Curves defined by x^3+y^3+z^3=nxyz for which I lack points as of 2003-09-27 Special thanks to Allan MacLeod, who single-handedly removed at least thirty curves from this list, using Silverman's method for rank-1 curves; to Tom Womack, who shows how to deal with the toughest cases using 4-descent; and to Denis Simon, whose ellQ smoothly provides 2-descent and in cases of low height gives fast searches and rank check. Following MacLeod's lead, we checked for the possibility that the 3-isogenous curve has a generator of smaller height (namely, 1/3 as large). In none of the remaining cases does that appear to be true. (That is, the isogenous curve has a generator whose height is LARGER by a factor of 3.) (These heights are as reported by MacLeod's program and APECS; heights reported by MWRank and PARI are double those. In any event, these are clearly generators of great height!) ============================================================================== RANK 1 ============================================================================== Seven curves of rank 1 remain (of the original 366). For all of these, we have been able to find covering spaces Y^2 = quartic in x, on which points (should) exist of half these heights. These are shown below (and of course these represent equivalence classes under fractional-linear transformations of x, so many variations are possible). On each of those, we have completed a 4-descent to an intersection of conics: v'Av = v'Bv = 0, further lowering the height; A and B are shown. We have conducted searches on these through 2^24 without success (that is, there are apparently no solutions v all four of whose coordinates are integers < 2^24; indeed the search algorithm used is efficient enough that it ought to have found any solutions whose coordinates are at most 2^36). Each search to that height take about a week of real time on my machines, and search time is proportional to the chosen bound, so I am reluctant to raise these upper bounds on the searches. (Moreoever, the search bound needed is proportional to height, so I expect searches to be fruitless unless begun with an upper bound of the form 2^k where we need k ~ 26 for n=901; k ~ 30 for n=609; k ~ 36 for n=802/748; k ~ 42 for n=907/697; and k ~ 46 for n=769. Unless we are very lucky, and find a solution early in the searches, these will take about 2 weeks, 1 year, 2 decades, 3 centuries, and 4 millenia, resp. I have resolved one or two of the curves only by searching at depth 2^24, and found the point with n=700 only by searching to a height of 2^26; this took over two weeks of contant CPU activity on a Sun workstation, finding the point [ 166646926928, 1248110850556, 410226312220, -19547336575 ] (whose coordinates are of magnitude 2^34.2 to 2^40.2 ) on the curve [SymmetricMatrix([ 0, 5, 0, 0, -2, 2, 13, 4, 4, -64 ]), SymmetricMatrix([ 1, 52, -6, 3, -10, 2, 102, 45, -2, 24 ])] This corresponds to the point (x,y)=(2828143130813538622953/1136992515626401307815, 5625528087517200411913724171516437717965864211/1136992515626401307815^2) on the homogeneous space y^2=1179100*x^2+2800926*x+2039383+155392*x^3+6356*x^4 and then finally to the point (526718819605581898643425825100306227196859268299433879268103549270603171\ 3967236993154491300987733/1265862650537797218276989830454271099405562851\ 04025133258033886094107510672218344992362610084 , -387380173726890978571471588991946154156917881994249276657329547548799353\ 874766027163513755541227784068601188668255884307132884168364804513170051/ 142422917910786971816240468619552731538206819618158939050179022466668632\ 2726494363507573542606654423762557343424739041115432503285366607448), of height approximately 107.72618 (or 215.4523587, according to convention) on the elliptic curve y^2+y = x^3+x^2-5002086483*x+136165942336906 itself, i.e. the curve [0, 1, 1, -5002086483, 136165942336906] (Remark: this isn't quite the quartic I thought I was pursuing; from the elliptic curve I passed to the homogeneous space y^2 = 150664384*x^4 + 920352*x^3 - 980240*x^2 - 2848*x + 1589 and then to the intersection of conics. Not quite sure what happened.) Shown are: value of n, expected height, conductor, and a0-a6 in minimal form. 901, 102,6 , 731432674, [1, 0, 1, -13729605380, 619204181234842] 1155354261148416*x^4 + 679447196288*x^3 - 459316592*x^2 + 14 [SymmetricMatrix([ 4, 1, -4, 17, -18, 7, 7, -10, -8, -10 ]), SymmetricMatrix([ 4, -39, 8, -3, 2, 7, 11, -17, -8, 0 ])] 609, 121.2 , 225866502, [1,-1, 0, -2865684327, 59046511507397] 2782720*x^4 + 180032*x^3 - 732492*x^2 - 21476*x + 49493 [SymmetricMatrix([ 0, 1, 3, 5, -5, -7, 1, 0, -7, -62 ]), SymmetricMatrix([ 8, 1, -13, -25, 31, 41, -17, 8, 64, -34 ])] 802, 141.3 , 515849581, [0, 1, 1, -8618990809, 307983862552785] 1168960*x^4 + 1032448*x^3 - 941456*x^2 - 513788*x + 295253 [SymmetricMatrix([ 6, 7, -12, 11, 6, 16, 12, -15, 6, -20 ]), SymmetricMatrix([ 3, 9, 23, 11, -16, 0, 4, -6, 18, -25 ])] 748, 145.6 , 418508965, [0, 1, 1, -6521768491, 202717213102626] 28353009*x^4 + 3916180*x^3 - 1530722*x^2 + 10148*x + 8185 [SymmetricMatrix([ 0, 15, 2, 3, 3, 9, 10, 7, 1, -16 ]), SymmetricMatrix([ 1, -4, 23, -11, -14, -15, 20, -7, -7, -25 ])] 907, 162.2 , 186535654, [1, 0, 1, -14098991107, 644360446364158] 1014997*x^4 + 87432*x^3 - 1643954*x^2 - 70504*x + 665597 [SymmetricMatrix([ 0, 1, 1, 1, -1, -1, 3, 0, -5, -20 ]), SymmetricMatrix([ 114, -18, -31, 27, 17, 31, -319, 74, 147, -164 ])] 697, 167.9 , 338608846, [1, 0, 1, -4916886147, 132703046307558] 519213*x^4 - 338512*x^3 - 811226*x^2 + 381200*x + 438317 [SymmetricMatrix([ 4, 9, 5, 2, 3, -6, 3, 2, 9, 7 ]), SymmetricMatrix([ 32, 4, 10, -8, 26, -24, 28, 17, -23, -24 ])] 769, 178.2 , 454756582, [1, 0, 1, -7285583301, 239355096410424] 25444672*x^4 - 704672*x^3 - 1177052*x^2 + 16564*x + 13673 [SymmetricMatrix([ 3, 1, -9, 4, 2, -12, 2, 30, 10, 17 ]), SymmetricMatrix([ 8, 2,-7, 7, -25, -41, -6, -12, 3, -12 ])] The most challenging rank-1 case will then probably be that last, for n=769. Good luck! The most dramatic example we have so far solves both the equations x^3+y^3+z^3 = n x y z and a/b + b/c + c/a = n where n=917. What is apparently the minimal solution has x = 2357912228167891363246796167158109772981336207612821236534306817531733 2723291772934017283089059756968677727558401148179736629123429381164223 6265544144881157772798127356466994113471028649073470780432192677724590 1228181403580811968107245828752698636076740015475273416486611265751710 221305737583492409749 y = -2438000973890426263310140365559854402543095022847272224432280317779489 6898670729714888702337615080129456490149390552714961140747971145199062 3050125720517682132709407486982372609208908675228645732084637119002770 3695494451412058453437629200745512098948185732770446432070343983979267 886448447612589586529 z = 2621107951825179693232506341803868762979310234554906682125633922703274 5123035963977570859358983369321367184755384021141076489483052092675301 3280356389620721946443996476203191288599869536096584257011198487600348 2321747990140265169300803198070918407093576490955173377314737514438793 05614407126728020 a = -1355467609925043908421203937674195888422765434664555604146502965278615 3223825829301908803158144415303498909504666128330833281621415457838334 4783459882218347808675247462015370065354808653069137541660684077913861 9081190161647357604081059386786767913008994673157322620226273208331460 2107843954993621280304127936106889714673958364886743856784476193614578 1078862619813112927904104449969088623474482206530177830928784279034045 8152048126019184191619856949281036873128383095454386500258942206784019 0521389279054236568905161464027371213175010029822680726622668874946417 4718497509479325946750681787523309541021813679847972288536325779476952 2913649484358030011627998434115860011535089557392865711456585915126182 7596554663224932270225469857767641706877382331355918913862710840607368 4748694494670553106956259992372111312993908653202489416491168243355855 46062567010467425816483739571180998645525541653421749506133529 b = 1557946921963925119702443338150852430091529741595741340393781749626365 7150417443530750633494232821410362628798826349710852850510526094367645 0672283667170961929838508889912230068629394378033406293347309991364284 4747252226088867710423143190562321163396629317113642225595209315282491 5487276482144248124298503823530131148169051455158766710002676542283375 6503144715629808036445774362551213152797056310134654665593558544394202 7228506834415930926190622373371470162450588796665871742575445255070807 9330893168291758508995505132022018798698389317610750876900269367370199 3190670942025244270745737452179334582998843150249358998884167432811899 6596056581688221351272964040860381824774834640812749055726185894700696 1657805006735919266220009098441191273130635744824229782601078492179135 7585893704471344691827748926400422580447354712091433915388672236863933 6122225575524744893595350299773637102717965260237519604820 c = 1619934484804961233469065690225356708731450548540504650055606018577542 8029841423972949278544360003485531313735098613580608002168618333227104 7717555398602414974580352291925221056789043622472167490213749801423434 2746170233961518252148357297850185997858922679523093591512594712750688 4370143915171659273419680907224110552169015797124202867152393502608352 3732170025794927597244508764654907969917993469348657547848704482641260 1871800049784710908341640439817640931845551636284461021155712889759262 7316549523557697351241972704829441161534488782280801107974590047245246 1087850717133067066242031169990922204794888156581953741339770664079076 8650413997101285415412814076430360484690822782241032732299814235764952 1759543527071292950349244123349142326300412243306141730239986877310930 7248389641377157201945850415851299725952713254321060664637119646848349 762856775803097830562617879275881686964964181830779600 Note that a,b,c are approximately 900 digits each. ============================================================================== RANK 2+ ============================================================================== ** Previous versions of this page listed some curves as being of rank 2 ** ** based on the supposed (by APECS) vanishing of L(1); a search of ** ** homogeneous spaces by ellQ shows the rank to be zero (so that L(1) ** ** must be small but nonzero). ** These values are n = 289, 295, 340, 373, 545, 571, 584, 614, 631, 647, 720, 761, 824, 854, 914, 920, 927, 979, 989 Fortunately Sha is zero in all these cases, so that ellQ's enumeration of a set of equivalence classes of quartics shows that the rank is zero. For comparison we have also computed the analytic rank, that is, we found L(1) to be nonzero, showing (conjecturally) that the rank is 0. The computed value of L(1) is small enough to confuse APECS; for example when n=989 we have L(1) = 0.1255212322003497601569345208, according to PARI/GP's ellanalyticrank. For all the other rank-2 (and the one rank-3) curves, I have a number of generators equal to the rank previously reported, so the reported ranks are correct.