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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

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    $\begingroup$ 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? $\endgroup$
    – MyNinthAccount
    Commented Jan 31, 2020 at 4:44
  • $\begingroup$ Thank you, MyNinthAccount! Resolved. $\endgroup$
    – Maksym Voznyy
    Commented Jan 31, 2020 at 11:41

1 Answer 1

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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
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