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](https://arxiv.org/pdf/1306.0027v1.pdf)
> 
> 
> 



> 
> 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](https://arxiv.org/abs/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](https://mathoverflow.net/questions/384358/z-8z-elliptic-curve-with-a-missing-generator)).
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](https://mathoverflow.net/questions/385814/3-6-12-descent-for-z2xz6-elliptic-curves)). (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](https://web.math.pmf.unizg.hr/%7Eduje/tors/z2z6old5.html) when the fifth generator is uncovered for the curve(s).



Z2xZ6 elliptic curves with missing generators
By implementing the techniques described in



> 
> T. Fisher, [Finding rational points on elliptic curves using 6-descent and 12-descent](https://www.dpmms.cam.ac.uk/%7Etaf1000/papers/sixandtwelve.pdf), *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](https://mega.nz/file/HsxiyZCD#v27IEn9KMvlWuXNnT4sOxMpq6ZBZZavhh_04dgXYLpY) (33 kB) and [example1.m](https://mega.nz/file/zgp2SBSK#N3AiSr-vO8BXvmM-U6gpzjpBLT03ZFHZg2IgPR7Cg0M) (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](https://web.math.pmf.unizg.hr/%7Eduje/tors/z2z6old5.html).


