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.

According to Siverman's book[1], rank is known below.

rank is 0:     p=7,11 mod 16
rank is 0,1:   p=3,5,13,15 mod 16
rank is 0,1,2: p=1,9 mod 16


2. Search results

 [ ]: Rank 0.

 p [x, y]

 2 []
 3 [(1, -2)]
 5 [(1/4, -9/8)]
 7 []
11 []
13 [(9/4 , 51/8 )]
17 []
19 [(9 , 30 )]
23 []
29 [(25/4, 165/8 )]
31 [(25/9 , 280/27 )]
37 [(22801/900 , -3540799/27000 )]
41 []
43 []
47 [(289/25 , 5712/125 )]
53 [(49/4 , 399/8 )]
59 []
61 [(11881/8100 , -7016221/729000 )]
67 [(2401/225, 148274/3375 )]
71 []
73 [(36 , 222 ), (657/4, 16863/8 )]
79 [(711, 18960 )]
83 [(747 , 20418 )]
89 [(25/16 , 765/64 ), (801/4 , 22695/8 )]
97 []
101 [(69872881/2102500 , -610189287279/3048625000 )]
103 []
107 []
109 [(68121/100 , -17781669/1000 )]
113 [(16/9 , 388/27 ), (49/16, 1239/64 )]
127 [(42849/30625 , 71989632/5359375 )]
131 [(25/9 , 530/27 )]
137 []
139 []
149 [(83521/4900 , -29688681/343000 )]
151 []
157 [(7896277321/249956100 , -754849218749869/3951805941000 )]
163 [(81 , 738 )]
167 []
173 [(169/4 , 2301/8 )]
179 [(49/9 , 910/27 )]
181 [(7129492752321/434584192900 ,-24619226475431132631/286490937485467000 )]
191 [(7225/121 , 630360/1331 )]
193 []
197 [(47995604297578081/7389879786648100 ,-25038161802544048018837479/635266655830129794121000 )]
199 []
211 [(1225/1089 , 555310/35937 )]
223 [(1089/49 , 43296/343 )]
227 [(93876721/9828225 , -1698746655394/30811485375 )]
229 [(225/4 , 3495/8 )]
233 [(196/25 , 6006/125 ), (169/16 , 3861/64 )]
239 [(14161/169, 1713600/2197 )]
241 []
251 []
257 []
263 []
269 [(109561/12100 , -75034059/1331000 )]
271 [(65025/361, 16650480/6859 )]
277 [(4512287297420721/269721826704100 ,-427555620570872645438769/4429698201545042161000 )]
281 [(100/9, 1810/27 ), (175561/400 , -73613691/8000 )]
283 []
293 [(289/4 , 5049/8 )]
307 [(811801/50625 , -1083385826/11390625 )]
311 []
313 []
317 [(9110859798270041392441/51276692442358684900 ,-873994228512638323467064295836011/367180835274953261650562993000 )]
331 []
337 [(144/25 , 5772/125 ), (102394161/6760000 ,-1628016453591/17576000000 )]
347 []
349 [(25/36 , 3365/216 )]
353 [(289/16 , 7089/64 ), (1024/49 , 44064/343 )]
359 []
367 [(494209/1625625 , 21895930448/2072671875 )]
373 [(49/36 , 4879/216 )]
379 []
383 [(121/25 , 5544/125 )]
389 [(128881/16900 , -128295471/2197000 )]
397 [(408386379490401068761/27192982096777464900 ,-13711249785729025890926318611259/141802946520771128436338643000 )]
401 []
409 []
419 [(169/9 , 3250/27 )]
421 [(18227430081/47171496100 , -130695434930980521/10245177237959000)]
431 [(47541025/3308761 , 575998481520/6018636259 )]
433 []
439 []
443 []
449 []
457 []
461 [(225661613535254521/57969119070772900 ,-600895814953270614429279819/13957095527534463309433000 )]
463 [(81/25 , 4896/125 )]
467 [(443410823881/2840357025 , -298078830961874846/151376827647375 )]
479 [(961/49 , 44640/343 )]
487 []
491 []
499 [(5929/1521 , -2655730/59319 )]
503 []
509 [(2156209225/520478596 , -554380820755155/11874198689144 )]
521 []
523 []
541 [(56836521/30360100 , -5340940626819/167284151000 )]
547 [(4004001/3025 , -8013256626/166375 )]
557 [(652393828681/476548900 , -527022665382389421/10403062487000 )]
563 [(22801/9 , -3443102/27 )]
569 []
571 []
577 []
587 []
593 [(256/9 , 5392/27 ), (529/16, 15111/64 )]
599 []
601 [(900/49 , 45030/343 ), (292681/14400 , -248085829/1728000 )]
607 [(9/25, 1848/125 )]
613 [(289/36 5929/216 )]
617 [(4/25, 1242/125 ), (236821321/2560000 ,-3773535597381/4096000000 )]
619 []
631 []
641 []
643 [(9/25, 1902/125 )]
647 []
653 [(17530969/357604 , -82775721339/213847192 )]
659 [(289/9 , 6290/27 )]
661 [(25303583041/1386072900 , -6952300370898839/51603494067000 )]
673 []
677 [(19463319121/1820728900 , -7145241014093919/77690502163000 )]
683 []
691 [(8205642225/3961569481 , -9462516729612210/249345144703621 )]
701 [(807386359657781509321/12488740334378616100 ,-24790884300537496225354127701869/44134473878874964490987159000 )]
709 [(731025/1493284 , -34002044415/1824793048 )]
719 [(841/49 , 45240/343 )]
727 []
733 [(729/4, 19899/8 )]
739 [(29929/9 , -5177890/27 )]
743 []
751 [(47128225/480249 , 335913896240/332812557 )]
757 [(3677559478737567201/128229486563102500 ,-9773037730242719350780384551/45917882776136890723625000 )]
761 []
769 []
773 [(21743569/195364 , -104505929631/86350888 )]
787 [(81/25 , 6354/125 )]
797 [(44352773425738202562487170601/93882365289124698123568900 ,-9357400448379130876262700661108617883580349/909653582278003417635545675509156363000 )]
809 []
811 []
821 [(169973973841/60905304100 , -722883300059430111/15030819998839000 )]
823 []
827 []
829 [(184041/28900 , -365597661/4913000 )]
839 []
853 [(529/36 , 27071/216 )]
857 []
859 []
863 [(6889/121 , 643416/1331 )]
877 [(375494528127162193105504069942092792346201/6215987776871505425463220780697238044100 ,-256256267988926809388776834045513089648669153204356603464786949/490078023219787588959802933995928925096061616470779979261000 )]
881 [(400/9 , 9620/27 ), (344139601/40000 , -6384171730599/8000000 )]
883 [(441 , 9282 )]
887 []
907 []
911 [(319225/82369, 1416192840/23639903 )]
919 []
929 []
937 [(925444/99225 , 3054524122/31255875 ), (502681/14400 ,-474015421/1728000 )]
941 [(64090408676120688976921/1383526426973634904900 ,-19460134840296185916047442440786469/51461351616376374479310289843000 )]
947 [(3194397361/10989225 , -181553043920734/36429280875 )]
953 []
967 []
971 []
977 []
983 []
991 [(265225/729, 137101240/19683 )]
997 [(860880224051357946383150694640081/454952888893307669728214841912900 ,-422244329715862751254691678207402818016091512405729/9703974366148359222222683619255690411904398867000 )]


3.Reference

[1]: J.H. Silverman, The Arithmetic of Elliptic Curves, Springer, New York, 1986.



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