1.Introduction

We searched the generators of elliptic curve y^2 = x^3 - px where p is prime number with p<1000. 

We used the command e.gens() of SAGE.

It seems that rank of y^2 = x^3 - px is 0,1,2.   
rank is 0,2:   p=1 mod 16
rank is 0:     p=3,11,13 mod 16
rank is 1:     p=5,7,9,15 mod 16

2. Search results

 [ ]: Rank 0.

 p [x, y]

2 [(-1, 1 )]
3 []
5 [(-1 , 2 )]
7 [(4 , 6 )]
11 []
13 []
17 [(-4 , 2 ), (-1 , 4 )]
19 []
23 [(324/25 , 5418/125 )]
29 []
31 [(16 , 60 )]
37 [(-1 , 6 )]
41 [(-4, 10 )]
43 []
47 [(64/9, 136/27 )]
53 [(1325 , 48230 )]
59 []
61 []
67 []
71 [(36 , 210 )]
73 [(1825/16 , 77745/64 )]
79 [(160000/17689 , -11871600/2352637 )]
83 []
89 [(-64/25, 1816/125 )]
97 [(-9, 12 ), (-4, 18 )]
101 [(-1, 10 )]
103 [(195364/625 , 86305362/15625 )]
107 []
109 []
113 []
127 [(64 , 504 )]
131 []
137 [(-4 , 22 )]
139 []
149 [(-289/25, 1666/125 )]
151 [(7396/121 , 623070/1331 )]
157 []
163 []
167 [(3364/81, 185426/729 )]
173 []
179 []
181 [(-9 , 30 )]
191 [(1296/49 , 39780/343 )]
193 []
197 [(-1, 14 )]
199 [(100 , 990 )]
211 []
223 [(9193024/207025 , 26249846232/94196375 )]
227 []
229 [(-2025/169 , 70290/2197 )]
233 [(-256/25 , 4528/125 )]
239 [(14400/169, 1699320/2197 )]
241 [(-4 , 30 ), (121, 1320 )]
251 []
257 [(-16 , 4 ), (-1, 16 )]
263 [(855177358564/39674664225 , -520947907941053762/7902597993656625 )]
269 []
271 [(678976/41209 , 23323320/8365427 )]
277 [(-9 , 42 )]
281 [(-16 , 20 )]
283 []
293 [(-277729/34225 , -271838194/6331625 )]
307 []
311 [(196/9, 1610/27 )]
313 [(-459684/42025 , -396198114/8615125 )]
317 []
331 []
337 [(-16 , 36 ), (-9 , 48 )]
347 []
349 []
353 []
359 [(167184900/1615441 , -2125163667270/2053225511 )]
367 [(440896/1225 , -292339944/42875 )]
373 [(-1792921/297025 , -7296374358/161878625 )]
379 []
383 [(6350814864/59367025 , 497567821065612/457422927625 )]
389 [(-121/25 , 5258/125 )]
397 []
401 [(-2500/169, 114050/2197 ), (-1 , 20 )]
409 [(-2500/289 , 264150/4913 )]
419 []
421 [(-441/25 , 5502/125 )]
431 [(256/9 , 2800/27 )]
433 [(-324/25 , 7326/125 ), (-169/25 , 6396/125 )]
439 [(1464100/67081 , 496173810/17373979 )]
443 []
449 [(-529/25 , 644/125 ), (225 , 3360 )]
457 [(-4 , 42 )]
461 []
463 [(367004389393912384/6872105760367225 ,203489260419706675311874152/569684957220372678642125 )]
467 []
479 [(21160000/439569 , -86694171400/291434247 )]
487 [(1444/49 , 36366/343 )]
491 []
499 []
503 [(84798838445832596484/671031463275625 ,-780880986315776148553683352602/17382583558344190796875 )]
509 []
521 [(-84100/3721 , -3442010/226981 )]
523 []
541 []
547 []
557 []
563 []
569 [(-100/289 , 68930/4913 )]
571 []
577 [(-1 , 24 ), (289 , 4896 )]
587 []
593 []
599 [(1067907237322500/26620264827289 ,-27650853303324553530150/137346803831895531587 )]
601 [(-5184/841 , -1436760/24389 )]
607 [(868624/25921 , -548745756/4173281 )]
613 [(-2289169/93025 , -383935854/28372625 )]
617 [(-16 , 76 )]
619 []
631 [(89226916/724201 , -825134123730/616295051 )]
641 [(-25 , 20 ), (-4 , 50 )]
643 []
647 [(324 , 5814 )]
653 []
659 []
661 [(-25 , 30 )]
677 [(-1 , 26 )]
673 []
683 []
691 []
701 []
709 [(-81225/3721 , -16170330/226981 )]
719 [(400/9 , 6380/27 )]
727 [(676/25, 1326/125 )]
733 []
739 []
743 [(4288883605444/141695780625 , 3861550506672014122/53337834221765625)]
751 [(65536/361, 16584960/6859 )]
757 [(-9 , 78 )]
761 [(-100/169 , 46610/2197 )]
769 [(-25 , 60 ), (354025/841 , 210187320/24389 )]
773 [(-18769/4225 , -15886246/274625 )]
787 []
797 []
809 [(-1198682884/231192025 , -223852071993914/3515274740125 )]
811 []
821 [(-25 , 70 )]
823 [(43919200130322724/329183144994025 ,8988814898053547815656882/5972503118800318155125 )]
827 []
829 []
839 [(2088993753638928900/21341007494270689 ,-2884080018334289277061089030/3117614247202157567580113 )]
853 [(-945993049/33466225 , -7561913497926/193602111625 )]
857 [(-4 , 58 )]
859 []
863 [(66522650055701904/818577018316225 ,15997257122879332904309292/23420131076986575486625 )]
877 []
881 [(-25 , 80 ), (-16, 100 )]
883 []
887 [(484/9 , 8866/27 )]
907 []
911 [(491242896/11417641 , -7759278066540/38580208939 )]
919 [(4210710802200100/138769036120729 ,11795806214364097029390/1634702992266162879683 )]
937 [(-576/169, 123384/2197 )]
929 []
941 []
947 []
953 [(-515524/34225 , -662170474/6331625 )]
967 [(484, 10626 )]
971 []
977 [(-4 , 62 ), (529/9, 10304/27 )]
983 [(58920968623470724/93151934825625 ,14284684039746654829154618/899058227767890328125 )]
991 [(3154394896/70576801, 125761619195940/592915705201 )]
997 [(-361/25, 13338/125 )]





HOME