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The analytic rank of the Mordell elliptic curve $y^2=x^3-86069^5$ indicates that it has rank 2. However, deriving a set of generators, and hence the regulator, is proving to be a little bit of an intractable problem.

I'm posting this question in the hope that someone may have investigated this curve already and, as such, may be able to help me out with some of the points on the curve.

Any help with this would be very much appreciated.

As further background, I have been involved in a small group searching for values of #Sha > 1 up through the ranks of Mordell curves, currently from ranks 0 to 9.

To date we have been able to progress to a rank 9 curve were #Sha would seem to be at least equal to 9. The details of which may be seen at the NMBRTHRY Archives.

Kevin.

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You might be able to prove that your elliptic curve has analytic rank exactly 2 by finding an exact formula for L(E,1). The point is that your elliptic curve has CM and for some CM elliptic curves like $x^3+y^3=p$ with $p$ prime, you have an effective lower bound on $L(E,1)$ when nonzero. See for example the article 'Which primes are sums of two cubes ?' by Rodriguez-Villegas and Zagier. –  François Brunault Aug 31 '12 at 19:35
    
Why this particular curve? Do we already have generators for all curves $y^2 = x^3 \pm p^5$ with prime $p<86069$ for which the analytic rank exceeds $1$? –  Noam D. Elkies Sep 1 '12 at 4:37
1  
@Kevin thanks, but I still wonder why these particular curves. If the $k=443^5$ curve would do as well then it should be tried first because the generators ought to be be much more accessible: the real period is proportional to $k^{-1/6}$, so if all else that goes into BSD is about the same (conjecturally $L^*(E,1)\sim k^{\pm\epsilon}$) then the regulator should grow as about $p^{5/6}$, and (for rank $2$) the generators' heights as $p^{5/12}$; since $(86069/443)^{5/12}$ is about $9$, this would reduce the heights from about $10^3$ to $10^2$ which should be feasible with present descent tools. –  Noam D. Elkies Sep 2 '12 at 2:26
7  
Well mwrank does only 2-descents, which usually aren't enough to bring a point of height ~100 within reach. But (following J.Weigandt's magma computation) here the ConjecturalRegulator is only 2612.64 (those $k^{\pm\epsilon}$ factors can make a big difference in practice), so you can expect to get at least one point. Meanwhile I ran FourDescent and PointsQI, and got $(46160353/42^2, 438188195663/42^3)$, which mwrank may have found by now (height $18.29$), and ... –  Noam D. Elkies Sep 2 '12 at 3:09
7  
... a second generator with $x$-coordinate $N/D^2$ where $$N = -18508992109435642648272765287494073191969714723796327, $$ $$ D = 857276411592231840843144 $$ (height $120.9$), which mwrank surely could not find in reasonable time. –  Noam D. Elkies Sep 2 '12 at 3:11
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3 Answers

up vote 23 down vote accepted

This particular curve, which I'll call $E$, may be quite challenging. The analytic rank is probably 2, but as far as I know, the only way to prove this is to show that the algebraic rank is not 0. Assuming the analytic rank is 2 and the full BSD formula, the product of the Regulator and the order of the Tate-Shafarevich group is approximately 1435241.110225344. (Using the command ConjecturalRegulator(E); in MAGMA).

Since the the rank is probably 2, I would guess that the Shafarevich-Tate group is trivial, so that the regulator is quite large.

In this case a 4-Descent was feasible. Below are the genus 1 curves in $\Bbb P^3$ that represent the elements of the $4$-Selmer group modulo torsion. If we can find points on these curves, they will map to points of infinite order on $E$.

> E := EllipticCurve([0,-86069^5]); 
> ConjecturalRegulator(E); 
1435241.11022534407264592039437 2
> TD := TwoDescent(E); 
> SetClassGroupBounds("GRH"); 
> time FD := [FourDescent(C) : C in TD]; 
Time: 3.890
> FD;
[
    [
        Curve over Rational Field defined by
        15*x1^2 + 111*x1*x2 + 4*x1*x3 + 10*x1*x4 - 38*x2^2 - 95*x2*x3 + 67*x2*x4
            - 54*x3^2 - 14*x3*x4 - 71*x4^2,
        31*x1^2 - 71*x1*x2 - 69*x1*x3 - 23*x1*x4 + 9*x2^2 - 92*x2*x3 - 24*x2*x4 
            + 35*x3^2 - 148*x3*x4 + 168*x4^2,
        Curve over Rational Field defined by
        35*x1^2 + 26*x1*x2 + 41*x1*x3 + 61*x1*x4 + 54*x2^2 + 11*x2*x3 + 25*x2*x4
            + 68*x3^2 + 3*x3*x4 - 78*x4^2,
        36*x1^2 - 138*x1*x2 - 38*x1*x3 + 127*x1*x4 + 25*x2^2 + 42*x2*x3 + 
            47*x2*x4 + 81*x3^2 - 12*x3*x4 + 60*x4^2
    ],
    [
        Curve over Rational Field defined by
        21*x1^2 + 13*x1*x2 + 13*x1*x3 + 44*x1*x4 - 32*x2^2 - x2*x3 - 18*x2*x4 - 
            45*x3^2 + 24*x3*x4 - 238*x4^2,
        5*x1^2 - 14*x1*x2 + 122*x1*x3 + 268*x1*x4 + 26*x2^2 + 6*x2*x3 + 
            149*x2*x4 - 57*x3^2 - 23*x3*x4 - 78*x4^2,
        Curve over Rational Field defined by
        4*x1^2 + 49*x1*x2 + 34*x1*x3 + 26*x1*x4 + 26*x2^2 - 33*x2*x3 - 74*x2*x4 
            + 53*x3^2 - 74*x3*x4 + 111*x4^2,
        38*x1^2 - 84*x1*x2 + 3*x1*x3 - 88*x1*x4 - 29*x2^2 + 27*x2*x3 - 154*x2*x4
            + 5*x3^2 - 234*x3*x4 - 120*x4^2
    ],
    [
        Curve over Rational Field defined by
        7*x1^2 + 78*x1*x2 + 106*x1*x3 + 62*x1*x4 - 21*x2^2 - 26*x2*x3 + 22*x2*x4
            + 34*x3^2 - 25*x3*x4 - 118*x4^2,
        33*x1^2 + 2*x1*x2 - 14*x1*x3 + 106*x1*x4 + 48*x2^2 - 33*x2*x3 + 
            165*x2*x4 + 69*x3^2 + 31*x3*x4 - 26*x4^2,
        Curve over Rational Field defined by
        7*x1^2 + 46*x1*x2 + 33*x1*x3 + 23*x1*x4 + 13*x2^2 + 36*x2*x3 - 108*x2*x4
            - 69*x3^2 - 88*x3*x4 + 145*x4^2,
        19*x1^2 - 28*x1*x2 - 14*x1*x3 + 8*x1*x4 + 150*x2^2 - 52*x2*x3 + 
            190*x2*x4 - 46*x3^2 + 33*x3*x4 + 248*x4^2
    ]
]

I searched for points on these curves up to height $10^9$ and didn't find any, which isn't that surprising given that the regulator is probably so large. Maybe an $8$- $9$- or $12$-descent would help, but I'm not sure since the regulator looks to be so large.

I've come across a lot of curves like this one, and honestly I don't know what to do other than spend a great deal of time implementing higher and higher descents, which will require better and better architecture for working with algebraic number fields.

Finding points on curves with rank at least 2 is much harder than the rank 1 case, where a non-torsion Heegner point can be constructed. I've often wondered if the notion of visibility of Mordell-Weil groups could be useful here to prove that the rank is 2. There might be some hope since this curve is just a sextic twist of a very simple elliptic curve, but I have no idea what other abelian variety one would try to use to "visualize" the Mordell-Weil group.

EDIT: Found a rational point!

Since this is a Mordell curve, it has a 3-isogeny. So ThreeDescentByIsogeny(E); works in magma. This, together with the 4-Descent above, can be patched together with Tom Fisher's 12-Descent code. Running PointSearch(C, 10^15) on all the 12-covers, I found one rational point on a 12-cover. This maps back to the HUGE point $P = (r/t^2, s/t^3) \in E(\Bbb Q)$ where

$$t = 1064315237527062416197829497356636659645269584461099593669088031745089037672135800872679057089531240249644479256790731579010363138838881840905955995370777995601208434305913644468575606,$$

$$r = 280330358182289626756155234063598323321079390462341827366518987565228009421332474964681906370265513552399459534408574388270067641670339702167066942312699829200035334349465315027488431491497462250163493458119988220147933027219004129214673758569410719992144923411319943555067857245550021843686621974171705862665984304126540657354963027685841117202627482318291749321065520239017,$$

and

$$s = -3894169495021322706664690332034015776766200723677334488117565017206832575610394605401029654682981124772659459298812689312208671041841435640288530899795853801784491292694434680924857836480106291120205903428248497270005655966231499766892812223437056078000735443855989892506051021805114412318184420256429821942583648074219767873044937972437645324252633026228735728785740079828040374108087622426272823877167554852034435686184476368398239806496596492675365799418315372050294056476779196364271993469777717240193414071588739725730054647942389914059081838403974737198837$$

Magma (and sage) says the canonical height of $P$ is 862.589016739449. $P$ has infinite order and is not a multiple of another point.

I'll keep looking for more points on 12-covers, but there should be a smarter way to look for a second generator since we know conjecturally what the regulator is supposed to be, at least up to division by an integer square (the order of Sha).

Second Edit: Second generator found by Tom Fisher!

I received email from Tom Fisher who got wind of this example thanks to Kevin Acres and Mark Watkins. He used some more careful minimization techniques for 12-covers which made it much easier for him to find the second generator $Q = (u/w^2, v/w^3)$ where

$$w = 182114807484521200807106885195433046250677033531559359037472093313104122418454868285361195199297225042175078422885282555317033956772998359062866824462985656413093380882725866443379879289096885109176415930832786071510257551923931848041894889001647111552913336376307990237676516910188887260245394301062683134535223719019999940520792408074037957510825778941886955164545102$$

$$u = 6409818948420000148009253515371033320765674658334995509069337486116156301741657451140832523716129416946553724548364793698626405777843956622076380997023235142487299844326278568312577933328949209325079504842164930964499614652180484249884310193722360960206856049327696413372088438499522988880226096525828673972002212550418445757426791463589511241682169507191857404060127705996714164059762587639453970388047710151991644286292989545241436751637355865317439825495449344823977758797766025325026099986880721224779341838599953004314077314458400203457162779096001177667625960285991372461677348744634118572760595736047817251613300845834732694598202906328385669708044981270667425075403152129513088853355196780998723571290752400352784381236905647783762282233$$

$$v = 9541721267275526440336656075081968921795973155189044266523956895286489666750185996900611973660561530367870689843326743006933458752307224811373451969681421106849987021781748740069623362388923316032840282833074591791294185661671902447470420594105932470690611184875968605655279379274234568020070054315280341961704988604471445169462057158108739193383808091705876229256447451432215635271166605138776098284834213663487614216411244364417456782497135614069447487775516105502956819954881548171764933506408613533330700718054166278531808088754093650937927404096948372079871885150489168059803000137727292308698902319300888120656605108547686297443217509881949442132801485412503095358513515604711600740777338910658971774981045705755311582771691405245163713860189075823123694400673272454942433819801867842344237200886745700934179591344744409814700762619336056693409131813436072830610006822421568399180878828865624130779460438617259828853398661927306856293129558704629256045989464076110208592569978237366668166654822438426674566555490387538893656476099175185704061248678944708728574308512767720849000403950411614279621166869036373101$$

This height of $Q$ is 1715.46805605884712533431816634 and the regulator is actually 1435241.11022534407264592039437 as BSD predicts if Sha(E/Q) is trivial.

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An 8-descent took about 1500 seconds in Magma with class group bounds set to GRH. I searched up to height 10^12 and the 8-covers (this took about another 1000 seconds) and didn't find anything yet, but the coefficients of the 8-covers are very very large. –  Jamie Weigandt Aug 27 '12 at 2:25
1  
@Kevin - I am interested in the long term project of implementing higher descents in Sage. Jeechul Woo has some GP scripts for descent via 3-isogeny, which is applicable to Mordell curves. –  Jamie Weigandt Sep 6 '12 at 1:21
1  
Congratulations! Not a record because Heegner-point techniques go well beyond descent when they're available, but possibly a record or near-record height for a curve of rank $>1$. $$ $$ My point-search techniques aren't that abstruse -- basically all I'm using (besides LLL) is that the image of an $\epsilon$-interval under an analytic map is contained in a box of size $\ll\epsilon\times\epsilon^2\times\epsilon^3\times\cdots$. Descent technology is much more complicated. (And Jeechul Woo's $3$-descent formulas won't help here: they need a rational $3$-torsion point, not just a $3$-isogeny.) –  Noam D. Elkies Sep 6 '12 at 4:44
1  
UPDATE: PointSearch(C,10^24) for all 12-covers C took about 3 days and found no new points. I'll try to run something bigger on my department's servers. –  Jamie Weigandt Sep 10 '12 at 14:40
1  
UPDATE: PointSearch(C, 10^27) for all 12-covers C took about 11 days. Nothing to report. Trying 10^30. –  Jamie Weigandt Sep 24 '12 at 22:46
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Tom A. Fisher has kindly identified the second generator for this curve, at a height of $1715.46805605884712533431816634$, and with a regulator of $1435241.11022534407264592039437$.

Many thanks to Tom for this impressive workout of his 12 descent code.

P2:= E![6409818948420000148009253515371033320765674658334995\ 509069337486116156301741657451140832523716129416946553724548\ 364793698626405777843956622076380997023235142487299844326278\ 568312577933328949209325079504842164930964499614652180484249\ 884310193722360960206856049327696413372088438499522988880226\ 096525828673972002212550418445757426791463589511241682169507\ 191857404060127705996714164059762587639453970388047710151991\ 644286292989545241436751637355865317439825495449344823977758\ 797766025325026099986880721224779341838599953004314077314458\ 400203457162779096001177667625960285991372461677348744634118\ 572760595736047817251613300845834732694598202906328385669708\ 044981270667425075403152129513088853355196780998723571290752\ 400352784381236905647783762282233/33165803105124219179549823\ 283035686736456675469805935767692444023551658331570872880192\ 989613282702564119725223095907264020401187622726619827641022\ 070312051693797896281152446136738277971480729817095517412821\ 500028443753503375569153920181215330943019021452618523429306\ 154890615692258509116181263907011892227215259838198394721409\ 844161749103601296315808769202168882630133040496609244967027\ 573328564357038779635203477380317914882271609699042932772428\ 958728775525114597825318245131110925771677312456670934287090\ 438610192382893431121092876711361026457071789640268657157836\ 261872615922281539370618784278612205624356072354848973487567\ 580677872395212361402429095761079520397611920420171112702594\ 975559992623826579984003799616507192963910592190404, 9541721\ 267275526440336656075081968921795973155189044266523956895286\ 489666750185996900611973660561530367870689843326743006933458\ 752307224811373451969681421106849987021781748740069623362388\ 923316032840282833074591791294185661671902447470420594105932\ 470690611184875968605655279379274234568020070054315280341961\ 704988604471445169462057158108739193383808091705876229256447\ 451432215635271166605138776098284834213663487614216411244364\ 417456782497135614069447487775516105502956819954881548171764\ 933506408613533330700718054166278531808088754093650937927404\ 096948372079871885150489168059803000137727292308698902319300\ 888120656605108547686297443217509881949442132801485412503095\ 358513515604711600740777338910658971774981045705755311582771\ 691405245163713860189075823123694400673272454942433819801867\ 842344237200886745700934179591344744409814700762619336056693\ 409131813436072830610006822421568399180878828865624130779460\ 438617259828853398661927306856293129558704629256045989464076\ 110208592569978237366668166654822438426674566555490387538893\ 656476099175185704061248678944708728574308512767720849000403\ 950411614279621166869036373101/60399838475592326331362752791\ 944763132549609694041832059854653129731539771658328168942361\ 906568116204357279114901731638612673017179392637587993370268\ 468122751057317804766338855060441224492531078338764563745984\ 361634534757917619567424588718030226164336198810659807830928\ 894950054794919605549933531685874245309484554173084344795097\ 778895379979816703495608964775949909264616094661840599063186\ 011543131081083532339588386281198484224591916139857808728711\ 337072945237794430249025950156217838942697194548707342214906\ 621985660080150918372835213775235759078837614090802954692490\ 955558620434772673319859076612900166605163593210918785877475\ 268380688613826217264936422758722358549436515729050089664820\ 517090881544523313666983596465352462319026753145000093740537\ 178177489596826835806901560570709618578068241711965931639634\ 809586146278457253570314819921037995164438751876277090234639\ 821432381946017446571917916673348009411849176259307380401023\ 993294177103548682416638846438507436013936091382661874934845\ 500838177629920215935612770414110473755964022901968320874940\ 92431878821676880656612635392756954758754490670429601208 ];

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Take a look at this question and answers. I am conjecturing that Cremona's mwrank (which, I believe, is available in sage) will find what you are looking for.

EDIT Yes, available in sage.

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Hi Igor, thanks for the link. Unfortunately this particular curve seems to be out of the range achievable (in any reasonable time) by mwrank, or even findinf. Kevin. –  Kevin Acres Aug 26 '12 at 22:52
    
@Kevin: very sad... –  Igor Rivin Aug 26 '12 at 23:26
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