Tables of fundamental integer solutions (x,y,z) of x3 + y3 = p z3
with p a prime congruent to 4 mod 9 and less than 5000
or congruent to 7 mod 9 and less than 10000

Which numbers n other than zero can be written as the sum of two rational cubes? Equivalently, for which nonzero n does the elliptic curve

E(n): x3 + y3 = n z3

have a rational point other than the point at infinity (1:-1:0)?

For n=1, the curve E(n) is the Fermat cubic curve, known [Euler?] to have no rational points except (1:-1:0) and the 3-torsion points (-1:0:1) and (0:1:-1). For general n, the curve E(n) is a ``twisted Fermat cubic''. It has no nontrivial rational torsion points except when n is a cube or twice a cube, when the torsion group has order 3 or 2 respectively (for n=2c3, take x=y=c and z=1). In these two cases, it is known that there are no rational non-torsion points. We may disregard the cases n=c3 and n=2c3, and then our question is equivalent to: for which n does E(n) have positive rank?

Since x3+y3 factors as (x+y)(x2-xy+y2), one might expect that the answer depends in part on the factorization of n; the same is suggested by a ``3-descent'' on E(n). So, the simplest case (past the known n=1) is n=p, a prime. Part of a conjecture of Selmer asserts that the elliptic curve E(p) has rank 1 for each prime p congruent to 4 or 7 mod 9. [See E.S.Selmer, ``The Diophantine equation ax3+by3=cz3'', Acta Math. 85, 203-362 (1951). The conjecture is often attributed to Sylvester, but I could not find it anywhere in his four-part paper ``On certain ternary cubic form equations'' in volumes 2 and 3 of the American Journal of Mathematics.] The full conjecture is that for prime p, the curve E(p) has rank 1 if and only if p is of the form 9k+4, 9k+7, or 9k+8. These are the cases in which the curve has odd analytic rank, and since the 3-descent yields an upper bound of 1, the arithmetic rank should be exactly 1. [For p=9k+2 and 9k+5, the rank is zero by 3-descent; for p=9k+1, the rank is at most 2, so presumably either 0 or 2, and it is hard to say anything more for general p=9k+1. This leaves the case p=3, for which E(p) is known to have rank zero.]

I have proved Selmer's conjecture in the 9k+4 and 9k+7 cases, using a novel variation of the Heegner-point construction. The proof also yields an algorithm to compute a nontrivial point on E(p) in time polynomial in p. While the standard Heegner-point method also does this, the power of p that my algorithm requires is smaller, conjecturally p5/3+o(1) rather than p8/3+o(1) [without fast-multiplication tricks that reduce both exponents by 1/3]. This makes it feasible to carry out the computation for a considerably larger range of values of p.

The following tables list the resulting points on E(p) for p=9k+4<5000 and p=9k+7<10000. [Added 11/2002: Experimental data for p=9k+8<10000, for which no analogous construction is known but one can still search for rational points and compute heights and L'(E(p),1).] Note that the point obtained on E(p) need not be a generator; in general it will be S*g for some positive integer S, where g generates the group of rational points on E(p). Analogy with other CM-point constructions suggests that S is proportional to the square root of the order of the Tate-Shafarevich group SHA of E(p). The computational data suggests a more precise conjecture: the order of SHA is S2 when p=9k+4 and S2/4 when p=9k+7. In support of this conjecture, it can be shown that neither S nor the order of SHA can be a multiple of 3. Moreover, when p=9k+4<5000 the only cases where S is even are the primes 877, 2281, 2659, 4099, 4297, 4657 for which SHA has nontrivial 2-torsion, as determined by using Cremona's MWRANK to perform a 2-descent on E(p). When p=9k+7, I do not quite understand why I always get a point divisible by 2; but granting that, the only p<104 where S/2 is even are 547, 1879, 1951, 2437, 2797, 3967, 5227, 5857, 6091, 6199, 6343, 6793, 7351, 9241, 9439, and 9907 -- again the same primes for which MWRANK finds nontrivial SHA[2]. [Added 11/2002: Numerical computation of L'(E(p),1) now confirms in each case the guessed proportionality between L'(E(p),1) and the canonical height of the point on E(p) produced by our CM-point construction.]

Caveat: I have checked only that the points g listed below are not divisible by 2 in the group of rational points of E(p). By construction, they are also not divisible by 3. It is conceivable, though unlikely, that one or more g's might be divisible by 5, 7, 11, or some larger integer coprime to 6.

It will be noted that the generators for p=9k+4 seem to grow faster than those for p=9k+7. In both cases, the BSD formula for the regulator has a factor of p1/3 coming from the real period; but the Tamagawa number c3 is 1 for p=9k+4 but 2 for p=9k+7, so ``on average'' we expect the p=9k+7 generators to have half as many digits as those for 9k+4 primes of the same size.

A natural variation of the construction yields nontorsion points on the curves E(p2) for the same primes p. Generators for the first handful of such curves are tabulated in this page. Those tables are shorter because the regulator of E(p2) can grow much more quickly than on E(p), roughly as p2/3 rather than p1/3.


p=9k+4

p S*g
13 (7, 2, 3)
31 (137, -65, 42)
67 (1208, 5353, 1323)
103 (592, -349, 117)
139 (16, -7, 3)
157 (19964887, -19767319, 1142148)
193 (135477799, -116157598, 16825599)
211 (66458, 74167, 14925)
229 (745, -673, 78)
283 (20824888493, -8780429621, 3090590958)
337 (53750671, -53706454, 1043511)
373 (1604, -1595, 57)
409 (22015523, 21425758, 3687411)
463 (403, -394, 21)
499 (80968219, 17501213, 10242414)
571 (44391008927477, -23911621134284, 5056073480199)
607 (29, -20, 3)
643 (18147157, 12227843, 2298150)
661 (2140498649652379, 3566251148564900, 437001508520199)
733 (1150783283, -1141783283, 36443100)
751 (1289, -1280, 39)
769 (4057041904, -1880802601, 427602903)
787 (7216943312, -2486921825, 770868189)
823 (142, 983, 105)
859 (488102289797, -268828281506, 48311408847)
877 2*(34, -25, 3)
967 (33205563567547, -33201628358236, 237872527101)
1021 (21622561, -21498145, 553896)
1039 (131597244744273801421, -97089659597438911546, 10949027164884016635)
1093 (101401155583, -50472101671, 9421432218)
1129 (38, -29, 3)
1201 (73359135128653103, -72823084618436210, 1926807554028903)
1237 (6481042, 19836821, 1869153)
1291 (2140292807674207, 3576426735553922, 350417260676301)
1327 (1672301454563700931825069, -1087300214320016905438165, 136720113253815524227998)
1381 (12751, -12742, 147)
1399 (3431924795, -3411225083, 80397108)
1453 (6624460432493830843558, -4381133795572218309259, 521950792185133082121)
1471 (589409033267, 419667548965, 57433014462)
1489 (43, -34, 3)
1543 (172543581807202622, -172535521743945551, 775574824638159)
1579 (3279944328204175, -3212757696133519, 110393325581784)
1597 (156989, 86011, 14130)
1669 (351523006005178106420447449932418114, -249018482378218647681291198015004093, 25598257699893539852091901829180247)
1723 (156062477, 112056523, 14458710)
1741 (1820923555858, 3747268486991, 322978509867)
1759 (43301, -12926, 3555)
1777 (1129628108783, -415063104455, 91693991886)
1831 (277023759242353, -134874535181353, 21737108814330)
1867 (1566087180708674, -1538450389485299, 47475544932135)
1993 (43196096, -27252047, 3117081)
2011 (1493461483, -1267077670, 86394273)
2029 (3155560607762723, 1479622268906314, 257544943343919)
2083 (1833826149263, -518805826796, 142499311773)
2137 (9317211958897, 39091015246952, 3048520437417)
2281 2*(79, -7, 6)
2371 (92353237129586, -12606507356573, 6920002683969)
2389 (1024, -781, 63)
2551 (46650528497766636598241057505394188431, -4693290971310701626269901995946387370, 3413020765298467283616684975869976081)
2659 2*(56, -47, 3)
2677 (101147989491184384667095712583241482722378852620814, -1090511126634181763163979336242624375879979922423, 7284631983398771119113860579850891711470579124501)
2713 (562637905575231047186396, 1400404470141455786116357, 102533717638623666312477)
2731 (35678804871750649, 21233772663558383, 2720566678345386)
2749 (14029, -9421, 888)
2767 (8510542021073, -8435030430161, 180465295152)
2803 (38548520016286279368153336071824071077852, 16559312643821353422873184285353123823027, 2804421281186398237777671906439363102713)
2857 (2031007246, 1601243525, 163483797)
3001 (2024791, -1204666, 129735)
3019 (13915978, -13915735, 36027)
3037 (248397382518133346062, -127229047069363115929, 16347150123479542563)
3109 (11638940186598044200, 4006441455083578757, 808153768661288553)
3163 (33999361, -33962497, 343056)
3181 (9438794346312826121, -2398926467326866185, 638261914547430936)
3217 (345051561802437173812763, -345038312257820805568124, 1137297684016656392139)
3253 (2280441850, -1443804757, 139597839)
3271 (30471800915, 10548239893, 2080758498)
3307 (8163776470688569206144398834817863593621651, -7246778190429810912969071213650506627114892, 367041189256673720749496234141171990775969)
3343 (98125682311185131921598955382137, -86324633899654659691531785966241, 4484687164635861880768197686574)
3361 (169653045245789, 1178828499179284, 78775650116157)
3433 (1458211345091568513540918059257541, 1669191267000540546647826950695948, 131190484154065933871637591442221)
3469 (63721748860840698052030252, -51457318328811028937463151, 3280675326525458108702637)
3541 (43363054968023, -41736345771647, 1356387311994)
3559 (24944620096288154, 46883796384934735, 3217781348691441)
3613 (25785093332066413781730859, 27994713787041364649341118, 2211606560337602178255063)
3631 (117775051, 140098493, 10647882)
3739 (458074363293544223, -443384232261844034, 13378902618716973)
3793 (21155832835379543, 15049271448773746, 1502950636248351)
3847 (1354711550083745079993728897, -1354134432512141746176799604, 9381177120806539885103463)
3919 (2490250540654609354663, -802589744699670045670, 156166769605404702417)
4027 (34794267887, -34772612951, 269244726)
4099 2*(769662245, -708900824, 28971999)
4153 (97, -25, 6)
4243 (1814993570306, -1526939710931, 82917058575)
4261 (7766153023432466459, -7651508257756283834, 168648696302309535)
4297 2*(70, -61, 3)
4423 (46722166988617379, -46647144864356195, 480449077149732)
4441 (3187420763861872945808133878038079517432429122123819821, -549618151017067240354662945083286248930737498409735812, 193583506256487091776899934893190281472805578036882117)
4513 (4313028093317203396238593, 1964398171404502619284223, 268965455279860239838272)
4549 (8588497880193099869, -1135692848817851861, 517936816845346638)
4567 (390077219639373783310327957, -255628424817944753281077229, 21058588407131964530795058)
4603 (1606021274010682557756259221728465081023024658415989, -864964801592399435950345437618983214690358282280168, 91231976195178405461900815262587300793670311639559)
4621 (15829, 45902, 2793)
4639 (3083577679043255720816882066, -2271259276807059829521594119, 155976120975659419340613327)
4657 2*(101, -29, 6)
4729 (171280900256860608383659817, -155245260106605047339884241, 6474224247740561423014722)
4783 (26814111785497349189027138009, -2024874917648513663250735776, 1591234801041889220644542231)
4801 (3246383, -3226610, 50583)
4909 (160905564112507, -66167090182843, 9242862584484)
4999 (192272780602499314151, 1768901270713419902866, 103497137970758304537)

p=9k+7

p (S/2)*g
7 (2, -1, 1)
43 (1, 7, 2)
61 (5, -4, 1)
79 (13, -4, 3)
97 (14, -5, 3)
151 (338, -95, 63)
223 (509, 67, 84)
241 (292, -283, 21)
277 (209, -145, 28)
313 (22, -13, 3)
331 (11, -10, 1)
349 (23, -14, 3)
367 (42349, 526, 5915)
421 (19690, 4699, 2639)
439 (571, -563, 26)
457 (31, 41, 6)
547 2*(14, -13, 1)
601 (49, 23, 6)
619 (1378346, -844475, 148239)
673 (14480, -14137, 679)
691 (19, -11, 2)
709 (31, -22, 3)
727 (36491, 60845, 7222)
853 (114782, 2867, 12103)
907 (1123814369, -1107620092, 40550587)
997 (624271, -381271, 57330)
1033 (68423, -65336, 3423)
1051 (35, 29, 4)
1069 (3065, 1543, 312)
1087 (6499, -1586, 629)
1123 (5809046126, -5804937329, 71799189)
1213 (7843, -7834, 111)
1231 (13294, -11963, 803)
1249 (396177623257, -322106067526, 28450704177)
1303 (35924, -5549, 3285)
1321 (56264, -56263, 193)
1429 (3626, 11999, 1075)
1447 (3853, -3728, 155)
1483 (4328008, -3756221, 266513)
1609 (9149, -8906, 333)
1627 (45274063, -5575567, 3846948)
1663 (1816250129434, -1746874262405, 73513104077)
1699 (47, 17, 4)
1753 (13961543168731, -9139495882099, 1037557129854)
1789 (38119538057820221, -24606633997841365, 2828707454055574)
1861 (12803, -6242, 999)
1879 2*(464, -463, 7)
1933 (195565, 118867, 16796)
1951 2*(26, -25, 1)
1987 (49, -40, 3)
2113 (77, -5, 6)
2131 (16326102641, -9699273698, 1172972619)
2203 (31, -23, 2)
2221 (985064, -954689, 33795)
2239 (11719, -6806, 833)
2293 (11117829730618, 40108190586001, 3063016704461)
2311 (2468, -2125, 133)
2347 (53, 11, 4)
2383 (10697, 11255, 1036)
2437 2*(29, -28, 1)
2473 (16304, 27913, 2193)
2617 (815048995, -806316304, 18738027)
2671 (4572520331, -2155626643, 317620534)
2689 (16451, -16387, 268)
2707 (20646518, -18798107, 927303)
2797 2*(566, -565, 7)
2833 (85, -13, 6)
2851 (1507, -1382, 65)
2887 (35134673279462, -31871147931925, 1561792275589)
3049 (1143433867, 889364894, 89670987)
3067 (565573, 6214, 38927)
3121 (1023220750505, -1019159791538, 15967000257)
3229 (2138657, 17634343, 1193790)
3301 (29794944328072394315, 8315461958928542518, 2015459161416995343)
3319 (184, 1147, 77)
3373 (625, -617, 14)
3391 (136921086355, 1079551853357, 71905586172)
3463 (800683, 140509, 53018)
3499 (1030670087, -996341962, 31153525)
3517 (10019474742712, -76938912185, 658851339419)
3571 (35, -34, 1)
3607 (15573037, -14155861, 638766)
3643 (217926556047482, 6354491339479, 14163177158949)
3697 (74150, -71953, 2119)
3733 (1869332, -1702957, 75295)
3769 (42307298578862707, -42273634287116704, 363231237887931)
3823 (1250784379, -892873379, 68799710)
3877 (1669, -544, 105)
3931 (332794465191903278069, 2223549920413153006688, 141046961959243083751)
3967 2*(67, 58, 5)
4003 (155489558, -154869269, 2237001)
4021 (2271872107454, -2271856941023, 3879822093)
4057 (2353280440963, 116063760965, 147555719766)
4093 (73483330, 46325147, 4949271)
4111 (81994, 373883, 23421)
4129 (1703273, -1703201, 5334)
4201 (63193157, -20684608, 3870061)
4219 (38, -37, 1)
4273 (1299536141559504007, -307520147528838940, 79728378515948631)
4327 (11966356652, 8780435797, 820535187)
4363 (44062, -42731, 1199)
4507 (2933081, -2932738, 12523)
4561 (24362820201160157, -325030396567532, 1469063354905965)
4597 (2206471667, -2132289995, 61078134)
4651 (16683035870344636660663, -3784387249372778365295, 995542809275796017138)
4723 (46, 79, 5)
4759 (21943899181, -21210028373, 599444846)
4813 (548778731017456760, 1381267707844835569, 83485537750646157)
4831 (694567932226, -596020823567, 29446185047)
4903 (697325350729596731410, -697304718218806813273, 1831006278161720121)
4957 (56425693, -11423794, 3300129)
4993 (130, 113, 9)
5011 (112, 131, 9)
5101 (21870341, 75465659, 4419220)
5119 (76, -67, 3)
5209 (633982831, 510462505, 42069074)
5227 2*(139, 104, 9)
5281 (226394054325554753696662, 635409218388951797725213, 37029870379705591216325)
5407 (706152581461322358389835167, 138051400463468126753184170, 40332709467453806599663369)
5443 (9250623932705702, -8621487531367999, 302574930299467)
5479 (26354, 24299, 1813)
5569 (258053597170637793108041094861649, 37800278803136075395982400837743, 14573766434991053317272216061524)
5623 (148, 95, 9)
5641 (5706886, -5086597, 212667)
5659 (7503706765094, 2600606554565, 426841839151)
5749 (81690962, -51919421, 4130727)
5821 (138687565, 161788667, 10584462)
5839 (2237, -1894, 91)
5857 2*(88, 37, 5)
6037 (868110908829204526153010917088983, -542620396003040640833775460845695, 43427878639692116012890579250426)
6073 (4953119, 4881424, 339591)
6091 2*(49, -41, 2)
6163 (9329157266974417, 19836052581294727, 1118196797224598)
6199 2*(2359142, -1064113, 124369)
6217 (4205803, -4205731, 8502)
6271 (2327, -383, 126)
6343 2*(91, 34, 5)
6361 (113, -41, 6)
6379 (11148656, -10528367, 324843)
6397 (35904957607, -33763257038, 1068275507)
6451 (1834798354675933668583, -1454573208238879041247, 78320773727693735154)
6469 (167292249042074, 145124508129475, 10615459631991)
6577 (84310075566662348369500409668, -84223101641624995334156299201, 655537864624260370749094287)
6703 (309164478284813476749566, 92920174027052198845577, 16544147666354782433407)
6793 2*(8311, -5224, 399)
6829 (66524166918925, 957089297760616, 50452422769049)
6883 (31, 94, 5)
6991 (76, 167, 9)
7027 (46786544384, -4606010743, 2441890171)
7207 (25692129837314626170099666571, -7494755731539239022959715344, 1318994341579555098758967821)
7243 (274126, 181751, 15429)
7297 (33790507549, -27113975162, 1367206757)
7333 (214651058723644044697333, -212315631669389876948333, 3518136904334151182510)
7351 2*(50, -49, 1)
7369 (2362814894291151163, -1120524953028251171, 116941002446164514)
7459 (91615874302, 143612025415, 7937990333)
7477 (28, 97, 5)
7549 (1839778838809, -1836799071290, 15876226541)
7603 (155610561192698038, 259750464000575819, 14095722339643353)
7621 (24620, -24611, 129)
7639 (308475337, -196495169, 14177066)
7873 (2817843793361, 341853980663, 141729988246)
7927 (6567683610791718654121, -4009689331620024673217, 302225355122867850674)
7963 (179147640766910618255, -81094977288034181527, 8684803540805875294)
8017 (94, -85, 3)
8053 (35297943787175, -17675258513008, 1683993011491)
8089 (29396153802078213477116683, -19800152984592221082465874, 1296775986440186825522523)
8161 (114750244809662, -44666194287713, 5585208375591)
8179 (36929067858211898516565556648, -15624107171948768081750102387, 1785394599416461839338237831)
8233 (784171841, -631041466, 30383185)
8269 (53, -52, 1)
8287 (94007805725365, 10652436791011, 4647741287342)
8377 (95099, -79547, 3492)
8431 (24320115389944552939328, -21144540714567411399175, 836281843322211629623)
8467 (79106564, -71225965, 2508793)
8521 (3389, 8590, 429)
8539 (330113, -29350, 16147)
8629 (306786088873436593, -304604605622561593, 4138383396850050)
8647 (60468378681554, -60403945708825, 433829673433)
8719 (2077417726345839529, -2069440744234517417, 22766209192275584)
8737 (18630381057401289020, -17810305263917145581, 453871704552206643)
8863 (20678148637, -10572433109, 952536122)
8971 (23846990456906032971862523, -16716562220375295704905492, 996979243946524215405049)
9007 (59236, -34847, 2639)
9043 (55483743595, -8234830147, 2660240154)
9133 (8149, 851, 390)
9151 (22526159, -4090712, 1074801)
9187 (610024512097859098157, -55143992767354164781, 29119399000933328416)
9241 2*(56, -55, 1)
9277 (8814795164389259123, 47297282527025941, 419510718543060414)
9349 (7875563536249920690529919840, -5500151677664775332886439033, 325389577275916294246344883)
9403 (714011, 1430941, 70494)
9421 (59520824716, -2142360923, 2818144049)
9439 2*(126179833, -78767617, 5440806)
9511 (13169, -12169, 370)
9547 (3004618298438067718, 3415574238976225157, 191428966614581535)
9601 (400199338712389496684, -398391205745103233801, 4482717899766530487)
9619 (283725893787705429705697, -213801748514324311291016, 11075100395904434357027)
9781 (80629355, -13705939, 3764026)
9817 (1552599880267527069381405447989, -1059438237422300008230689412917, 63834224357162374655929960032)
9871 (129437, -122578, 3211)
9907 2*(104, -95, 3)