# Hard: One more generator needed for a Z/6 elliptic curve

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

A. Dujella, J.C. Peral, P. Tadić, Elliptic curves with torsion group $$\mathbb{Z}/6\mathbb{Z}$$, Glas. Mat. Ser. III 51 (2016), 321-333 doi:10.3336/gm.51.2.03, 1503.03667

and came across a curve

[1,0,1,-728177856117250596635013323700992100749546784263413,7725511368374502384905062271934799362136250437256099440874528514783779254988]


Both Magma Calculator and mwrank return $$7$$ generators for this curve:

SetClassGroupBounds("GRH");
E:=EllipticCurve([1,0,1,-728177856117250596635013323700992100749546784263413,7725511368374502384905062271934799362136250437256099440874528514783779254988]);
MordellWeilShaInformation(E);


Both Magma and mwrank return $$8$$ for the upper bound on rank:

E:=EllipticCurve([1,0,1,-728177856117250596635013323700992100749546784263413,7725511368374502384905062271934799362136250437256099440874528514783779254988]);
TwoPowerIsogenyDescentRankBound(E);

8 [ 4, 4, 4, 4, 4 ]
[ 6, 6, 6, 6, 6 ]

mwrank -v0 -p200 -s
[1,0,1,-728177856117250596635013323700992100749546784263413,7725511368374502384905062271934799362136250437256099440874528514783779254988]

Version compiled on Oct 29 2018 at 22:35:09 by GCC 7.3.0
using NTL bigints and NTL real and complex multiprecision floating point
Enter curve: [1,0,1,-728177856117250596635013323700992100749546784263413,7725511368374502384905062271934799362136250437256099440874528514783779254988]
Curve [1,0,1,-728177856117250596635013323700992100749546784263413,7725511368374502384905062271934799362136250437256099440874528514783779254988] :       selmer-rank = 9
upper bound on rank = 8


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.

A bounty of $$100$$ will be awarded soon for obtaining it.

Also, if you can compute an extra generator, your name will be published at the bottom of the page here: https://web.math.pmf.unizg.hr/~duje/tors/z6.html