Z2xZ6 elliptic curves with missing generators

Question

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:

1. 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).
2. Mwrank (Jul 28 2020) runs with bounds $-b14$.
3. 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).
4. 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 by ThreeDescentCubic(E, <1,1,1>) was used to create $6$- and $12$-coverings, a full run of ThreeSelmerGroup(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).

Answer 1

By implementing the techniques described in

T. Fisher, Finding rational points on elliptic curves using 6-descent and 12-descent, Journal of Algebra 320 (2008), no. 2, 853-884,

Tom Fisher himself used $12$-descent to find the missing generators for both mentioned curves. The Magma files are available for download: example2.m (33 kB) and example1.m (30 kB). The fifth generators have heights $446.24$ and $375.99$.

He added, "As I indicate in the attached files, it turns out that these points could in principle have been found using $4$-descent, but not in practice since the product of Tamagawa numbers is so large. This is typically the case for curves in these torsion families.

The reason I can use $12$-descent on these curves is that the rational $3$-torsion point makes the $3$-descent practical (although not using Magma's ThreeSelmerGroup, which was never designed to take advantage of this extra structure)."

EDIT: Both curves were added to Andrej Dujella's $\mathbb{Z}/2\mathbb{Z}\times\mathbb{Z}/6\mathbb{Z}$ rank $5$ page.