518177993549795503618517876831409622955500693667580084239734271000] 918, [-3524/9, 7620364/27] 922, [404388436241040964/497003850225, 367723054847259360660799412/350380259350871625] 923, [46633787906884130219565678163197535284696/13145558418358315364036200598325601, 11214710469370481960246204836080931545784840109521558363039252/1507191927049693457476780175325512428613212538967151] 924, [451188, 515414628] 925, [-778916334864251675991961592360726264118969899026366523587134341729037505104814733/6251088986431737720824675246762396662783512260374411992060240292032617689681, 52654137155883124238740364729212112329112901254676079415417227422382017824089717339662794324015066422080322060980660495074/494235027931272130910523880125737855592343176395293104897386795793855321291485091392394465659577153737540662650329] 929, [1020933/1156, 31009443983/39304] 932, [351084459251311077756/13065739328866225, 37974501642775537488481149606796/1493485409554841771419625] 933, [-715159521/3025, 31320947855592/166375] 934, [2952408244/729, 176857774526132/19683] 935, [232323504760668175069569956/1955679051595455512041, 10238111396256334971549325043131913103968/86486112389515475071906402115339] 938, [41001889689615341088359876/8093619814481006521, 284433192315953318666335512428742290372/23025772017437854545289633069] 940, [933690030487396/574623125521, 658286196528106749793252/435586739446813319] 944, [858251612/12769, 94940637876476/1442897] 946, [-32677269329584760128600827675084682123451/520126296327619047694272239829236209, 21497210456087984960813014502771446162573518885196707859852561/375113951308356953637783982812635313931431745373632073] 949, [-1342338648352922067023532821633979915350465388077/11017536772157448896415793238762083342141441, 3931936879104525525575251458054421795223670696772536687097406154626280542/36570151796962076376930050475593835984928343316846679291275455489] 951, [23391040, 115295385268] 953, [-12196831129319881027576/430186141976721361, 7503514816990050779147080824930268/282152968543545069958211609] 955, [49850428384/218089, 24862960518554116/101847563] 956, [13667153135903616004269569513079310843848748600408386414480377242769/4762321802197426697050334080920209622514366730060959901250649, 58016397521408986648955506781885188873213460017831320276876785450414809189683078661208638885433541881/10392693221155528576972285005985068113439310023703170832556256957385130044239878442879563357] 957, [-2336560918450545695979926559389183111929337792048381064164356694503178692057196234860596120/660979928327942814828310916569714645477748564086405858753967093683133445186989987801881, 57237018780507344753237963684127255371326654623072622946840577512382487177240363724015437182672721159471544121873831804780293140079202156/16993483238989187160112666431408616458845432701196785363598108008230122113765678881329211814989931174783934869170561510033592460029] 958, [-23722914254219934387322181753300/28272967111625617372330089, 35386105938097456908947869721754453246208470740/150333954625381888268417428725421696437] 959, [356, 245216] 961, [-138454268977164359438342956679778590209216686887268505304/445792993466590523336316051712594223651124571650625, 2288665227311020340758201507731920779965648412885710482825727205745337604825988857244/9412388782445798463318853630549888411744366836231575987020321440315473609375] 965, [-493, 410774] 969, [692631421286634497664/17074050697979521, 89571639583490059319898056554620/2231027122202936333946431] 970, [-107242174299255346532320236679025474856485936688277320092161416403723083583284884912703590146073230692350429198710468171185910615760276/333472415012131487580464086226905004860418840208638711082403132121166513662363728617506630202774884321755467982553877881292921260529, 37974767387535377968601176183945271444165789457688408373073761054617986924128952201870965597121514709075651725749806332462025299732950473534853975305462766292885453359820349717614018168217681533342390196/192570550560152253072875624853571441240677309076903862415427531187539289773600239475282016248584188792888480198818264630022885712743846992592173747072780087030784521086461093519307817212801233482167] 971, [6773568, 18815905108] 972, [1258849/25, 6278847407/125] 974, [-62868808470777948741965418883/1267588494189196151729689, 67114721662041932571439495406556894302301539/1427142806215319901598017927934240387] 975, [97467197604/494209, 73409928892880544/347428927] 981, [1787705027435989/1485177444, 101395844946042391363049/57235768336872] 982, [-1432213665524/230888025, 21284526970673282924/3508343539875] 983, [900216, 1229875892] 985, [7797999523008607352379280612350078522791776/217205284254771546046729130157051535081, 115274707111532659910489476358297433911260178923922631050524841612/3201146721671449177801628797795640405573854850471138691979] 986, [-27353557300/57121, 4591752589468268/13651919] 987, [32333764383573400605946528850483611042616653470910899402542888369211144514752872945576940607320/6454018395951006486762247409826643114628220079242480522236413373181857589731689602612289, 6354310229864178256826682558198125582306639791589258757302823857336179579345327305849746569070628273870527022945869944701244142249915427940908/518495866397972148593893398080345794104913209407302054730416490978040900610618634669024208128889255942910027579317376256068871389087] 990, [-162698155476/419635225, 2509162363250993268/8596227584125] 992, [537054156/25, 12727795497004/125] 994, [3580545604/729, 234814567605148/19683] 995, [146104/529, -1359004116/12167] 997, [10388194380153371390547170861543145854905/36286418748393082348806437004421029904, -29362022284626714330133325476301341214514935262433929600002293/218582889188182370011024196342851718226213177641555017792] 998, [264937380/3721, 16695297214100/226981] 92 Curves of rank 2. Two independent generators are shown; no claim as to minimality in or completeness of the full Mordell-Weil group is made. The cases for which there is a point on the curve having X<0 are precisely these: [ 41, 69, 77, 94, 105, 106, 126, 129, 149, 154, 161, 174, 186, 195, 261, 262, 274, 323, 329, 366, 370, 381, 405, 406, 409, 418, 446, 478, 493, 497, 517, 526, 550, 561, 563, 570, 581, 622, 653, 659, 681, 701, 714, 718, 726, 766, 777, 782, 809, 810, 825, 845, 858, 893, 897, 941, 966, 978]: 41, [[-1000, 26068], [4320, 334628] 69, [[-72, 1908], [-344, -22436] 76, [[1156, 96068], [4684, 478996] 77, [[-4600, 167684], [-765, 54746] 94, [[-116, 8060], [-467, 42133] 105, [[-504, -51012], [-648, 65484] 106, [[-10724, 242668], [13393/16, 5857543/64] 126, [[-596, -72836], [-8052, 712188] 129, [[-14376, 684180], [-152, -15652] 136, [[121, 10507], [1444, 203476] 149, [[-16072, 1258228], [-696, -101060] 151, [[100, -2752], [1024, 157324] 154, [[-13451, 1362775], [-1652, 244948] 160, [[40084/9, 20841436/27], [316, -48188] 161, [[-4041, 597556], [-23080, 1230188] 174, [[-6852, 1048572], [-804, 136836] 186, [[2916, 564372], [-18900, 2367828] 195, [[-13068, 2064384], [-27404, 2824192] 196, [[12340, 2779972], [5168884/9, 12138216724/27] 215, [26212, 7054656], [402692, 269808736] 220, [577, 124961], [26873212/729, 211866148900/19683] 248, [380, 89180], [1556, 389452] 261, [-21512, 4644172], [-1176, -302436] 262, [3844/25, 2717428/125], [-26726045540/35724529, 40996208785227332/213525509833] 274, [-1095428/49, 1760612204/343], [-66596, 6132620] 304, [710465318068/102434641, 2266029449327708428/1036741001561], [104110611533023586224444/453767562621333650881, 535911201274554677251076239176639620/9666075231663385231284773634721] 311, [7408, 2390052], [265316/1849, 646292192/79507] 316, [11569, 3861481], [95351524348/5067001, 73938141647321188/11405819251] 323, [-9484, 2920448], [-91404, 10391552] 329, [-1176, 381860], [-102376, 7840300] 366, [-1620, -586692], [-3278916/25, 867423348/125] 370, [-13124, 4616948], [-2340772692347/48260809, 4834753694625672223/335267840123] 376, [74402332/82369, 7934751360796/23639903], [574081540/49729, 50057746067924/11089567] 381, [2256, 864108], [-138360, 11410308] 383, [1944676, 2812303392], [7844, 3082912] 405, [-1784, -715652], [-12273, 4780584] 406, [-332660/49, 925486276/343], [-1073396/625, 10790282308/15625] 409, [-46856, 16260020], [7111974256/1042441, 3028982908769636/1064332261] 418, [-164, 5068], [102694159489/327184, 41059818068110561/187149248 446, [-1620, 715652], [7284, 3306716] 478, [-49412, 20909500], [-88462979/444889, 13412903893309/296740963 493, [-17624, 8367268], [-17284616/81, 26790334228/729] 497, [-635208456/24025, 46239271946972/3723875], [7873766516400/397723249, 81106023767562409652/7931794754807] 517, [-134264/25, 343371316/125], [6299298628514647736148578320/457548855376720283533449, 2258757459324887520774884425502547211446428/309496833357219089190840701388993957] 526, [-136715, 51146903], [486640660/729, 12767594421844/19683] 527, [7440, 3971836], [4351300/9, 11389156768/27] 550, [1277613972457/232324, 1483293854899480445/111980168], [-30973203662866724/4241997471321, 34648575395192381425774868/8736864653904916131] 561, [-304680, 30530412], [-3701, 2061314] 563, [-4044, 2259712], [113217640/4489, 4437212464020/300763] 570, [-579, 311283], [-228, 70956] 581, [-2520, -1454372], [-101240/441, -670113548/9261] 587, [39600, 24544324], [15148935/361, 178946903996/6859] 616, [9637684/9, 34819643204/27], [65063358916/9, 16596484997447132/27] 622, [-153620/9, 280241108/27], [-75682955/1521, 1713779506999/59319 653, [2344396448/121, 114755558456500/1331], [-292904/1369, 2088072444/50653] 656, [48444, 33520004], [8502108000668/54922921, 48198299503911738412/407033767531] 659, [-1356, 881920], [38928, 26778268] 664, [12003748, 42345582188], [152476/9, 309489740/27] 681, [-2936, -1988036], [-252504, 116057628] 690, [4868280756/841, 353367465992892/24389], [3190715794752369683944462342948/87012819416626357029568321, 21312462694837028626085303569172916720815773300/811661341562793618976400038444110777119] 699, [2248, 1567972], [326186856/966289, 174733539617700/949862087] 701, [-69080/289, 381472356/4913], [-478840, 53666428] 711, [2244, -1591776], [11236/25, 35173216/125] 714, [-98436, 63134100], [-19829028/529, 313427260404/12167] 718, [-382004, 139585732], [17419346437156369/20301120324, 2908825326302899425385951/2892544226004168] 724, [9361, 6835801], [192916/625, 2503204964/15625] 726, [-73251, 49346451], [20280681/16, 108674736597/64] 735, [3064/9, 5209156/27], [14199188832/2809, 1780098161334660/148877 766, [-50088356/4489, 2545574995796/300763], [1592598100/729, 71584462526444/19683] 777, [-589800, 69608916], [749587891039046485/8578724670916, 1825177023549282188699781769/25126621309980734536] 779, [4256, 3322444], [485222584/625, 14266683804948/15625] 782, [-88980, 64320956], [-442642175940/620059801, 8180488935674577676/15440109104701] 796, [6545044, 17536150436], [967956182470823821363497328512514536167305/33604697658515631661360018391809733449, 4566769263989773623105472344104568897492458427696564397537782299/194804950220036861979607140181975548349837235185845012293] 799, [156004, 139045280], [9968176324/126025, 2997454208246432/44738875] 809, [-229, -20894], [432460016450227364416/410459330241, 8996089915935394089772910364604/262969389564831711] 810, [1012, 799316], [-110056308/529, 1694367973620/12167] 816, [801, 626823], [435549452433597655415075334776958900/1889402530824901241429639521849801, 2026709785494339041046066127782456897007840418031760932/82127147551112446370319456197936219723639394210299] 825, [-215144, 146784428], [-553464, 197363268] 827, [379440, 391277636], [430129742395587448625340710091978736443520/3899325990272117289308155399176222921, 756952356783987183778441722094679245620050374183584785985696356/7699886379096326770138852475597327266087472095155876219] 845, [-698568, 86850316], [5566627123928896848/1347115386409, 14223251006867525417874366812/1563533514581765077] 858, [-1524, 1290708], [1825627652164/18774889, 7221490004909503196/81351594037] 876, [1727041/36, 9356651137/216], [1165658762436/3721, 1260051872397361812/226981] 886, [692260, 841389508], [2421124/9, 7456936276/27] 893, [-36776, 32074092], [1920, 1703260] 897, [-28440, 25054884], [-328599137345354451261/791628024733225, 5770184075195232087228485836962/22273155066671174279125] 908, [119396, 115995908], [363500/961, 7886340140/29791] 919, [163723, 164399558], [6244/25, -6856928/125] 941, [16336, 15511572], [-21117, 19631402] 966, [-4116, 3960036], [-47231841540/85951441, 377112148972863228/796855809511] 968, [3252564/5329, 209926159444/389017], [21842289/16, 132564290167/64] 978, [-203787, 176801625], [-101412, 93775500] 991, [94123/81, 820011842/729], [147460132/10609, 15155345624224/1092727] One curve of rank 3; it has no points with X<0: 484, [79444, 44495620], [36988, 19263532], [27364/25, 65511212/125] This concludes the list of 451+456+92+1 = 1000 values of n in this range. (A few of the most "impressive" rank-1 cases; sign shown is "+" iff a solution exists with x,y,z>0; generator is (a/d^2, b/d^3) with d shown 917 - 1877689269402280129109691277052028781356882910937076200519251307689212\ 361171313591466221066449910 769 ? ? 697 ? ? 907 ? ? 598 + 1512780637140911068803584543618074007712244927224574087650592939460059 970 + 577470704895176035440166633804275667368101896621009118612104810023 814 - 402921301995594752780791011792786336884927238303136354099152555472 748 ? ? 802 ? ? 442 + 100935097121075635543932032635351093665366077702297224168107 609 ? ? 661 - 19308578885398229814893009243976148264648399715417 987 - 80336905566190477254448568041091462432981983 700 - 46070434484783010046785936421506813148228893 957 + 25709529912620783172488633265183553490655109 901 ? ? 645 + 162726049371756505272809063893733344631527 (Others are <10^40. Positions of missing seven are estimated.)