``Congruent number curves'' ny2=x3-x
with |n| congruent to 6 mod 8 and less than a million:
Positivity of rank, and list of curves of conjectural rank 3

For which numbers n is there a rational Pythagorean triangle of area n? This problem is known to be equivalent to: for which n does the elliptic curve

E(n): n y2 = x3 - x

have a rational point other than the point at infinity and the 2-torsion points with y=0? Such n are said to be ``congruent'' because they are also known to be the numbers that occur as the common difference (``congruum'' in Latin) of a three-term arithmetic progression of squares.

We may, and henceforth do, assume that n is a positive squarefree integer. It is conjectured that if n is congruent mod 8 to 5, 6, or 7, then n is congruent. See this page, concerning the 8k+7 case, for more background on this problem and the relevant conjectures.

Theorem. If n=8k+6 and n<106 then n is a congruent number.

Together with the corresponding results in the 8k+7 and 8k+5 cases, this shows:

E(n) has positive rank for all positive integers n<106 for which the sign of the functional equation for L(E(n),s) is -1.

The proof, combining computation and theory, proceeds as described in the 8k+7 and 8k+5 pages, except that we regard E(n) as a quadratic twist of E(2) rather than of E(1), and split the 8k+6 case into two subcases according to the parity of k, or equivalently to the residue of n modulo 16. [The curve E(2) has Weierstrass equation y2=x3-4x, and is 2-isogenous with the curve y2=x3+x; these curves have conductor 64.] If n is congruent to 14 mod 16, the rational prime 2 is split in the imaginary quadratic field of discriminant -n/2, so Heegner points yield a rational point P(n) on E(n). If n is congruent to 6 mod 16, the prime 2 is inert in that quadratic field, but a ``mock Heegner'' construction applies as in the 8k+5 case, and we again get a rational point P(n). In either case we're done unless P(n) is numerically indistinguishable from a torsion point on E(n) -- which happens for 360 values of n=16m+6, and 295 values of n=16m+14. For each of these 360+295=655 values, we guess that P(n) actually is torsion, and that E(n) has analytic rank at least 3 [by Gross-Zagier in the Heegner subcase, and by the conjectural generalization of Gross-Zagier in the ``mock Heegner'' subcase]. We then expect that E(n) has a rational non-torsion point small enough to find by direct search or by 2-descent with Cremona's MWRANK. In each case we did find at least one such point. In the table at the end of this page we list the 655 n's, and for each one give the numerator and denominator of the x-coordinate of a nontrivial rational solution of ny2=x3-x. This completes the proof.

We believe that all 655 of these curves have rank exactly 3. This is because of an upper bound on the rank, obtained by 2-descent on E(n) and/or on a curve 2-isogenous with E(n). In almost all cases, we obtain an upper bound of 3; in a handful of cases the 2-descent bound is 4 but the parity conjecture reduces this to 3. In all but 30 cases, the 2-part of the Tate-Shafarevich group SHA of E(n) is trivial if E(n) has arithmetic rank 3 as expected. The 30 exceptional cases are the following 22 in the 16k+6 subcase:
42486 = 2*3*73*97 88502 = 2*17*19*137 112326 = 2*3*97*193 202742 = 2*17*67*89
275286 = 2*3*11*43*97 295222 = 2*17*19*457 372742 = 2*17*19*577 392502 = 2*3*11*19*313
487302 = 2*3*241*337 496774 = 2*17*19*769 503206 = 2*11*89*257 563686 = 2*17*59*281
632886 = 2*3*313*337 660822 = 2*3*241*457 667318 = 2*17*19*1033 668166 = 2*3*193*577
767814 = 2*3*73*1753 774598 = 2*11*137*257 820374 = 2*3*73*1873 831878 = 2*17*43*569
875526 = 2*3*337*433 964374 = 2*3*97*1657
and the following 8 in the 16k+14 subcase:
142222 = 2*17*47*89 287246 = 2*31*41*113 564094 = 2*17*47*353 634382 = 2*7*113*401
646366 = 2*7*137*337 679582 = 2*31*97*113 771374 = 2*23*41*409 820046 = 2*17*89*271
In each of these cases, MWRANK finds 3 independent points and confirms arithmetic rank 3 by 2-descent on an isogenous curve, but finds a subgroup of SHA isomorphic with (Z/2Z)2.


nn, factored numerator and denominator of x
12542*3*11*19 19, 3
27742*19*73 171, 25
35022*17*103 128, 25
62862*7*449 961, 63
72302*3*5*241 1445, 1
77662*11*353 1331, 81
94302*5*23*41 23, 18
96542*3*1609 1225, 384
121662*7*11*79 81, 77
122702*3*5*409 507, 98
125342*3*2089 1225, 864
162062*3*37*73 147, 1
179582*3*41*73 98, 25
190462*89*107 98, 9
197262*7*1409 2809, 9
203662*17*599 1849, 599
207742*13*17*47 81, 13
214142*3*43*83 169, 3
221342*3*7*17*31 31, 3
234462*19*617 1225, 9
241902*5*41*59 4761, 41
244142*3*13*313 625, 1
288062*3*4801 2401, 2400
292942*97*151 3872, 97
296142*13*17*67 117, 17
302702*3*5*1009 5043, 2
323182*11*13*113 112225, 113
332862*11*17*89 299209, 9
352862*3*5881 8281, 3481
363662*3*11*19*29 1859, 3
378622*11*1721 2450, 729
389822*3*73*89 146, 121
396302*3*5*1321 841, 480
404062*89*227 13778, 13689
407102*3*5*23*59 2883, 8
408942*7*23*127 1863, 169
412302*5*7*19*31 76, 14
415822*17*1223 2448, 2
424862*3*73*97 243, 49
438702*5*41*107 107, 98
451182*17*1327 1352, 25
462462*19*1217 1369, 152
470942*3*47*167 841, 334
486222*7*23*151 10764800, 161
513022*113*227 908, 900
515902*5*7*11*67 72, 5
555102*5*7*13*61 63, 2
564062*3*7*17*79 175, 17
566302*5*7*809 1089, 280
569902*5*41*139 90, 49
573102*5*11*521 961, 81
583262*3*9721 19321, 121
600062*3*73*137 242, 169
605742*31*977 1225, 729
651982*7*4657 5058001, 582624
653102*3*5*7*311 343, 32
674382*7*4817 40328, 3025
675422*3*11257 29403, 15625
676062*7*11*439 4489, 439
693262*17*2039 2048, 9
698702*3*5*17*137 272, 2
707742*11*3217 321651, 49
716542*11*3257 1849, 1408
728542*73*499 4491, 2809
741022*7*67*79 578, 25
741662*3*47*263 216, 47
751742*3*11*17*67 50, 17
754542*31*1217 2209, 225
769582*7*23*239 207, 32
770462*3*12841 29403, 3721
774862*17*43*53 37553, 1089
784222*113*347 23762, 18225
785262*7*71*79 86528, 497
841342*23*31*59 3751, 25
843902*3*5*29*97 121, 24
854702*3*5*7*11*37 529, 11
857022*73*587 34322, 13225
860862*7*11*13*43 99, 13
867902*3*5*11*263 2888, 5
882062*3*61*241 11881, 169
884222*3*14737 49923, 9025
885022*17*19*137 323, 225
892382*3*107*139 139, 75
892862*3*23*647 1225, 69
901742*3*7*19*113 57, 56
919102*5*7*13*101 1805, 13
931262*3*11*17*83 5329, 17
974222*3*13*1249 2523, 25
987982*7*7057 5041, 2016
1019262*11*41*113 338, 113
1036702*5*7*1481 2601, 361
1040062*7*17*19*23 3042, 17
1063582*7*71*107 2178, 497
1123262*3*97*193 242, 49
1124862*11*5113 8281, 3168
1140862*7*29*281 4176, 242
1142942*3*43*443 486, 43
1151022*13*19*233 8125, 729
1155982*7*23*359 184, 175
1157022*17*41*83 722, 25
1169662*233*251 242, 9
1190942*3*23*863 21025, 1176
1204142*3*7*47*61 54, 7
1206942*7*37*233 529, 63
1225182*11*5569 292681, 19800
1232222*3*11*1867 1875, 1859
1235102*3*5*23*179 363, 5
1237102*5*89*139 1780, 722
1238462*3*20641 77763, 36481
1246702*5*7*13*137 72, 65
1280302*5*7*31*59 45, 14
1283342*3*73*293 147, 146
1296302*3*5*29*149 147, 2
1320462*103*641 5408, 361
1342462*7*43*223 567, 121
1368302*3*5*4561 3481, 1080
1385742*193*359 27889, 8975
1422222*17*47*89 800, 1
1427022*7*10193 67712, 24025
1435262*3*19*1259 867, 392
1441422*97*743 13225, 743
1445902*5*19*761 1521, 1
1452542*3*43*563 2209, 43
1479342*17*19*229 475, 441
1487742*73*1019 6006578, 18769
1508542*11*6857 480491, 205209
1509902*3*5*7*719 1445, 7
1528622*3*73*349 1323, 73
1534542*7*97*113 55112, 113
1535102*3*5*7*17*43 121, 51
1537982*3*25633 56307, 46225
1546222*13*19*313 475, 162
1550622*31*41*61 2511, 2450
1580702*3*5*11*479 139445, 1014
1587742*7*11*1031 7569, 352
1593102*5*89*179 90, 89
1608542*3*17*19*83 8281, 19
1614462*89*907 3728761, 13689
1617582*31*2609 871200, 790321
1624542*43*1889 2337841, 23409
1635902*3*5*7*19*41 128, 5
1648862*3*27481 87723, 22201
1659902*3*5*11*503 2662, 147
1670782*139*601 3759721, 1000800
1679102*3*5*29*193 169, 24
1680622*17*4943 19584, 14641
1693342*11*43*179 3179, 43
1697822*3*28297 57963, 55225
1702542*7*12161 6561, 5600
1704302*3*5*13*19*23 92, 38
1705262*3*97*293 293, 289
1713902*3*5*29*197 10609, 29
1723342*199*433 2048, 1849
1723742*3*28729 24025, 4704
1736142*7*12401 27889, 3087
1756302*5*7*13*193 1681, 56
1764702*5*7*2521 5041, 1
1779262*7*71*179 1775, 1089
1791582*7*67*191 3481, 2144
1793982*19*4721 227529, 3800
1817182*43*2113 781250, 36481
1820062*11*8273 7921, 352
1837022*3*17*1801 110450, 22201
1847102*3*5*47*131 141, 121
1877262*7*11*23*53 99, 7
1898542*7*71*191 145161, 1528
1903902*5*79*241 160, 81
1914062*3*19*23*73 46, 27
1922702*3*5*13*17*29 68, 10
1928942*3*13*2473 3721, 1225
1970302*5*17*19*61 153, 152
1993102*5*19*1049 20691, 289
1993742*3*7*47*101 54, 47
2009742*17*23*257 25281, 257
2021342*3*59*571 1083, 59
2022862*7*14449 29241, 343
2027422*17*67*89 5508, 1058
2041262*3*13*2617 3146, 529
2043582*7*11*1327 22528, 10647
2044702*5*7*23*127 81, 46
2057822*3*34297 34322, 25
2058862*113*911 5197088, 152881
2076942*113*919 968, 49
2085902*3*5*17*409 392, 17
2090942*3*34849 70227, 529
2096862*59*1777 48841, 38232
2116142*3*13*2713 7514, 625
2136702*5*23*929 3721, 5
2142142*7*11*13*107 99, 8
2144302*5*41*523 605, 441
2150702*3*5*67*107 67, 40
2153262*23*31*151 279, 23
2157102*5*11*37*53 2401, 1521
2157982*11*17*577 27889, 961
2162062*17*6359 6465608, 1168561
2184542*3*23*1583 38809, 2400
2189262*17*47*137 103041, 17
2200942*7*79*199 8649, 199
2201102*3*5*11*23*29 40, 29
2211902*3*5*73*101 1083, 73
2212542*11*89*113 6889, 3168
2228382*7*11*1447 29929, 19800
2233902*5*89*251 84681, 4361
2266142*3*179*211 111747, 128
2279582*3*37993 258847800363, 241462269913
2284862*3*113*337 338, 1
2288062*233*491 28440882, 6039593
2293262*3*37*1033 49729, 888
2311662*13*17*523 523, 361
2317102*5*17*29*47 1681, 376
2325262*7*17*977 3025, 1071
2330062*113*1031 163592, 47089
2369102*3*5*53*149 245, 53
2390462*3*39841 21025, 18816
2392462*7*23*743 1472, 14
2417102*3*5*7*1151 1183, 32
2436382*43*2833 5929, 3096
2454942*131*937 1058, 121
2477982*19*6521 3864619, 95481
2502062*3*11*17*223 57121, 33
2505182*3*43*971 235225, 243
2518702*5*89*283 283, 162
2519662*11*13*881 5733, 2209
2522462*3*17*2473 3829849, 2941225
2523742*257*491 243049, 3928
2527262*3*73*577 87723, 3481
2559902*3*5*7*23*53 160, 1
2572942*103*1249 2601, 103
2587742*3*17*43*59 38328481, 43
2596702*5*23*1129 1127, 2
2616542*3*43609 41209, 2400
2618702*3*5*7*29*43 29, 14
2696462*3*13*3457 2209, 1248
2704702*5*17*37*43 7938, 17
2718302*3*5*13*17*41 147, 17
2752862*3*11*43*97 24843, 11
2772062*3*47*983 109561, 983
2787182*3*11*41*103 1849, 824
2787262*7*43*463 3362, 121
2811982*23*6113 422970696, 12794929
2829822*7*17*29*41 343, 82
2846942*3*23*2063 51529, 46
2860142*3*73*653 507, 146
2871662*3*11*19*229 916, 338
2872462*31*41*113 72, 41
2903662*47*3089 1568, 1521
2912702*3*5*7*19*73 54, 19
2914142*3*17*2857 4489, 1225
2946302*3*5*7*23*61 343, 23
2951182*41*59*61 3101121, 1661521
2952222*17*19*457 1682, 1225
2974622*3*11*4507 117238, 4563
2979662*3*53*937 3481, 937
2980862*3*49681 1424163, 1324801
3000142*11*13*1049 5625, 4576
3021582*17*8887 32912, 15138
3058302*5*7*17*257 338, 257
3071662*383*401 392, 9
3074062*11*89*157 539, 89
3103102*5*7*11*13*31 117, 7
3145742*3*13*37*109 8427, 361
3165262*7*23*983 22201, 983
3202142*3*83*643 1369, 83
3206062*11*13*19*59 891, 59
3227662*71*2273 197192, 77841
3241742*3*97*557 363, 194
3266302*5*89*367 3481, 3125
3316062*11*15073 761378, 22801
3345102*5*11*3041 3481, 2601
3415022*3*7*47*173 5000, 329
3452782*31*5569 52398721, 27007200
3465662*3*11*59*89 30899, 2225
3485582*3*7*43*193 343, 43
3500142*7*23*1087 575, 512
3506702*3*5*11689 51483, 6962
3522942*353*499 966289, 225
3551102*3*5*7*19*89 54, 35
3561742*7*13*19*103 175, 72
3603262*67*2689 5427, 5329
3610542*23*47*167 7225, 3384
3613022*3*60217 96721, 36504
3636302*3*5*17*23*31 147, 23
3642062*3*101*601 57963, 2738
3645022*59*3089 1638050, 2209
3674782*43*4273 134689, 78961
3700782*31*47*127 174375, 512
3724702*5*7*17*313 160, 153
3727422*17*19*577 729, 425
3764142*17*11071 14792, 3721
3767742*31*59*103 1194649, 271872
3778462*7*137*197 2809, 343
3798462*257*739 1028, 450
3812542*19*79*127 20691, 625
3812622*7*113*241 124609, 16641
3812702*3*5*71*179 125, 54
3838462*281*683 683, 441
3848542*337*571 2738, 2401
3850142*3*7*89*103 96, 7
3853662*43*4481 48680128, 7529536
3866702*3*5*12889 13467, 578
3886542*7*17*23*71 343, 71
3902862*13*17*883 442, 441
3925022*3*11*19*313 294, 19
3925742*3*7*13*719 847, 128
3942702*5*89*443 43245, 169
3947902*5*11*37*97 12482, 7857
3954142*211*937 1899, 1849
3966862*241*823 10830681, 823
3973342*7*101*281 2527, 2
3974542*37*41*131 2358, 676
3992542*89*2243 1158242, 638401
3994062*7*47*607 29929, 1400
4005822*7*13*31*71 175, 104
4008622*7*11*19*137 175, 99
4022542*17*11831 21632, 9801
4026702*5*67*601 623045, 101761
4032062*3*17*59*67 169, 67
4039902*5*71*569 73960, 22201
4056542*3*17*41*97 36481, 2328
4057902*5*7*11*17*31 125, 62
4074782*3*113*601 3051, 1849
4075262*3*7*31*313 338, 313
4096382*3*67*1019 2401, 1675
4127422*11*73*257 514, 289
4181822*3*69697 70590962, 36869713
4195662*7*23*1303 736, 567
4212702*5*103*409 42849, 41405
4215902*3*5*13*23*47 128, 13
4218942*11*127*151 22201, 151
4222942*19*11113 12482, 1369
4257582*193*1103 13939200, 3508129
4278542*3*7*61*167 175, 8
4330862*3*19*29*131 289, 262
4342942*7*67*463 11449, 126
4357102*5*11*17*233 7921, 1
4381262*3*13*41*137 5929, 312
4430862*7*31649 37249, 5600
4438942*11*20177 49379, 31329
4446862*11*17*29*41 99, 17
4468942*7*137*233 44521, 12600
4488062*3*131*571 116162, 108241
4509262*7*31*1039 5041, 2232
4521662*3*11*13*17*31 496, 54
4533902*3*5*7*17*127 169, 85
4536302*3*5*15121 15123, 2
4540622*3*7*19*569 625, 513
4562142*11*89*233 305762, 4361
4661342*3*77689 312987, 157609
4682142*17*47*293 1175, 882
4697662*11*131*163 361, 163
4702702*5*31*37*41 242, 37
4707582*113*2083 15334722, 4264225
4718142*7*67*503 19363, 12321
4718222*3*13*23*263 2311247, 3
4734782*3*23*47*73 3481, 3381
4776542*3*79609 41209, 38400
4778622*7*11*29*107 225, 203
4803262*7*11*3119 46475, 2809
4826942*3*80449 227138, 146689
4834462*17*59*241 5618, 1521
4843342*23*10529 15129, 5929
4860942*7*34721 57121, 12321
4861742*3*13*23*271 13697401, 23
4873022*3*241*337 723, 625
4890622*7*181*193 813967, 1458
4943262*7*17*31*67 4225, 536
4956702*5*7*73*97 22898, 97
4959422*3*82657 8233682, 8151025
4967742*17*19*769 1700, 162
4984702*5*7*7121 94249, 5445
4996862*3*11*67*113 289, 113
5001182*3*19*41*107 164, 50
5007742*31*41*197 361, 197
5014862*3*19*53*83 121, 38
5019902*3*5*29*577 722, 145
5032062*11*89*257 1058, 801
5047102*5*41*1231 54760, 43681
5105102*3*5*7*11*13*17 88, 3
5111662*431*593 512, 81
5121342*7*157*233 37303, 722
5125822*7*19*41*47 450, 329
5130062*3*13*6577 16250, 3481
5136782*3*11*43*181 30625, 507
5155262*3*11*73*107 1250, 73
5161982*3*227*379 18723, 227
5162062*199*1297 7569, 4975
5164302*5*43*1201 8153645, 212521
5173982*3*7*97*127 1875, 1681
5194862*3*11*17*463 512, 49
5217742*11*37*641 172225, 13024
5226142*17*19*809 8281, 5472
5229662*3*43*2027 6241, 2187
5239342*241*1087 28912338369, 22198019407
5243422*7*13*43*67 225, 43
5330942*3*23*3863 183027, 5329
5400142*193*1399 1568, 169
5423662*11*89*277 6135529, 1260329
5436382*67*4057 22080601, 12201049
5472302*3*5*17*29*37 128, 17
5472462*7*39089 33489, 5600
5494302*5*7*47*167 167, 162
5497662*7*107*367 1575, 107
5518542*17*16231 36397512, 22877089
5522462*3*92041 351122, 259081
5542622*3*92377 2380849, 2145624
5556222*353*787 49977648, 39690000
5566542*7*39761 80089, 567
5575742*3*19*67*73 3698, 121
5579262*83*3361 717409, 418609
5614702*5*7*13*617 2809, 2744
5619662*3*229*409 867, 49
5619902*3*5*11*13*131 98, 33
5636862*17*59*281 722, 281
5637662*3*7*31*433 867, 1
5640942*17*47*353 200, 153
5651742*19*107*139 10259209, 23491
5682302*3*5*13*31*47 125, 31
5747262*67*4289 365397938978, 58367804369
5749262*3*11*31*281 529, 33
5771822*3*19*61*83 3050, 1681
5773742*3*7*59*233 289, 177
5791662*7*41*1009 648, 361
5833102*5*7*13*641 641, 529
5838622*353*827 157922, 140625
5858302*5*7*8369 74263, 1058
5865342*13*17*1327 134784, 112225
5867582*3*19*5147 845899, 399675
5872382*3*97*1009 211250, 82369
5880702*5*7*31*271 65596, 14
5881262*3*7*11*19*67 1369, 38
5908702*5*7*23*367 1849, 14
5934462*7*19*23*97 1682, 161
5938142*3*13*23*331 363, 299
5947342*7*23*1847 31329, 18400
5954062*7*71*599 4761, 568
5971742*3*99529 395307, 2809
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