# Resolved: Two more generators needed for a Z/6 elliptic curve

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) but our software chokes when we are trying to follow his instructions.

Max

• Doesn't Magma's TwoPowerIsogenyDescentRankBound (Fisher's code) prove at step 5 (just beyond 4-descent, but not yet 8-descent) the rank is at most 6 in about 20 seconds and 120 MB of memory? – MyNinthAccount Jan 31 at 4:44
• Thank you, MyNinthAccount! Resolved. – Maksym Voznyy Jan 31 at 11:41

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