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Generator points of E(Q)/2E(Q) on E : y^2=x^3+(n^2-6n-3)x^2+16nx (-500 <= n <= -1) (using Cremona's mwrank.exe) This curve is birationally equivalent to (x+y+z)(1/x+1/y+1/z)=n Ranks of some of the curve are not completely determined yet. ("x" in front of "n") n |rank| (xp, yp, zp), height -----+---+---------------------------------------------------------------------- -1 | 0 | -2 | 0 | -3 | 0 | -4 | 1 | (64, -640, 1), 1.1705 -5 | 0 | -6 | 0 | -7 | 0 | -8 | 0 | -9 | 0 | -10 | 1 | (-8, -104, 1), 1.58 -11 | 1 | (100, -1680, 1), 1.8679 -12 | 1 | (48, -496, 27), 2.9355 -13 | 0 | -14 | 0 | -15 | 0 | -16 | 0 | -17 | 0 | -18 | 1 | (-6, -171, 8), 1.0649 -19 | 1 | (484, -14960, 1), 2.9608 -20 | 1 | (144, -3696, 1), 2.0019 -21 | 0 | -22 | 1 | (896, 19648, 343), 5.4893 -23 | 0 | -24 | 0 | -25 | 1 | (2500, -143000, 1), 2.712 -26 | 0 | -27 | 0 | -28 | 2 | (-16, -496, 1), 2.1368 | | (12288, -460288, 27), 4.2539 -29 | 1 | (242, -7755, 8), 4.1382 -30 | 1 | (-756, -27180, 343), 5.4912 -31 | 1 | (196, -7168, 1), 1.9552 -32 | 1 | (5780, -204204, 125), 4.3495 -33 | 0 | -34 | 0 | -35 | 0 | -36 | 1 | (18928, 746096, 343), 6.8257 -37 | 0 | -38 | 0 | -39 | 1 | (64, -2720, 1), 2.6836 -40 | 1 | (32, 1376, 1), 2.3439 -41 | 0 | -42 | 1 | (-32856, -927368, 27), 7.6639 -43 | 1 | (1887876, 129335536, 729), 8.4583 -44 | 1 | (256, -12672, 1), 2.5563 -45 | 0 | -46 | 0 | -47 | 0 | -48 | 1 | (704, -23744, 1331), 3.831 -49 | 0 | -50 | 1 | (89100, -4758300, 1331), 5.2143 -51 | 1 | (7396, -750608, 1), 6.6564 -52 | 2 | (-1, -62, 1), 3.6863 | | (67280, -4008496, 125), 5.2029 -53 | 0 | -54 | 1 | (36, -2052, 1), 1.9724 -55 | 0 | -56 | 1 | (2268, -131076, 343), 6.7702 -57 | 0 | -58 | 1 | (-2052855000, -117940167800, 5000211), 15.0715 -59 | 2 | (1444, -104880, 1), 3.4165 | | (3364, -285360, 1), 4.1437 x -60 | 0-2 | -61 | 1 | (28938650, 2190541865, 17576), 14.1665 -62 | 0 | -63 | 0 | -64 | 0 | -65 | 0 | -66 | 0 | -67 | 1 | (16428, -1217744, 27), 4.8959 -68 | 0 | -69 | 1 | (3468, -252280, 27), 3.8419 -70 | 2 | (648, 50040, 1), 5.2289 | | (10, 665, 8), 4.4589 -71 | 0 | -72 | 1 | (-84, -8532, 343), 4.5198 x -73 | 0-2 | -74 | 0 | -75 | 0 | -76 | 2 | (18845008, -2293095056, 2197), 12.1983 | | (400, -32560, 1), 2.9726 -77 | 0 | -78 | 1 | (-12990877176, -1063298090184, 1647212741), 13.5095 -79 | 0 | -80 | 0 | -81 | 0 | -82 | 0 | -83 | 0 | -84 | 1 | (236992, -21520640, 343), 5.9326 -85 | 1 | (-60, -5480, 27), 3.6766 -86 | 1 | (96160712, 42542055789, 512), 13.331 -87 | 0 | x -88 | | -89 | 1 | (100, -9240, 1), 2.1175 -90 | 0 | -91 | 1 | (-7, -665, 1), 2.833 -92 | 1 | (66660617744, -10537038516432, 4173281), 14.8107 -93 | 0 | -94 | 0 | -95 | 1 | (484, -48576, 1), 2.8587 -96 | 1 | (35208152148, 3448011074516, 4716275733), 18.125 -97 | 0 | -98 | 0 | -99 | 0 | -100 | 1 | (10543167251152, -1079242332937648, 861985991413), 18.0515 -101 | 2 | (38332, -4002216, 343), 7.3004 | | (12, -1016, 27), 4.229 -102 | 1 | (-1507038, -157673121, 17576), 8.7197 -103 | 2 | (12996, -1372256, 729), 6.5157 | | (20, -608, 125), 3.9891 -104 | 1 | (7396, -1014972, 1), 5.6478 -105 | 1 | (-518940, -30892275, 64), 10.474 -106 | 0 | -107 | 1 | (345845, -42150747, 125), 10.4792 -108 | 2 | (-36, -3996, 1), 2.3914 | | (646912, -77072512, 343), 9.7474 -109 | 1 | (1682, -189805, 8), 5.1226 -110 | 0 | -111 | 0 | -112 | 1 | (-300352, -34362944, 2197), 7.3188 -113 | 1 | (66631303840, -11485214672793, 4096000), 19.208 -114 | 0 | -115 | 2 | (180, 21360, 1), 3.1112 | | (153228, -21444688, 27), 6.497 -116 | 1 | (400, -48240, 1), 3.3058 -117 | 0 | x -118 | 0-2 | -119 | 0 | -120 | 1 | (-18816, -2259712, 27), 6.9995 -121 | 0 | -122 | 2 | (-128, -15936, 1), 3.7059 | | (1469412, -185318196, 4913), 8.4918 -123 | 1 | (16820, -2126512, 125), 7.1159 -124 | 1 | (646416, -84286000, 729), 6.5227 -125 | 1 | (3364, -472584, 1), 4.3818 -126 | 0 | -127 | 1 | (42284, -5489728, 1331), 6.9686 -128 | 1 | (1862295186089404, 262639143582600444, 679383257671), 24.3311 -129 | 1 | (2116, -295688, 1), 6.2389 -130 | 2 | (-682328, -90517000, 6859), 10.2223 | | (79380, 24730020, 1), 9.1193 -131 | 1 | (1764, -247632, 1), 4.0449 -132 | 1 | (93104, -12578608, 1331), 5.3424 -133 | 1 | (-473458614804, -70575730626744, 831379493827), 16.8258 -134 | 1 | (104188, -14381604, 343), 7.0542 -135 | 1 | (252, -31968, 343), 3.4768 x -136 | 0-2 | -137 | 1 | (29877156, 7353162008, 729), 11.1328 -138 | 1 | (-5595450, -747174345, 2744), 9.4977 -139 | 2 | (98283, -15159655, 27), 8.3669 | | (43732, -6193936, 2197), 6.8294 x -140 | 0-2 | -141 | 1 | (3950628, 608081160, 1331), 9.9179 -142 | 0 | -143 | 1 | (7220, -1054272, 125), 4.6723 -144 | 0 | -145 | 0 | -146 | 1 | (2692868, 406014924, 4913), 9.0976 -147 | 1 | (3, -109, 27), 6.0869 -148 | 2 | (-714420, -93351636, 125), 10.713 | | (1684108617040, 301855601429936, 180362125), 16.6004 -149 | 1 | (1596587436, 280497978360, 205379), 14.1524 -150 | 0 | -151 | 1 | (9409, -1712147, 1), 7.3904 -152 | 1 | (30260236, 5101718292, 6859), 9.872 -153 | 1 | (612, -41832, 4913), 5.3528 -154 | 1 | (56, 8792, 1), 3.1582 -155 | 1 | (14558408, 3363789507, 512), 12.0638 -156 | 1 | (73721480348594190000, -11853428056711568477200, 125464824978676659), 31.4178 -157 | 0 | -158 | 0 | -159 | 0 | -160 | 0 | -161 | 0 | -162 | 0 | -163 | 1 | (547118270465858, 90730317458383933, 16190436763592), 25.928 -164 | 2 | (23104, -5216640, 1), 4.6816 | | (2088968, -373431135, 512), 9.4398 -165 | 1 | (-7535360, -595395520, 343), 8.7862 -166 | 1 | (9226822472, 742013097400, 76711450249), 14.7364 -167 | 1 | (144, -24528, 1), 3.737 -168 | 1 | (1332396, 224727516, 389017), 11.4 -169 | 1 | (114244, -43324840, 1), 4.3019 -170 | 1 | (-577368, -99148536, 1331), 8.1903 -171 | 1 | (637, 91931, 2197), 9.2688 -172 | 2 | (-505141, -90146602, 226981), 12.8684 | | (1489664, -265351552, 1331), 7.0172 -173 | 0 | -174 | 0 | -175 | 0 | -176 | 0 | -177 | 0 | x -178 | 0-2 | -179 | 0 | -180 | 2 | (-48, -8784, 1), 2.4303 | | (9851472, -1807327152, 50653), 10.68 -181 | 2 | (68644, -21975512, 1), 7.633 | | (96100, -34641880, 1), 6.658 -182 | 1 | (-59763816, -11091733352, 5000211), 12.8455 -183 | 1 | (3288781467972, 613448307261152, 14832537993), 19.3899 x -184 | 0-2 | -185 | 0 | -186 | 1 | (17956, -4159628, 1), 5.0882 -187 | 0 | -188 | 1 | (671744, -127755264, 68921), 8.2791 -189 | 0 | -190 | 2 | (47432, 8173880, 117649), 10.3218 | | (39690, 8153145, 8), 9.471 -191 | 1 | (676, -132288, 1), 3.4179 -192 | 1 | (755318954936765891431092060, -230224801955810996475855818468, 13759843824727599693375), 43.5834 x -193 | 0-2 | -194 | 0 | -195 | 0 | -196 | 1 | (2888693392, 460801009904, 13027640977), 17.626 -197 | 1 | (81559363396, 69838399738008, 117649), 16.145 -198 | 1 | (2213072169408306405000, 570164413294608578897400, 85161350807660681), 29.9312 -199 | 2 | (33708, -6911200, 27), 5.1578 | | (37004, -7465760, 1331), 6.3687 -200 | 0 | -201 | 1 | (583182762516898820, 36150970000491755224, 6847580693068258625), 28.0388 -202 | 0 | -203 | 1 | (99304336460, -152507671354704, 42875), 18.7255 -204 | 1 | (799071495376, 214618969621232, 27270901), 17.2749 -205 | 1 | (-878140, -187079240, 571787), 10.7838 -206 | 1 | (2018068, 426113364, 2197), 8.4224 -207 | 0 | -208 | 2 | (4762880, 1368961024, 125), 11.1037 | | (12544, -2996224, 1), 3.3434 -209 | 1 | (492804, -361375560, 1), 8.9003 -210 | 2 | (-24, -5320, 27), 4.9994 | | (-9444708, -2017115316, 753571), 12.4664 x -211 | 0-2 | x -212 | 0-2 | x -213 | 0-2 | -214 | 0 | -215 | 0 | -216 | 0 | -217 | 1 | (7266381077817368, -1587046662621027401, 1444736517166592), 28.3257 x -218 | 0-2 | -219 | 0 | -220 | 2 | (-20, -5356, 125), 4.3821 | | (1024, -230656, 1), 3.8215 -221 | 1 | (146068, -32719656, 2197), 8.7217 -222 | 2 | (-4848120, -527268024, 125), 11.5367 | | (56316, -12192452, 59319), 10.5089 -223 | 2 | (6724, -1616384, 1), 4.0752 | | (27380, -6199424, 125), 5.0514 -224 | 0 | -225 | 1 | (47315196432523342567500, -9163814909479781492143000, 190074096188717948450307), 33.4125 -226 | 0 | -227 | 0 | -228 | 2 | (-7884, -3808476, 389017), 10.7795 | | (240, -10864, 3375), 7.4425 -229 | 0 | -230 | 0 | -231 | 1 | (19808800284, 4647433035232, 63521199), 16.087 x -232 | 1-3 | (-39200, -4960480, 1), 8.1947 -233 | 0 | -234 | 1 | (120, 27432, 125), 4.1691 x -235 | 0-2 | -236 | 0 | -237 | 1 | (461216220140, -113877920252824, 136590875), 18.0165 -238 | 1 | (-56, -13496, 1), 3.4267 -239 | 1 | (20164, -5657280, 1), 4.8466 x -240 | 0-2 | -241 | 0 | -242 | 0 | -243 | 1 | (711957765660, -10342052331106672, 3375), 19.5322 -244 | 0 | -245 | 0 | -246 | 1 | (97468, -24320036, 343), 7.8301 -247 | 0 | x -248 | 0-2 | -249 | 0 | -250 | 0 | -251 | 1 | (1156, -296208, 1), 4.16 -252 | 1 | (60212939664, 16204698701488, 8120601), 14.1416 -253 | 1 | (13162170022293820, 2389900835439375544, 105854023288566125), 24.0466 -254 | 1 | (2215779934159116, 570197704005948660, 10851105871619), 24.4115 -255 | 0 | -256 | 0 | -257 | 0 | -258 | 0 | -259 | 1 | (525532, -139202960, 343), 6.9415 x -260 | 0-2 | -261 | 0 | -262 | 1 | (2267523400, 180539053640, 34550415593), 14.9114 -263 | 1 | (49284, -17074464, 1), 6.557 -264 | 1 | (616340609436, 162943910373572, 202365391311), 20.986 -265 | 0 | -266 | 1 | (-207576, -55984760, 19683), 9.5203 -267 | 1 | (27911161431296892, -7566286809360937456, 46063660590423), 23.4035 -268 | 2 | (-13328, -3650192, 4913), 8.5837 | | (20121304147330928, -28509837617160780592, 10403062487), 24.0473 -269 | 1 | (1420794788, 856213500744, 4913), 14.712 -270 | 2 | (72, 19656, 1), 2.9555 | | (-4024566, -1106009073, 941192), 12.6701 -271 | 1 | (1369, -378473, 1), 8.087 -272 | 0 | -273 | 1 | (7961898700, 283983337960, 136511322949), 15.1271 -274 | 0 | -275 | 0 | -276 | 1 | (1223117085621202180382656, -325260864659845453335484672, 1969842675630924992704591), 36.0729 -277 | 1 | (1992758, -557624389, 54872), 13.2954 -278 | 1 | (226286201448, 63587572511373, 3036027392), 20.9276 -279 | 1 | (196, -55328, 1), 4.1191 -280 | 1 | (2847520, 812265440, 2197), 8.548 x -281 | 0-2 | x -282 | | -283 | 1 | (169983713940, 50747005808848, 23149125), 15.4885 -284 | 2 | (898704, -890153040, 1), 8.6038 | | (192892, -55541940, 343), 6.2659 -285 | 0 | -286 | 0 | -287 | 1 | (131044, -60781248, 1), 7.0892 -288 | 1 | (224238003048, 65408055397803, 527514112), 17.293 -289 | 0 | -290 | 0 | x -291 | 0-2 | -292 | 1 | (272, -13936, 4913), 7.8153 -293 | 1 | (586762348, 180836474952, 79507), 12.672 x -294 | 0-2 | -295 | 1 | (140645060054750524, -41305917090637737824, 75703608616538791), 27.9037 -296 | 1 | (16904683652, -6395949503361, 314432), 15.6698 -297 | 0 | -298 | 0 | -299 | 0 | -300 | 0 | -301 | 0 | -302 | 0 | x -303 | 0-2 | -304 | 1 | (-64, -19648, 1), 3.5783 -305 | 0 | -306 | 0 | -307 | 0 | x -308 | | -309 | 1 | (145906913729252, 45653896901903641, 251000770112), 22.8043 -310 | 1 | (45017098820, 12149404554500, 227987153029), 15.8687 -311 | 1 | (2, -561, 8), 5.1498 -312 | 1 | (303314132804490247348, -79511150113147111030669, 1852884881832938635072), 33.2399 x -313 | 0-1 | -314 | 2 | (62500, -25231500, 1), 6.8411 | | (31684, -11518380, 1), 6.6476 -315 | 1 | (-32448, -8502624, 1), 7.5113 -316 | 2 | (-196, -62468, 1), 5.5773 | | (165888975155200, 54791284297387520, 22593183283), 19.3507 -317 | 1 | (1959380, -673247976, 125), 7.9723 -318 | 0 | -319 | 1 | (1444, -468160, 1), 3.8362 x -320 | 0-2 | -321 | 0 | -322 | 1 | (-83824200, -37655357480, 1564936281), 16.7512 -323 | 0 | -324 | 1 | (300304, -191635600, 1), 6.7141 -325 | 0 | -326 | 1 | (2554428174821968055446768136343287896, 902399088043295938151055986291293980603, 154160100685588614025425943320064), 55.5855 -327 | 2 | (1182558238400, -481448219437024, 20796875), 18.4663 | | (10894232580, -3747018863456, 1157625), 15.9434 -328 | 2 | (542944, 178883936, 103823), 9.3192 | | (956484, -316238419, 46656), 9.1768 -329 | 1 | (49284, -19682520, 1), 5.1661 x -330 | 0-2 | x -331 | 0-2 | -332 | 0 | -333 | 0 | -334 | 0 | -335 | 0 | -336 | 1 | (2544709374846253422677810104472, 866168547813136621789209767221769, 2663860337688680312405898752), 46.1738 x -337 | 0-2 | x -338 | 0-2 | -339 | 2 | (1859, -625105, 1331), 8.673 | | (114244, -54931760, 1), 6.4642 -340 | 2 | (80, 27440, 1), 3.6477 | | (35344, -13824016, 1), 6.3159 -341 | 0 | -342 | 1 | (-588, -202356, 1), 4.8946 -343 | 1 | (2548, -863968, 2197), 4.4968 -344 | 1 | (8145316, 4053614685, 64), 12.8059 -345 | 1 | (92080661167288260130860, 17859016807494222698451000, 1392479442125088453157861), 34.7195 -346 | 0 | x -347 | 0-2 | x -348 | 0-1 | -349 | 1 | (62669648092, 24632162282120, 2048383), 17.066 -350 | 0 | -351 | 1 | (7396, -2694208, 1), 4.5291 -352 | 1 | (-2368456000000, -18623872259584000, 25949321221297193), 21.0126 -353 | 1 | (3505087579820902965910248, -249090505004065964604920693, 75509412349333585583135232), 41.33 -354 | 1 | (384, 137280, 1), 2.8627 -355 | 1 | (49650919688535172, -17351916940284992653, 52604544192781888), 30.5437 -356 | 3 | (258064, -160525968, 1), 7.8094 | | (4415349712, -2052247266384, 50653), 14.6076 | | (84872, -30483159, 512), 7.1826 -357 | 1 | (2558312474976435012, 913595637245870112744, 924497180235202211), 26.8519 x -358 | 0-1 | -359 | 1 | (12100, -4577760, 1), 5.5747 -360 | 1 | (-13380704, -4858540448, 300763), 12.2653 -361 | 1 | (5936857857463772, -5761620357265333096, 7335308807), 23.4515 -362 | 1 | (1175674382716706060, -412829849250602131524, 2011995844908162875), 28.752 -363 | 1 | (174724, -97072976, 1), 4.4259 -364 | 2 | (-96721, -18843490, 1), 7.5858 | | (9922832, -3668682160, 4913), 8.1471 -365 | 1 | (940372707967644670215975558500, 349509850547830680581532897681864, 344467125588829445931640625), 44.1188 x -366 | 0-1 | -367 | 0 | -368 | 0 | -369 | 0 | -370 | 1 | (-7911025901205080, -3167045917043949640, 28263089785136543), 23.4291 -371 | 0 | x -372 | 0-2 | -373 | 1 | (2596607080, 753199750543, 24897088000), 18.5348 -374 | 1 | (-30118, -11202675, 8), 5.888 -375 | 0 | -376 | 2 | (23716, -9701692, 1), 7.2768 | | (4700, -486980, 103823), 8.964 -377 | 0 | -378 | 1 | (-72, -27432, 1), 3.1641 x -379 | 0-2 | x -380 | 0-2 | -381 | 1 | (100597222950, -38783602071235, 84027672), 20.3297 x -382 | | -383 | 0 | -384 | 0 | x -385 | 0-2 | -386 | 1 | (90589417170913412, -182451217690585205979, 23197894208), 27.6706 -387 | 1 | (3265195615667864158973571944005576557403482, -1195552379897872513757885427992417842517738231, 9503604759659338539609458935137196705484328), 70.6727 -388 | 2 | (-20, -8756, 125), 6.4299 | | (544444339487568, 218022838494658352, 72668325717), 20.086 -389 | 0 | -390 | 0 | -391 | 0 | x -392 | 0-2 | -393 | 0 | x -394 | | -395 | 1 | (1764, -705936, 1), 4.5315 -396 | 1 | (9695136274863233202330736, 2570390009429964384703361072, 136031224116102606712396711), 38.4962 -397 | 1 | (4628438, 1851358099, 54872), 13.9296 -398 | 2 | (465124, -367981284, 1), 9.0031 | | (112148, -43771668, 148877), 8.7536 -399 | 1 | (3319684, -6193925440, 1), 10.4448 -400 | 1 | (7519032499419212395503090514636, 3033377795784906392815657732044916, 20148541273935685072498733299), 45.5352 x -401 | 0-2 | -402 | 0 | -403 | 2 | (10812672, 4583039552, 729), 11.7388 | | (1602756, -655014896, 729), 7.6071 -404 | 0 | -405 | 1 | (2723220, 1181505528, 125), 8.2011 -406 | 0 | -407 | 1 | (7368732, 3208326624, 343), 11.4344 x -408 | | -409 | 0 | -410 | 1 | (-35912, -13177560, 1), 8.245 -411 | 0 | -412 | 1 | (127018546497286460042745079601968, 53195673238430056852936835204220784, 39886149729350473252250308867), 50.6922 -413 | 0 | -414 | 0 | -415 | 1 | (10159449828, -6871075923392, 35937), 16.3102 x -416 | | -417 | 1 | (16272506386, 12503433588437, 39304), 16.4664 -418 | 1 | (88, 37048, 1), 3.7759 -419 | 1 | (2523, -1064735, 27), 6.5666 -420 | 2 | (6077916, 2571331932, 50653), 12.1559 | | (1291584, 546461056, 9261), 10.9582 x -421 | 0-1 | -422 | 0 | -423 | 1 | (118847906496, -65522803279904, 970299), 16.9889 x -424 | | -425 | 0 | -426 | 1 | (7220, -3096772, 125), 7.0476 -427 | 2 | (-14, -6083, 8), 5.5866 | | (1843274608028, 808754675126896, 241804367), 16.9625 -428 | 1 | (900, -232716, 15625), 7.9215 -429 | 0 | -430 | 2 | (930250, 512700425, 8), 12.3687 | | (74880, 15035840, 1601613), 11.4326 -431 | 1 | (256, -111168, 1), 4.2003 -432 | 0 | -433 | 0 | -434 | 0 | x -435 | 0-2 | -436 | 2 | (615766564788242032, 273918117730228340336, 118972078011487), 24.7148 | | 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