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Generator points of E(Q)/2E(Q) on E : y^2=x^3+(n^2-6n-3)x^2+16nx (1 <= n <= 500) (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 ----+---+----------------------------------------------------------------------- s 1 | | 2 | 0 | 3 | 0 | 4 | 0 | 5 | 0 | 6 | 0 | 7 | 0 | 8 | 0 | s 9 | | 10 | 0 | 11 | 1 | (-4, -8, 1), 0.6736 12 | 0 | 13 | 0 | 14 | 1 | (-4, -28, 1), 1.5187 15 | 1 | (-12, -120, 1), 1.0338 16 | 0 | 17 | 0 | 18 | 1 | (-36, -468, 1), 1.6879 19 | 0 | 20 | 0 | 21 | 0 | 22 | 0 | 23 | 0 | 24 | 0 | 25 | 0 | 26 | 1 | (-1, -10, 1), 1.7217 27 | 0 | 28 | 0 | 29 | 1 | (12, 496, 27), 3.0678 30 | 1 | (-216, -4824, 1), 3.3908 31 | 1 | (484, -17160, 1), 2.7547 32 | 0 | 33 | 0 | 34 | 1 | (-24, -440, 27), 1.3026 35 | 1 | (-16, -496, 1), 2.8624 36 | 0 | 37 | 1 | (20, 1392, 125), 3.2385 38 | 1 | (-676, -15652, 1), 5.2514 39 | 0 | 40 | 0 | 41 | 0 | 42 | 1 | (-8664, -298376, 27), 6.5992 43 | 1 | (6724, -613032, 1), 4.9754 44 | 1 | (14896, 665392, 6859), 7.5273 45 | 0 | 46 | 0 | 47 | 0 | 48 | 1 | (9199680, -423343936, 1157625), 10.7298 49 | 0 | 50 | 0 | 51 | 0 | 52 | 1 | (8, 1887, 512), 4.9072 53 | 1 | (3468, -177616, 27), 5.5634 54 | 1 | (-1764, -50652, 1), 4.5107 55 | 1 | (-80, -4080, 1), 2.2263 56 | 0 | 57 | 1 | (588, 32032, 27), 3.9936 58 | 0 | 59 | 1 | (-36, -1992, 1), 3.0962 60 | 0 | 61 | 0 | 62 | 1 | (-637, -4382, 2197), 8.582 63 | 1 | (-3, -171, 1), 2.6459 64 | 1 | (16384, -2322432, 1), 3.3275 65 | 0 | 66 | 0 | 67 | 1 | (1764, -134904, 1), 4.2628 68 | 0 | 69 | 1 | (96100, -30456880, 1), 9.1698 70 | 1 | (-1620, -86580, 1), 5.3901 71 | 1 | (3364, -300440, 1), 4.06 72 | 0 | 73 | 1 | (1444, -114912, 1), 2.9769 74 | 2 | (-1, -62, 1), 3.8249 | | (1764, 131068, 729), 7.1024 75 | 1 | (2429900, 203275000, 1331), 9.1089 76 | 1 | (704, 61440, 1331), 4.4213 77 | 0 | 78 | 0 | 79 | 0 | 80 | 0 | 81 | 0 | 82 | 1 | (-3872, -187968, 1), 5.6003 x 83 | 0-2 | 84 | 1 | (108, 8820, 1), 3.0929 85 | 1 | (3, 413, 27), 4.5789 86 | 1 | (196, 16492, 1), 4.8221 x 87 | 0-2 | 88 | 0 | 89 | 0 | 90 | 1 | (-67032, -5766264, 6859), 7.7499 91 | 0 | 92 | 1 | (19600, 1868048, 15625), 6.7171 93 | 1 | (45783380, 28012359824, 125), 11.6324 94 | 1 | (1532644, -275094123, 64), 12.17 95 | 1 | (-560, -48560, 343), 5.8257 96 | 1 | (868332, -175389076, 27), 9.263 97 | 0 | 98 | 1 | (-7396, -297388, 1), 7.5524 99 | 0 | 100 | 1 | (4624, -547536, 1), 4.0396 101 | 1 | (44, 10640, 1331), 4.8649 102 | 1 | (-6048, -595008, 343), 6.6354 103 | 1 | (6074166502404, -769389821735592, 1003003001), 18.9549 104 | 0 | 105 | 1 | (-108, -10944, 1), 2.2909 106 | 1 | (1444, -158460, 1), 4.5111 107 | 0 | 108 | 1 | (115596, 12196548, 6859), 7.9038 109 | 0 | x 110 | 0-1 | 111 | 1 | (11939452, 2573525240, 343), 9.8727 112 | 1 | (18321419200, -2265352767168, 5359375), 12.8258 113 | 0 | 114 | 0 | 115 | 0 | 116 | 1 | (137904, 16630640, 132651), 8.8874 117 | 1 | (116487, -13448637, 343), 9.5178 118 | 0 | 119 | 0 | 120 | 0 | 121 | 0 | 122 | 1 | (-281048, -33325688, 6859), 7.6959 x 123 | 0-2 | 124 | 0 | 125 | 1 | (10095404, 1512865936, 1331), 11.5993 126 | 1 | (-9, -1098, 1), 3.7901 127 | 1 | (57854108, 8206198728, 12167), 10.929 128 | 1 | (23328058788082980, -2925683010645735313, 247909857768000), 24.8727 129 | 1 | (3844, -539648, 1), 3.6956 x 130 | 0-1 | 131 | 1 | (-20, -1144, 125), 3.4585 132 | 0 | 133 | 1 | (13689, 2393703, 1), 9.6501 134 | 0 | 135 | 1 | (-9769572, -1235228616, 6859), 7.357 136 | 0 | 137 | 0 | 138 | 1 | (-19872908413475696984, -2498517272017587681928, 21649218681701829763), 28.8979 139 | 0 | 140 | 1 | (5602068141120, 789310996575360, 2732843155141), 17.3893 141 | 2 | (2, 335, 8), 4.3589 | | (192721, -88683707, 1), 10.8847 142 | 1 | (7502174316, -1054210778244, 17373979), 16.7093 143 | 1 | (-19855, -2766533, 125), 8.3848 144 | 0 | 145 | 1 | (20164, -4048704, 1), 4.9026 x 146 | 0-2 | 147 | 0 | 148 | 1 | (26356432, 3868014864, 50653), 9.9142 149 | 0 | 150 | 0 | 151 | 1 | (1764, -271320, 1), 4.8053 152 | 0 | 153 | 1 | (146692, 64446304, 10218313), 11.7323 154 | 0 | 155 | 0 | 156 | 1 | (12, 2044, 27), 4.0328 157 | 0 | 158 | 1 | (-21316, -1106972, 1), 8.597 159 | 1 | (-189321276, -29465517720, 3869893), 12.7057 160 | 0 | 161 | 1 | (28, 6656, 343), 4.0911 162 | 0 | 163 | 1 | (175211156, 31718730072, 24389), 13.0136 164 | 1 | (110746624, 17907185408, 493039), 13.3115 165 | 0 | 166 | 1 | (342412252, -70303327809, 21952), 14.1416 167 | 0 | 168 | 1 | (91875330618548832, 20537733070279692576, 777817379070886393), 28.5535 169 | 0 | 170 | 1 | (-21160, -1734200, 1), 5.3225 171 | 0 | 172 | 1 | (258428, 43780332, 12167), 7.4634 173 | 1 | (80528667947457028917237204, 13791172056924668097106449008, 185161626178861975847541), 37.9895 x 174 | 0-2 | 175 | 1 | (144, 47589, 4096), 7.3121 176 | 1 | (23104, -5320000, 1), 3.687 177 | 0 | 178 | 0 | 179 | 1 | (-100, -17560, 1), 4.3348 180 | 0 | 181 | 0 | x 182 | 0-1 | 183 | 1 | (100, 69992, 15625), 6.8793 184 | 0 | 185 | 1 | (4732, 864032, 343), 6.2676 186 | 0 | x 187 | 0-2 | 188 | 1 | (21780, 4711652, 91125), 8.5504 189 | 1 | (282807, 53218971, 343), 9.8004 190 | 1 | (-45, -8400, 1), 3.7663 191 | 0 | 192 | 1 | (178638219136483681280, -43451937094815662620672, 7615722528193625), 27.612 193 | 0 | 194 | 2 | (-33124, -1915732, 1), 9.0341 | | (4732, -1067261, 21952), 10.8814 195 | 1 | (-1269875210628, -67273648127400, 13856857998859), 18.7587 196 | 1 | (784, 181545, 4096), 4.2916 197 | 1 | (19112683068, 3735029512592, 3395290527), 15.7199 198 | 0 | 199 | 2 | (36, 7557, 64), 5.5661 | | (17956, -4262808, 1), 5.6074 200 | 0 | 201 | 1 | (131044, -54068320, 1), 7.5307 202 | 2 | (-1205064088, -210644833848, 3368254499), 13.6335 | | (225228, -49309588, 27), 7.3284 203 | 1 | (-636667503012, -120838946458056, 776911912057), 17.731 204 | 0 | 205 | 1 | (2809, -586551, 1), 4.8736 x 206 | 0-2 | 207 | 0 | 208 | 1 | (709907264, -289410301248, 26490242141), 14.315 209 | 0 | 210 | 1 | (-96, -19648, 27), 5.3665 211 | 0 | 212 | 1 | (3627444702445119191259889273640902800339237120, 761630479111499413153971818671086611528289702784, 435554668944905655572522996110453834317428625), 68.0897 213 | 1 | (1041672849351849, 607504685625150247, 3518743761), 24.6233 214 | 1 | (44, 16932, 1331), 6.7937 215 | 1 | (-11038555366698998727340, -797007783769105264100120, 127452823438297670385983), 33.7292 216 | 0 | 217 | 1 | (6156, 1371744, 6859), 7.4917 218 | 1 | (673224916, -240901777228, 15794358229), 18.4074 219 | 1 | (40362, -9298729, 74088), 11.922 220 | 0 | 221 | 1 | (7396, -1733072, 1), 5.3411 222 | 0 | 223 | 1 | (45188777288207735220, 10888073501226568876648, 4674635798140125), 31.6773 224 | 0 | 225 | 0 | 226 | 1 | (503756, 113019285, 85184), 10.4427 227 | 2 | (-180, -39288, 125), 5.995 | | (42284, 9484264, 1331), 8.767 228 | 2 | (27, 6084, 1), 5.1991 | | (53333736204, 12252723751268, 24642171), 18.2805 229 | 1 | (612, -173616, 4913), 7.6051 230 | 1 | (-104191661538345067236, -23429137504597145998116, 109847870450823961), 28.8919 x 231 | 0-2 | 232 | 1 | (297680, 72192219, 512000), 10.9296 233 | 1 | (13692454395, 3152675046763, 170953875), 16.681 234 | 2 | (-49284, -3142188, 1), 8.3303 | | (504, 119160, 343), 4.9769 235 | 0 | 236 | 1 | (4414877632, 1048791800320, 2048383), 14.8682 237 | 1 | (12120636629933057546875502465900, 2906272377665836937086226376964976, 8785287507590209466041689546875), 47.1845 238 | 1 | (-778726031384, -166028136793512, 1995141492469), 16.9319 239 | 0 | 240 | 1 | (5706921484992, -3094469217770048, 353498221683243), 23.7134 241 | 1 | (6724, -1692480, 1), 3.9317 242 | | 243 | 1 | (12167172, -2981977416, 4913), 7.519 244 | 1 | (9814694144, 4394463020928, 68921), 14.0356 245 | 0 | x 246 | 0-2 | 247 | 1 | (3991859334, -1023976778931, 636056), 18.0656 248 | 0 | 249 | 0 | x 250 | 0-2 | 251 | 1 | (-27556, -5076280, 1), 5.7991 252 | 0 | 253 | 1 | (20164, -5797008, 1), 4.87 254 | 1 | (1444, -366548, 1), 5.7238 255 | 1 | (-28842820, -7199501480, 24389), 11.9658 256 | 0 | 257 | 1 | (339637364, -86585969024, 39651821), 13.2042 x 258 | 0-1 | 259 | 0 | 260 | 0 | 261 | 0 | 262 | 1 | (13924, -3962676, 1), 5.2748 263 | 0 | 264 | 0 | 265 | 1 | (1468820, 416438112, 125), 9.2251 266 | 1 | (-154453, -40579174, 2197), 11.4331 x 267 | 1-3 | (484, 128216, 1), 5.5434 268 | 2 | (98496, -26157312, 6859), 9.3933 | | (173598271920, 46362967434928, 44928178875), 19.3725 269 | 0 | 270 | 1 | (-100940, -15803060, 1092727), 11.3179 271 | 1 | (4, 1503, 64), 5.3463 272 | 0 | 273 | 1 | (46747344, 13050518032, 9261), 13.7278 274 | 0 | 275 | 1 | (40459835000, -11066803835125, 7715442176), 18.7884 276 | 1 | (300304, -183854000, 1), 6.5477 277 | 0 | 278 | 1 | (-70756, -4931108, 1), 9.7887 279 | 1 | (2173040717513089771898100, 608448560844886509432140280, 1087138217686188451902301), 35.5836 280 | 1 | (23277280, 6956526240, 65450827), 11.6401 281 | 1 | (44, 20320, 1331), 5.0399 282 | 0 | 283 | 1 | (6416606, 1823133243, 2744), 10.4507 284 | 0 | 285 | 0 | 286 | 0 | 287 | 0 | 288 | 1 | (15162010220, -12317383292116, 1904318529875), 21.6974 289 | 0 | x 290 | 0-1 | 291 | 1 | (12295294, -3544270235, 97336), 11.0043 x 292 | | 293 | 1 | (10784171170942013163027080508751107004628, -3134500833676918197506079044118311147729741, 877656879327732016799540684256996053312), 62.5 294 | 2 | (-168, -48664, 27), 4.3269 | | (700284, 204261292, 59319), 11.5073 295 | 1 | (53792, 15968967, 32768), 9.482 296 | 1 | (37926980, 19826947868, 1501123625), 16.9786 297 | 1 | (-756, -219456, 343), 4.1106 298 | 1 | (-200, -58920, 1), 4.2211 x 299 | 0-2 | 300 | 0 | 301 | 1 | (8168164, -23471153520, 1), 11.5329 302 | 0 | x 303 | 0-2 | 304 | 0 | 305 | 1 | (784, 237776, 1), 6.3816 306 | 0 | 307 | 1 | (801090873327906, -318580915864235751, 10722807305156744), 28.9754 308 | 0 | 309 | 1 | (19604, 6192368, 24389), 7.8794 310 | 2 | (-7605, -2328560, 729), 9.141 | | (36, -15884, 729), 7.3487 311 | 1 | (99636, 33234040, 328509), 10.1052 312 | 1 | (42344495648256, 14737080105981696, 217613199487949), 19.6969 313 | 1 | (1867184, -583011312, 1331), 9.1317 x 314 | | 315 | 1 | (-467285, -141744785, 493039), 11.7521 316 | 0 | x 317 | 0-2 | x 318 | 0-1 | 319 | 1 | (-396, -124872, 1), 4.3191 320 | 1 | (462617249048832, -166201954471722752, 2583590982797313), 23.0873 321 | 0 | 322 | 1 | (-2592, -816192, 1), 6.8631 323 | 2 | (-816, -260848, 27), 4.1221 | | (2555564, -825374872, 1331), 7.9542 324 | 1 | (849977388, 286632846396, 79507), 13.612 325 | 1 | (1358300, 438212400, 103823), 8.4473 326 | 2 | (-20, -5356, 125), 4.5646 | | (24964, -8975980, 1), 5.9349 327 | 0 | 328 | 1 | (1700444607111364697660, -553823256781040838944484, 146274466515389300375), 31.3415 329 | 1 | (31425380781, 10770928296733, 2803221), 17.138 330 | 1 | (-45, -6580, 729), 4.0382 331 | 1 | (158404, -81693480, 1), 6.7856 332 | 1 | (74958495842740531092, 24676731753591975754148, 898065128266689573), 28.8349 333 | 1 | (670330154421828, -240502903838666064, 33792250337), 19.9924 334 | 0 | x 335 | 0-1 | 336 | 1 | (22186486182976, 7399681996552640, 64432972729), 17.2584 337 | 1 | (114244, -54285504, 1), 6.2381 338 | 2 | (-364, -119132, 343), 5.4489 | | (923668, 334558759, 3241792), 14.5939 339 | 1 | (547143200, -196957728695, 32768), 14.8421 340 | 0 | 341 | 1 | (211387740, 88899915536, 3375), 14.2915 342 | 0 | 343 | 1 | (3364, -1160232, 1), 6.0544 344 | 0 | 345 | 1 | (426026534087, 601055226872947, 144595657865303), 23.5442 346 | 1 | (18873148, 7842260964, 343), 10.7557 x 347 | | 348 | 0 | 349 | 1 | (28, 12144, 343), 5.2989 350 | 0 | 351 | 1 | (612, 249480, 4913), 4.032 352 | 0 | 353 | 2 | (15004, 5486272, 29791), 8.3107 | | (149212, 52317056, 343), 5.7892 354 | 0 | 355 | 0 | 356 | 1 | (554254768, 195777885680, 4330747), 13.83 357 | 0 | 358 | 1 | (10118176513142101668, -1166673327300337388911884, 761048497), 33.669 359 | 0 | 360 | 0 | 361 | 0 | 362 | 1 | (-9897361664226781536, -3083067790302174769664, 54398020955487562683), 30.6395 363 | 1 | (863075757090373487364788883839678270860, -310875609622114997047139330729169617627384, 11614809567825442518478263730424447875), 61.5759 364 | 1 | (586218416, 231806240688, 2622362939), 14.2532 365 | 1 | (12904428, -10070199152, 27), 11.2826 366 | 1 | (207690013274484, 77636532266143397, 26039617344), 25.4767 367 | 1 | (1905261824519676694887453156243586, -694304789790430909941441019772265759, 95344544139727797672148814358664), 54.2865 368 | 0 | x 369 | 0-2 | 370 | 1 | (-88, -18696, 1331), 2.5918 x 371 | 0-2 | 372 | 0 | 373 | 1 | (12321, 4564727, 729), 10.2639 374 | 1 | (-2978230320025, -355437450589490, 24137569), 19.1207 375 | 1 | (1887876, 708806296, 729), 7.7315 376 | 0 | 377 | 1 | (531760037596641280699472, 201849365282170259270583952, 372117533040860135301053), 37.258 378 | 1 | (-133956, -10929492, 1), 8.7766 379 | 2 | (12100, -4740120, 1), 4.2386 | | (155236, -58865373, 64), 7.6661 380 | 0 | 381 | 2 | (10580, 4001264, 125), 7.5315 | | (2500, -953200, 1), 6.3452 382 | 0 | 383 | 1 | (2009743460196498237964, -7322587466405478100343512, 153053107249771), 36.3179 384 | 0 | 385 | 0 | 386 | 0 | 387 | 2 | (-507, -194337, 1), 6.2508 | | (2366, 930371, 2744), 10.1941 388 | 1 | (424388, 165899289, 314432), 10.0483 389 | 0 | x 390 | | 391 | 1 | (1141233530028, -523587721281976, 19034163), 18.7898 392 | 0 | x 393 | 0-2 | 394 | 0 | 395 | 1 | (-633124635895243213380, -248031707244558703679112, 12857389879308052625), 30.286 396 | 1 | (3679312, 1453749584, 2197), 11.4507 397 | 0 | 398 | 1 | (-182535500052, -71744914921716, 119823157), 15.3616 399 | 0 | 400 | 0 | 401 | 0 | x 402 | 0-1 | 403 | 0 | 404 | 0 | 405 | 0 | 406 | 1 | (-53816, -17733240, 1), 5.4305 407 | 0 | 408 | 1 | (136945763559100, 56233089559024741, 29791000000), 21.1929 409 | 0 | 410 | | 411 | 1 | (1297890246004080652, -634476575293136132920, 14313984267214055683), 28.5643 412 | 0 | 413 | 0 | 414 | 0 | x 415 | 1-3 | (20, 9192, 125), 5.0895 416 | 1 | (48807148, 27535092116, 1083206683), 16.9805 417 | 0 | x 418 | 0-1 | 419 | 0 | x 420 | 0-1 | 421 | 1 | (36, 20080, 729), 5.5229 422 | 0 | 423 | 0 | 424 | 1 | (42811441564, 54393929988228, 29791), 15.4582 425 | 1 | (118300, -49972000, 343), 6.9605 426 | 0 | 427 | 1 | (976615189728701730466253670, 438237732657025909010254569749, 45223301343951471663000), 45.1074 428 | 1 | (794254872560, 337831780156688, 2833148375), 17.0257 429 | 1 | (1055782106069284, -2123070565926783152, 273359449), 22.697 430 | 0 | 431 | 1 | (1548384, 669779585, 884736), 13.2295 432 | 1 | (10967738106543906296424595305486664028926941120, -4770495407814782941086520622913880754200168796352, 2126632487769175256612888787983935109191375), 65.7398 x 433 | 0-2 | 434 | 2 | (-3867425, -1655129510, 1442897), 10.7084 | | (-11552, -4821440, 1), 7.2863 435 | 0 | 436 | 1 | (487204, -400184340, 1), 7.6977 437 | 0 | 438 | 1 | (209764, -132496652, 1), 6.8926 439 | 2 | (42428207661937335172, -28897108034051755286616, 154966644138097), 28.9249 | | (11532, 5033656, 27), 7.9431 440 | 0 | 441 | 1 | (22932, 10061856, 2197), 5.1008 442 | 1 | (-1460680, -621472488, 125), 6.5334 443 | 0 | 444 | 1 | (4997858866572, 2411531490748996, 130323843), 21.8887 445 | 1 | (4515108853684, -1996727997682704, 25438123621), 17.6287 446 | 1 | (10527106154092999795706927380377514256689306691136, 26574607811825521389350044440029100566257816442996245, 1704422095360789303861231827626871845289984), 76.4022 x 447 | 0-2 | 448 | 1 | (244963924250737499136, -374898371430179626334208, 73276641703828357303159), 36.0818 449 | 1 | (484, 216128, 1), 4.3887 450 | 0 | 451 | 1 | (158404, -94923000, 1), 6.3478 452 | 1 | (2352634471375044, -1057123378193247244, 8146080231561), 24.219 x 453 | 0-1 | 454 | 1 | (7022522997420960187403389636366896281768822717809135788740, 4417516263339293078971407444557386528927042239584753132079039, 185921429000385741732235527921484761178046505263780562344000), 94.7834 455 | 2 | (-6227280, -2810283920, 531441), 12.4503 | | (448, 202720, 1), 5.2836 456 | 1 | (207998408, -162644103245, 512), 14.3384 457 | 1 | (777924, -771686496, 1), 10.8184 458 | | x 459 | 0-2 | 460 | 1 | (3364, -1549644, 1), 6.5883 461 | 2 | (870896876, 402872581040, 205379), 13.8878 | | (115258724, -64378467952, 1597509809), 14.3256 462 | 2 | (-715598268, -328312532548, 10503459), 13.8565 | | (-30927624, -14177320136, 59319), 12.362 463 | 0 | 464 | 2 | (62549051200, -29910773929280, 3869893), 17.4652 | | (19515328, -9012992576, 2048383), 12.7027 465 | 1 | (108303312, -50142751568, 13312053), 15.2789 466 | 1 | (22079596, -10299093420, 6859), 9.3343 467 | 2 | (-11424400, -5291638768, 15625), 9.3692 | | (10969296861667660878170952229233079940, 5499397701109644826805132965679634429304, 304177169445204630120643432939625), 55.1612 468 | 0 | 469 | 0 | 470 | 1 | (-315375, -143277400, 27), 9.9592 x 471 | | 472 | 0 | 473 | 1 | (39057776782846976, 18396154507800963712, 4869691184758499), 28.4301 474 | 1 | (6343674428732, -18351152572513651, 778688), 20.8536 475 | 1 | (1321179166950467500, -624121961929472469000, 3601488024516187), 27.2406 476 | 0 | 477 | 0 | 478 | 1 | (5551765295298222950873281651874156847172, 2904260750817815630534199784844808586733241, 34876662358866504421911031937997172675648), 63.3925 479 | 1 | 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