Mordell-Weil generators for elliptic curves $E : y^{2} = x^{3}

In the paper Three consecutive almost squares, we determine all positive integers $n$ so that the $n = ax^{2}$, $n+1 = by^{2}$ and $n+2 = cz^{2}$ with $a, b, c, x, y, z \in \mathbb{Z}$ and $a, b, c \leq 150$. The largest is $9841094$. This is related to the problem of finding integral points on the elliptic curve $y^{2} = x^{3} - (abc)^{2} x$. Existing software can do this provided generators for the Mordell-Weil groups are known.

We give a table below of the $37$ most difficult cases that occur, with some comments about them.

$a$ $b$ $c$ Rank of $E$ Canonical height of generator(s) Comments Largest height generator Smaller height generator

$19$ $43$ $109$ $1$ $253.7$ Generator found by $12$-descent $\frac{2071 \cdot (67857541272123363329139991690564833094219268965542724150)^{2}}{(10094030932912819985173678281702166862648545663116314889)^{2}}$

$19$ $83$ $109$ $1$ $163.3$ Generator found by $12$-descent $\frac{2071\cdot(2404729754563154628344913564267045850)^{2}}{(163868491487120348714920430025430129)^{2}}$

$19$ $131$ $109$ $1$ $380.5$ Generator found by $12$-descent $\frac{2071\cdot(468201144663906104503484829330659271574222442999797761100485972318886113004183589674)^{2}}{(16611698013193004330212054222237303174223331610135140868774003839243194087410261325)^{2}}$

$43$ $67$ $149$ $1$ $207.8$ Generator found by $12$-descent $\frac{6407\cdot(9179140092110860772826606768360672784144796818)^{2}}{(1009829760662074665415797111823721321160129317)^{2}}$

$59$ $1$ $149$ $1$ $161.1$ Generator found by $12$-descent $\frac{8791\cdot(82726193888154287262233411294982706)^{2}}{(76577936224526255686724020426012945)^{2}}$

$59$ $107$ $37$ $1$ $267.1$ Generator found by $12$-descent $\frac{-2183\cdot(8449695607332127972439157359125236392140213339952218595233)^{2}}{(820008170789612712889551818909262717913191381088587898398)^{2}}$

$59$ $107$ $61$ $1$ $305.7$ Generator found by $12$-descent $\frac{3599\cdot(20802632789316973880711107130364849118971207110999137350844739520390)^{2}}{(1899499116735354146218850075542544869356670944579780400402688546563)^{2}}$

$59$ $107$ $101$ $1$ $211.1$ Generator found by $12$-descent $\frac{5959\cdot(70506252108781530811763229218479819253407506770)^{2}}{(1989841069682594214362483175141488216789066097)^{2}}$

$59$ $107$ $149$ $1$ $564.1$ Generator found by $12$-descent $\frac{8791\cdot(2769650363410368157653416658210544646238022471205027157927824211286431250002069318974085985712689027083116289249059088156910)^{2}}{(266167030895573640460663298982568889556591100883936563310883398852171278482828574518525062373034383458407554683841720156267)^{2}}$

$59$ $137$ $101$ $1$ $220.1$ Generator found by $12$-descent $\frac{5959\cdot(525676213254891073277612046436176848798827245506)^{2}}{(44585080400568706913530976638769769808051525205)^{2}}$

$66$ $67$ $149$ $1$ $207.6$ Generator found by $12$-descent $\frac{9834\cdot(11875366583346175045308849517719563546947872905)^{2}}{(113316948313836379630979456641974661571109077)^{2}}$

$67$ $1$ $109$ $1$ $210.4$ Generator found by $12$-descent $\frac{7303\cdot(4806248515591894214558390288273183300072043058)^{2}}{(741817951830616322569413396183281471139846231)^{2}}$
$67$ $59$ $53$ $1$ $167.4$ Generator found by $12$-descent $\frac{-3551\cdot(1257350173204775834467149354775468199)^{2}}{(275025406234649402419681991929187806)^{2}}$

$67$ $131$ $101$ $1$ $309.2$ Generator found by $12$-descent $\frac{6767\cdot(149291284723636235859361462838539885889505562717236670728297282676622)^{2}}{(7824433831175789226647422526799670411545689928436971203757443741131)^{2}}$

$67$ $131$ $109$ $1$ $1692.7$ Heegner point method of Elkies $\frac{131\cdot(389234382246741691512295824144776312547348969430015137787778955490735603642964615455301304214850198212137150523901724928693584774619005789712986895438586759974184243850737614257459628095288255945499690917601118686171608635892963742412571815274892621734502985904872802600476151506411399262100892403060234020553347148573528914682872035879910711261765337662966792789078775)^{2}}{(4290491832147440825463221413465165519291830538673265365231248526470874762107665983564649520418298702114982624413079977429194160025077778025642249780887365775485305939249055731273154047460031807467231949288893009812197126666437015517675219680901419211890435622949863135034591635397177988904826895874272924510534621473865542628592613550707456387569466338342609467868817)^{2}}$

$83$ $131$ $53$ $1$ $275.4$ Generator found by $12$-descent $\frac{-4399\cdot(104542160754794385209479287007124768779354685111381314340421)^{2}}{(55688137642031069362449609538498451037829245426423061960058)^{2}}$
$83$ $131$ $101$ $1$ $297.3$ Generator found by $12$-descent $\frac{8383\cdot(406036484273712871398034282100610068073953957967349415518147956862)^{2}}{(1797783354990246623906044634421746283966593556228345612429618051)^{2}}$

$87$ $67$ $89$ $1$ $168.9$ Generator found by $12$-descent $\frac{7743\cdot(36361338701313278498207155775697115528)^{2}}{(2046002242334858420736663452028365323)^{2}}$

$131$ $107$ $149$ $1$ $740.1$ Generator found by $12$-descent $\frac{-19519\cdot(12080135327110395633475467624916527664551423857735434278750163258733329846851955293626115779273025021846545745990338974837566099493198153158196366605837213609999)^{2}}{(4863906769868489171979169831319919577776186738431272649467856553730733611643891754336202525709351046930323998862504316204403804191334026430186185565134248872490)^{2}}$

$134$ $103$ $38$ $1$ $215.0$ Generator found by $12$-descent $\frac{-103\cdot(52807710646670457937254031011497168157718229199)^{2}}{(5429238767645357073892731757050514694993498063)^{2}}$

$137$ $37$ $118$ $1$ $12.6$ Verified that $L'(E,1) \ne 0$. Analytic order of Sha is $16$. $\frac{137 \cdot (509)^{2}}{(4)^{2}}$

$137$ $93$ $118$ $1$ $199.8$ Generator found by $12$-descent $\frac{-16166\cdot(8589729076766496307168845597312916351256695)^{2}}{(2911548230640340326582690319638947183260091)^{2}}$

$137$ $101$ $118$ $1$ $11.5$ Verified that $L'(E,1) \ne 0$. Analytic order of Sha is $16$. $\frac{-16166 \cdot (85)^{2}}{(37)^{2}}$

$139$ $37$ $149$ $2$ $19.3, 209.7$ Large height generator found via $12$-descent $\frac{20711\cdot(3017021558615380446759125150515845058876860112)^{2}}{(435008295310643307693121524215059462939701215)^{2}}$ $\frac{-20711\cdot(12707)^{2}}{(2170)^{2}}$

$139$ $41$ $141$ $1$ $191.5$ Generator found by $12$-descent $\frac{-19599\cdot(51560745051703086035321848639796184211145)^{2}}{(60213728891839122003517229541334763963158)^{2}}$

$139$ $67$ $149$ $1$ $1079.7$ Generator found by $12$-descent $\frac{-20711\cdot(2401615883687082329458917767577674464355027916864324111377968309222317909434694455818252586818804529457022009033476957039652841223346611842213082699765688329048669070706612509435961139389873730004192838541377210267192383082141999813497)^{2}}{(293927652920529088347948257089327803176419165620306380966648895647754251404488320765406678970108968345517742087086480778614703443470783572928072035648982923058029951400470462266817633004637992091415778340727711073566652809824857397250)^{2}}$

$139$ $83$ $61$ $1$ $252.8$ Generator found by $12$-descent $\frac{-8479\cdot(5351404319489261031602931529609733497750717094946934063)^{2}}{(756031503590795081890079204406384240488767365905363110)^{2}}$

$139$ $89$ $109$ $1$ $8.9$ Verified that $L'(E,1) \ne 0$. Analytic order of Sha is $64$. $-(417)^{2}$

$139$ $107$ $53$ $1$ $414.7$ Generator found by $12$-descent $\frac{-7367\cdot(10201052413152433748847363925053833645555803740652176526193230266233760081064660209617809)^{2}}{(107604291563185497813757430924130324689666689030078865458045617567058800464015520134664634)^{2}}$

$139$ $107$ $61$ $1$ $1189.8$ Generator found by $12$-descent $\frac{-8479\cdot(1542976797264121368676627240515553875008375895558457529318893589206007812656851415332671864900866908952120358061379451101501751749633075920908481998014995436528109437980629134945551481130305020485946077574969498756571023563263076469676955341716417766202207523)^{2}}{(195386573974244898694815282113438126576733756551929646020017093465234954778904516255315135079782665807466990243374828554142388730959662931009886997767148676012291855270509163837513718526139585669684778444074390894421138375789004408000310723736468280841737870)^{2}}$

$139$ $107$ $101$ $1$ $1234.8$ Generator found by $12$-descent $\frac{-14039\cdot(1851495003814952396170748013111562596773243635333732094045304936364204024771699817685899344357070026081546684140917639864285420264121088638374915071081104171900077680130663727293193658977408625742386370380174266457233791711876859424449919571577221879894513306180780551)^{2}}{(1301421700907620472195186706767123160765386052792830074898473660127022033620519108248695601721558440460038659205402810380712272366431377160452154080509901761390306792286757767705853327742255425490817505548283606527488586652394276919622062793983848905119654371888362290)^{2}}$

$139$ $107$ $149$ $1$ $344.4$ Generator found by $12$-descent $\frac{-20711\cdot(434591749272223360724779541143652540136908822032911419364085915426823031367)^{2}}{(52966957777485298092255650355883279153830643472796270310036533563729013490)^{2}}$

$139$ $131$ $53$ $1$ $568.7$ Generator found by $12$-descent $\frac{-7367\cdot(1444095820769351855779233877938766458809285941721259798341634873836250491180632312301458667826315363086272766055008247161189)^{2}}{(260451937827318693681705167157500760167651067851339795642879366646639129342858482825017338398040733609100197154215165087054)^{2}}$

$139$ $131$ $61$ $1$ $160.3$ Generator found by $12$-descent $\frac{-8479\cdot(54481920001044334427930129629677065)^{2}}{(4784718638506573848135813974757802)^{2}}$

$139$ $131$ $101$ $1$ $1041.3$ Generator found by $12$-descent $\frac{-14039\cdot(4564980122088413809080969939772778659282789168364528070725204680649733159483887618092278913033998796037999486509666238239326869890308330984265376012709826161689071854652260072696128170967841678388945148775595059705509165419207)^{2}}{(1117322900636229245651658306666857517110601759313423112938049610003264504092849578876263150375779308639221868217866758159114428534781514333155995245425126615858719902371352739814146261648511527513727317963460654592848290617690)^{2}}$

$142$ $103$ $79$ $1$ $15.5$ Verified that $L'(E,1) \ne 0$. Analytic order of Sha is $16$. $2 \cdot (2753)^{2}$

$145$ $149$ $118$ $1$ $195.9$ Generator found by $12$-descent $\frac{149\cdot(26662359272076288974462910653684425185890157)^{2}}{(1101055762657820447681878365608649344213)^{2}}$

$a$	$b$	$c$	Rank of $E$	Canonical height of generator(s)	Comments	Largest height generator	Smaller height generator
$19$	$43$	$109$	$1$	$253.7$	Generator found by $12$-descent	$\frac{2071 \cdot (67857541272123363329139991690564833094219268965542724150)^{2}}{(10094030932912819985173678281702166862648545663116314889)^{2}}$
$19$	$83$	$109$	$1$	$163.3$	Generator found by $12$-descent	$\frac{2071\cdot(2404729754563154628344913564267045850)^{2}}{(163868491487120348714920430025430129)^{2}}$
$19$	$131$	$109$	$1$	$380.5$	Generator found by $12$-descent	$\frac{2071\cdot(468201144663906104503484829330659271574222442999797761100485972318886113004183589674)^{2}}{(16611698013193004330212054222237303174223331610135140868774003839243194087410261325)^{2}}$
$43$	$67$	$149$	$1$	$207.8$	Generator found by $12$-descent	$\frac{6407\cdot(9179140092110860772826606768360672784144796818)^{2}}{(1009829760662074665415797111823721321160129317)^{2}}$
$59$	$1$	$149$	$1$	$161.1$	Generator found by $12$-descent	$\frac{8791\cdot(82726193888154287262233411294982706)^{2}}{(76577936224526255686724020426012945)^{2}}$
$59$	$107$	$37$	$1$	$267.1$	Generator found by $12$-descent	$\frac{-2183\cdot(8449695607332127972439157359125236392140213339952218595233)^{2}}{(820008170789612712889551818909262717913191381088587898398)^{2}}$
$59$	$107$	$61$	$1$	$305.7$	Generator found by $12$-descent	$\frac{3599\cdot(20802632789316973880711107130364849118971207110999137350844739520390)^{2}}{(1899499116735354146218850075542544869356670944579780400402688546563)^{2}}$
$59$	$107$	$101$	$1$	$211.1$	Generator found by $12$-descent	$\frac{5959\cdot(70506252108781530811763229218479819253407506770)^{2}}{(1989841069682594214362483175141488216789066097)^{2}}$
$59$	$107$	$149$	$1$	$564.1$	Generator found by $12$-descent	$\frac{8791\cdot(2769650363410368157653416658210544646238022471205027157927824211286431250002069318974085985712689027083116289249059088156910)^{2}}{(266167030895573640460663298982568889556591100883936563310883398852171278482828574518525062373034383458407554683841720156267)^{2}}$
$59$	$137$	$101$	$1$	$220.1$	Generator found by $12$-descent	$\frac{5959\cdot(525676213254891073277612046436176848798827245506)^{2}}{(44585080400568706913530976638769769808051525205)^{2}}$
$66$	$67$	$149$	$1$	$207.6$	Generator found by $12$-descent	$\frac{9834\cdot(11875366583346175045308849517719563546947872905)^{2}}{(113316948313836379630979456641974661571109077)^{2}}$
$67$	$1$	$109$	$1$	$210.4$	Generator found by $12$-descent	$\frac{7303\cdot(4806248515591894214558390288273183300072043058)^{2}}{(741817951830616322569413396183281471139846231)^{2}}$
$67$	$59$	$53$	$1$	$167.4$	Generator found by $12$-descent	$\frac{-3551\cdot(1257350173204775834467149354775468199)^{2}}{(275025406234649402419681991929187806)^{2}}$
$67$	$131$	$101$	$1$	$309.2$	Generator found by $12$-descent	$\frac{6767\cdot(149291284723636235859361462838539885889505562717236670728297282676622)^{2}}{(7824433831175789226647422526799670411545689928436971203757443741131)^{2}}$
$67$	$131$	$109$	$1$	$1692.7$	Heegner point method of Elkies	$\frac{131\cdot(389234382246741691512295824144776312547348969430015137787778955490735603642964615455301304214850198212137150523901724928693584774619005789712986895438586759974184243850737614257459628095288255945499690917601118686171608635892963742412571815274892621734502985904872802600476151506411399262100892403060234020553347148573528914682872035879910711261765337662966792789078775)^{2}}{(4290491832147440825463221413465165519291830538673265365231248526470874762107665983564649520418298702114982624413079977429194160025077778025642249780887365775485305939249055731273154047460031807467231949288893009812197126666437015517675219680901419211890435622949863135034591635397177988904826895874272924510534621473865542628592613550707456387569466338342609467868817)^{2}}$
$83$	$131$	$53$	$1$	$275.4$	Generator found by $12$-descent	$\frac{-4399\cdot(104542160754794385209479287007124768779354685111381314340421)^{2}}{(55688137642031069362449609538498451037829245426423061960058)^{2}}$
$83$	$131$	$101$	$1$	$297.3$	Generator found by $12$-descent	$\frac{8383\cdot(406036484273712871398034282100610068073953957967349415518147956862)^{2}}{(1797783354990246623906044634421746283966593556228345612429618051)^{2}}$
$87$	$67$	$89$	$1$	$168.9$	Generator found by $12$-descent	$\frac{7743\cdot(36361338701313278498207155775697115528)^{2}}{(2046002242334858420736663452028365323)^{2}}$
$131$	$107$	$149$	$1$	$740.1$	Generator found by $12$-descent	$\frac{-19519\cdot(12080135327110395633475467624916527664551423857735434278750163258733329846851955293626115779273025021846545745990338974837566099493198153158196366605837213609999)^{2}}{(4863906769868489171979169831319919577776186738431272649467856553730733611643891754336202525709351046930323998862504316204403804191334026430186185565134248872490)^{2}}$
$134$	$103$	$38$	$1$	$215.0$	Generator found by $12$-descent	$\frac{-103\cdot(52807710646670457937254031011497168157718229199)^{2}}{(5429238767645357073892731757050514694993498063)^{2}}$
$137$	$37$	$118$	$1$	$12.6$	Verified that $L'(E,1) \ne 0$. Analytic order of Sha is $16$.	$\frac{137 \cdot (509)^{2}}{(4)^{2}}$
$137$	$93$	$118$	$1$	$199.8$	Generator found by $12$-descent	$\frac{-16166\cdot(8589729076766496307168845597312916351256695)^{2}}{(2911548230640340326582690319638947183260091)^{2}}$
$137$	$101$	$118$	$1$	$11.5$	Verified that $L'(E,1) \ne 0$. Analytic order of Sha is $16$.	$\frac{-16166 \cdot (85)^{2}}{(37)^{2}}$
$139$	$37$	$149$	$2$	$19.3, 209.7$	Large height generator found via $12$-descent	$\frac{20711\cdot(3017021558615380446759125150515845058876860112)^{2}}{(435008295310643307693121524215059462939701215)^{2}}$	$\frac{-20711\cdot(12707)^{2}}{(2170)^{2}}$
$139$	$41$	$141$	$1$	$191.5$	Generator found by $12$-descent	$\frac{-19599\cdot(51560745051703086035321848639796184211145)^{2}}{(60213728891839122003517229541334763963158)^{2}}$
$139$	$67$	$149$	$1$	$1079.7$	Generator found by $12$-descent	$\frac{-20711\cdot(2401615883687082329458917767577674464355027916864324111377968309222317909434694455818252586818804529457022009033476957039652841223346611842213082699765688329048669070706612509435961139389873730004192838541377210267192383082141999813497)^{2}}{(293927652920529088347948257089327803176419165620306380966648895647754251404488320765406678970108968345517742087086480778614703443470783572928072035648982923058029951400470462266817633004637992091415778340727711073566652809824857397250)^{2}}$
$139$	$83$	$61$	$1$	$252.8$	Generator found by $12$-descent	$\frac{-8479\cdot(5351404319489261031602931529609733497750717094946934063)^{2}}{(756031503590795081890079204406384240488767365905363110)^{2}}$
$139$	$89$	$109$	$1$	$8.9$	Verified that $L'(E,1) \ne 0$. Analytic order of Sha is $64$.	$-(417)^{2}$
$139$	$107$	$53$	$1$	$414.7$	Generator found by $12$-descent	$\frac{-7367\cdot(10201052413152433748847363925053833645555803740652176526193230266233760081064660209617809)^{2}}{(107604291563185497813757430924130324689666689030078865458045617567058800464015520134664634)^{2}}$
$139$	$107$	$61$	$1$	$1189.8$	Generator found by $12$-descent	$\frac{-8479\cdot(1542976797264121368676627240515553875008375895558457529318893589206007812656851415332671864900866908952120358061379451101501751749633075920908481998014995436528109437980629134945551481130305020485946077574969498756571023563263076469676955341716417766202207523)^{2}}{(195386573974244898694815282113438126576733756551929646020017093465234954778904516255315135079782665807466990243374828554142388730959662931009886997767148676012291855270509163837513718526139585669684778444074390894421138375789004408000310723736468280841737870)^{2}}$
$139$	$107$	$101$	$1$	$1234.8$	Generator found by $12$-descent	$\frac{-14039\cdot(1851495003814952396170748013111562596773243635333732094045304936364204024771699817685899344357070026081546684140917639864285420264121088638374915071081104171900077680130663727293193658977408625742386370380174266457233791711876859424449919571577221879894513306180780551)^{2}}{(1301421700907620472195186706767123160765386052792830074898473660127022033620519108248695601721558440460038659205402810380712272366431377160452154080509901761390306792286757767705853327742255425490817505548283606527488586652394276919622062793983848905119654371888362290)^{2}}$
$139$	$107$	$149$	$1$	$344.4$	Generator found by $12$-descent	$\frac{-20711\cdot(434591749272223360724779541143652540136908822032911419364085915426823031367)^{2}}{(52966957777485298092255650355883279153830643472796270310036533563729013490)^{2}}$
$139$	$131$	$53$	$1$	$568.7$	Generator found by $12$-descent	$\frac{-7367\cdot(1444095820769351855779233877938766458809285941721259798341634873836250491180632312301458667826315363086272766055008247161189)^{2}}{(260451937827318693681705167157500760167651067851339795642879366646639129342858482825017338398040733609100197154215165087054)^{2}}$
$139$	$131$	$61$	$1$	$160.3$	Generator found by $12$-descent	$\frac{-8479\cdot(54481920001044334427930129629677065)^{2}}{(4784718638506573848135813974757802)^{2}}$
$139$	$131$	$101$	$1$	$1041.3$	Generator found by $12$-descent	$\frac{-14039\cdot(4564980122088413809080969939772778659282789168364528070725204680649733159483887618092278913033998796037999486509666238239326869890308330984265376012709826161689071854652260072696128170967841678388945148775595059705509165419207)^{2}}{(1117322900636229245651658306666857517110601759313423112938049610003264504092849578876263150375779308639221868217866758159114428534781514333155995245425126615858719902371352739814146261648511527513727317963460654592848290617690)^{2}}$
$142$	$103$	$79$	$1$	$15.5$	Verified that $L'(E,1) \ne 0$. Analytic order of Sha is $16$.	$2 \cdot (2753)^{2}$
$145$	$149$	$118$	$1$	$195.9$	Generator found by $12$-descent	$\frac{149\cdot(26662359272076288974462910653684425185890157)^{2}}{(1101055762657820447681878365608649344213)^{2}}$