We are searching for the rank $6$ elliptic curves with the torsion subgroup $\mathbb{Z}/8\mathbb{Z}$ using the families similar to Allan MacLeod's as described in



> 
> 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)
> 
> 
> 


and came across a curve



```
[1,0,0,-885470498073167713002317184212739304,315787360224933400897171748517120652503410335938363968]

```

[Magma Calculator](http://magma.maths.usyd.edu.au/calc/), Magma V2.24-7, and mwrank (with $-b14$) return $4$ generators for this curve, e.g.:



```
SetClassGroupBounds("GRH");
E:=EllipticCurve([1,0,0,-885470498073167713002317184212739304,315787360224933400897171748517120652503410335938363968]);
MordellWeilShaInformation(E);

Using model [ 1, 0, 0, -885470498073167713002317184212739304, 315787360224933400897171748517120652503410335938363968 ]
Torsion Subgroup = Z/8
The 2-Selmer group has rank 6
New point of infinite order (x = 291590635879268240110317582512/87995882881)
New point of infinite order (x = -780802002478160103839088529264/1152485984521)
New point of infinite order (x = 526632508060857475043443664/1355049721)
New point of infinite order (x = 25006056691267829443814801295058/11996130112849)
After 2-descent:
    4 <= Rank(E) <= 5
    Sha(E)[2] <= (Z/2)^1
(Searched up to height 10000 on the 2-coverings.)

```

Both Magma and mwrank return $5$ for the upper bound on rank of the curve and all the related 2-isogenies ($\mathbb{Z}/4\mathbb{Z}$, $\mathbb{Z}/2\mathbb{Z}\times\mathbb{Z}/4\mathbb{Z}$, and $\mathbb{Z}/2\mathbb{Z}\times\mathbb{Z}/2\mathbb{Z}$ curves below) using all their 2-torsion points:



```
E:=EllipticCurve([1,0,0,-885470498073167713002317184212739304,315787360224933400897171748517120652503410335938363968]);TwoPowerIsogenyDescentRankBound(E);print(" ");
E:=EllipticCurve([1,0,0,6205127968803058885033015264909364456,-3406906009463658062905156451961488907125318013801796912]);TwoPowerIsogenyDescentRankBound(E);print(" ");
E:=EllipticCurve([1,0,0,-1810312155105870840254262815766788584,-457633210238033749039428724550432643513221271485300416]);G,h:=TwoTorsionSubgroup(E);T:=[h(g):g in G];TwoPowerIsogenyDescentRankBound(E,T[2]);TwoPowerIsogenyDescentRankBound(E,T[3]);TwoPowerIsogenyDescentRankBound(E,T[4]);print(" ");
E:=EllipticCurve([1,0,0,-24623218791538050601572671001307730104,-47007276920642307027416764645331974815957764878067127376]);G,h:=TwoTorsionSubgroup(E);T:=[h(g):g in G];TwoPowerIsogenyDescentRankBound(E,T[2]);TwoPowerIsogenyDescentRankBound(E,T[3]);TwoPowerIsogenyDescentRankBound(E,T[4]);print(" ");

5 [ 3, 3, 3, 3, 3 ]
[ 4, 4, 4, 4, 4 ]

5 [ 7, 7, 7, 7, 7 ]
[ 0, 0, 0, 0, 0 ]

5 [ 4, 4, 4, 4, 4 ]
[ 3, 3, 3, 3, 3 ]
5 [ 0, 0, 0, 0, 0 ]
[ 7, 7, 7, 7, 7 ]
5 [ 0, 0, 0, 0, 0 ]
[ 7, 7, 7, 7, 7 ]

5 [ 0, 0, 0, 0, 0 ]
[ 11, 7, 7, 7, 7 ]
5 [ 0, 0, 0, 0, 0 ]
[ 9, 7, 7, 7, 7 ]
5 [ 7, 7, 7, 7, 7 ]
[ 0, 0, 0, 0, 0 ]

```

We tried implementing [Jeremy Rouse's](https://mathoverflow.net/questions/337621/one-more-generator-needed-for-a-z-6-elliptic-curve) and Zev Klagsbrun's ([1](https://mathoverflow.net/questions/352295/hard-one-more-generator-needed-for-a-z-6-elliptic-curve) and [2](https://mathoverflow.net/questions/353987/a-generator-needed-for-a-z-6-elliptic-curve)) approaches to employ $2$- and $4$-coverings, but could not find a rational point for the intersection of two quadrics. We used different seeds in Magma Calculator and a bound up to $3\times10^8$, e.g.:



```
SetSeed(1);
SetClassGroupBounds("GRH");
E := EllipticCurve([1,0,0,-885470498073167713002317184212739304,315787360224933400897171748517120652503410335938363968]);
P1 := E![86609552610643851795318648321889070494,113996061058226165174165649863299331386757385142,41549112479447364007];
P2 := E![72445938379247589758672,7465810298311395376082337884312,117649];
P3 := E![36603887306503756637336310506125004116,9298633886669630228894900541541155002386034788,53504769840978593717];
P4 := E![29317967055802030746557715723363588500973254368,22661934910027931993184701249992172746172677836253523176,25876196859106536169778781677];
twocovers := TwoDescent(E : RemoveTorsion := true, RemoveGens := {P1,P2,P3,P4}); twocovers;
fourcovers := FourDescent(twocovers[1] : RemoveTorsion := true, RemoveGensEC := {P1,P2,P3,P4}); fourcovers;
_,m := AssociatedEllipticCurve(fourcovers[1] : E := E);
pts := PointsQI(fourcovers[1], 3*10^8 : OnlyOne := true);
pts[1];
m(pts[1]);

[
    Hyperelliptic Curve defined by y^2 = 4401556531432641*x^4 +
        189490115768444190*x^3 + 5942548110808516561*x^2 -
        11718931267663428016*x + 9974519367938837056 over Rational Field
]
[
    Curve over Rational Field defined by
    93*$.1^2 + 505\*$.1*$.2 + 715\*$.1*$.3 + 2737\*$.1*$.4 + 182\*$.2^2 +
        916*$.2\*$.3 + 1233*$.2\*$.4 + 690*$.3^2 - 1248\*$.3*$.4 - 3453\*$.4^2,
    43659*$.1^2 - 100751\*$.1*$.2 + 100262\*$.1*$.3 + 267364\*$.1*$.4 + 57470\*$.2^2
        + 183818*$.2\*$.3 - 93739*$.2\*$.4 - 647944*$.3^2 + 666862\*$.3*$.4 -
 133155\*$.4^2
]

>> pts[1];
      ^
Runtime error in '[]': Illegal null sequence

```


> 
> **Question.** Considering parity, there should be one more generator on the curve. Is there a way to find it?
> 
> 
> 


We would greatly appreciate any hint leading to the discovery of the extra generator.


If you can compute an extra generator, your name will be published at the bottom of the $\mathbb{Z}/8\mathbb{Z}$ [rank 5 page](https://web.math.pmf.unizg.hr/%7Eduje/tors/z8old5.html).



Z/8Z elliptic curve with a missing generator
By pts := PointsQI(fourcovers[1], 3\*10^12 : OnlyOne := true);
I found the fifth point with the first coordinate



```
184125172284095573254772251800166095132866268069994053071269775232967258156536458883390620097481326304558948736/556947891689070160911189283508448977897416986918076204554508778044285190076205830897454330201.  

```

A slightly smaller point has the first coordinate



```
31207315512700919990861160205048461703358374278620437702635535252739296354266705346529925962/4817243728090299166338796143704937948812120388662168566804089530091841. 

```

