We are searching for the rank 8 elliptic curves with the torsion subgroup Z/6 using newly discovered families similar to Kihara's (Kihara's family is described in <https://arxiv.org/pdf/1503.03667.pdf>).


Today we came across a curve


$[0,8169768624655967629114128598,0,-451787550647310420612086468536366715869054405951830599,0]$


Both Magma Calculator (<http://magma.maths.usyd.edu.au/calc/>) and mwrank with $-b12$ return 6 generators for this curve.
Magma V2.20-10 (STUDENT) runs out of memory running the following code:



```
SetClassGroupBounds("GRH");
E := EllipticCurve([0,8169768624655967629114128598,0,-451787550647310420612086468536366715869054405951830599,0]);
MordellWeilShaInformation(E);

```

Sagemath returns $8$ for the upper bound of analytic rank, even for max\_Delta=$3.3$ (we are still testing for higher max\_Delta):



```
E = EllipticCurve([0,8169768624655967629114128598,0,-451787550647310420612086468536366715869054405951830599,0])  
E.analytic_rank_upper_bound(max_Delta=3.3,root_number="compute")

```

Is there a way to find two more generators?


A similar question for the $6$ <= Rank(E) <= $7$ situation was successfully resolved by Jeremy Rouse ([One more generator needed for a Z/6 elliptic curve](https://mathoverflow.net/questions/337621/one-more-generator-needed-for-a-z-6-elliptic-curve)) but our software chokes when we are trying to follow his instructions.


Max



Resolved: Two more generators needed for a Z/6 elliptic curve
Just to give a more complete answer:



```
SetVerbose("cbrank",1);
E := EllipticCurve([0,8169768624655967629114128598,0,\
    -451787550647310420612086468536366715869054405951830599,0]);
TwoPowerIsogenyDescentRankBound(E);

  /---------------------------------------------------\
  |   SUMMARY TABLE      Step No :  6  5  4  3  2  1  |
  |---------------------------------------------------|
  |  dim_2 ( E'(Q)/phi E(Q) )   <=     4  5  5  5  5  |
  |  dim_2 ( E(Q)/phi'E'(Q) )   <=     4  5  5  5  5  |
  |     Therefore rank E(Q) <= 6                      |
  \---------------------------------------------------/

Total time: 22.980 seconds, Total memory usage: 96.16MB

```

