By implementing the techniques described in and similar to
A. Dujella, J. C. Peral, Elliptic curves with torsion group Z/8Z or Z/2Z x Z/6Z, arXiv, Number Theory [math.NT] (2013), arXiv:1306.0027v1
A. J. MacLeod, A Simple Method for High-Rank Families of Elliptic Curves with Specified Torsion, arXiv, Number Theory [math.NT] (2014), arXiv:1410.1662v1
we found two $\mathbb{Z}/2\mathbb{Z}\times\mathbb{Z}/6\mathbb{Z}$ elliptic curves with the suspected rank $5$. Magma or mwrank can uncover $4$ generators for each curve.
[1,0,0,-14578233419504842866768626074144915305633167440,677262813789165866285590491715440177836227774714086370178932369721600]
(-1632905513070906593633395/16 : 2124657596658008539974203142878778065/64 : 1)
(10622742675684263293819723408153580/141909670681 : 124725874840414877086253409903032599643909082039040/53458650132568829 : 1)
(-97457668455374061775698849271818397251850996320108070/699579678211711463845331372281 : 1251876399993507238708312434030333353053490882764701323776880839986798621571460/585134598109537035210690491594808002277939571 : 1)
(49337296288524457271016713026090973088733461258708686528/509102607359595399318340821876529 : 151796511564094491944492436957734460721901141802177579465531284308649800667607130768/11487035994099737573838589665510598244834003265833 : 1)
[1,0,1,-737894644327026219218841358570509913846652580014,190908291625785611127850212005200136993025927512858679851032108028536736]
Generator 1 is [-238715637021392602565547:-594518321028603238275438200237984267:1];
Generator 2 is [31360030622781021392151674068077015054:10671104634442837701763318571303490297384474168604:41410962494173];
Generator 3 is [1458463852616891911932692956390480548:-2127910767977542086809771217107258154280506806417:8268141253861];
Generator 4 is [-1269285019645294204766876727074196:-936846131419637244134703316207143480358743016:1672446203];
Each curve has $7$ isogenous curves: three $2$-isogenous $\mathbb{Z}/6\mathbb{Z}$, one $3$-isogenous $\mathbb{Z}/2\mathbb{Z}\times\mathbb{Z}/2\mathbb{Z}$, and three $3$-isogenous $\mathbb{Z}/2\mathbb{Z}$ curves each.
[ 1, 0, 0, -14578233419504842866768626074144915305633167440, 677262813789165866285590491715440177836227774714086370178932369721600 ]
[ 1, 0, 0, -16788121482662999494757620303458923799396647440, 458347722412840961402654372801843479314855058739320714200591470465600 ]
[ 1, 0, 0, -12390456997202053746439995758428243843549687440, 887546443368809714827401664475279559448327008402628774422894004977600 ]
[ 1, 0, 0, -14576851441951382397539853329545081741153167440, 677397680379739007790920569085466233004214499673105145976879825721600 ]
[ 1, 0, 0, -41980864276959314492810225577242671428565265440, -2468181464491007069038907228919975481228430981974833640494773900794000 ]
[ 1, 0, 0, -623018003626406527989379968810953241503944571740, -189260266529123210476476660352196009349286414306954926139458416124443260 ]
[ 1, 0, 0, 103607077681372989451472258964699809067497272860, -15814718849970915410189591344868440743288201743513473235175995725111140 ]
[ 1, 0, 0, -14765289440014632645792271909007165829857967440, 658984511291994677032788803916334626722274364249125813187527019161600 ]
[ 1, 0, 1, -737894644327026219218841358570509913846652580014, 190908291625785611127850212005200136993025927512858679851032108028536736 ]
[ 1, 0, 1, -241916501236723307413266334879564479597172572014, -43233279468511210455812058234371488153394628551933612572678486665396064 ]
[ 1, 0, 1, 1666422596383330350336832332119338002274496477386, 1176171118768273493852310932836981453901128598294298129440036617833140096 ]
[ 1, 0, 1, -11077862174482229377663715428315484777959481765414, 14190706014518294309757776770418575190667486067178220709161242656555664576 ]
[ 1, 0, 1, -19291288676001541738319227851451141863129028466429, -32597782170371077908961912239683117512235494535789630982779370974269425448 ]
[ 1, 0, 1, -19288797687595273046405834919359334932574148466429, -32606625123070618542609678065286321263560745277882760553634793344061425448 ]
[ 1, 0, 1, -15915357490227593061488748204997524479146121906429, -44371276360828247024588958692803998413327921261654610033506325574902129448 ]
[ 1, 0, 1, -22707075676275789485763994411373670135990015026429, -20258339007143308239877852937993242901252252662392631030077662487428721448 ]
All the attempts to uncover the fifth missing generators for the curves in question and all their respective isogenous curves are unsuccessful yet. The methods that were tried so far:
- A full run of
MordellWeilShaInformation
in the properly installed Magma V2.26-2 in Ubuntu 20.04 Linux with 16 Gb of RAM. (Note: a $\mathbb{Z}/2\mathbb{Z}$ curve[ 1, 0, 0, -623018..., -189260... ]
crashes Magma currently, a bug report is filed). - Mwrank (Jul 28 2020) runs with bounds $-b14$.
- Magma script searches on the $2$- and $4$-coverings with the bounds $2^{40}$. Only the $2$- and $4$-coverings where all known generators were excluded, were considered (as described in the question here).
- Magma script searches on the $6$- and $12$-coverings. Only the $6$- and $12$-coverings where all known generators were excluded, were considered (as described in the answer here). (Note: a random $3$-covering for the element
S3toA(0)
$=<1,1,1>$ produced byThreeDescentCubic(E, <1,1,1>)
was used to create $6$- and $12$-coverings, a full run ofThreeSelmerGroup(E)
was not performed as it would take too long).
// S3, S3toA := ThreeSelmerGroup(E); S3; S3toA; S3toA(0);
threecovers_1, maps3toE := ThreeDescentCubic(E, <1,1,1>); threecovers_1; maps3toE;
Question. We would greatly appreciate any hint leading to the discovery of the fifth generator(s) for the curve(s) in question or the respective isogenous curves.
Your name will be published at the bottom of Andrej Dujella's page when the fifth generator is uncovered for the curve(s).