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

Question

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 $100$ has been offered 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

Answer 1

A set of points that generate $E(\mathbb{Q})$ modulo torsion is given by

(1955516573881233507049678279 : -86467145649172260650105545143411861089140 : 1),

(49225691888888099223656060329/10201 : 67749663895993353685065159554645568700902610/1030301 : 1),

(61339810590192565389735634 : -440289331793622522908840423931186017125 : 1),

(301884243790342804873202050999/1681 : 164095919303197903219089875947912899634054060/68921 : 1),

(12495717670305680867142229 : -24031745881863415519418908823242701040 : 1),

(48812081421189741670987918753619270029/14228919471376 : -3895612939954697213016286372117889003488190324193605593985/53673248632044722624 : 1),

(5561842419887590167868100830494509281/162696869449 : 9905381606012663087305509196041719017978015930195439090/65624921170340293 : 1),

(-24644413733187137559835573003063695698428162289232517969749039/810893447144357785058346728220801409 : 30847724470076383865716266151756242512110696731502256770076024073253839003102120576612459770/730206486187013450403786627354716551758061149557632577 : 1)

My guess is that you were missing the final one.

We can find the final point by applying $4$-descent to the $2$-covering $C_2$ given by

$$y^2 = 3600489235862039958255625x^4 - 26108156374576368607091450x^3 + 135553629286468859778411799x^2 + 184563701310722380421312754x + 49111306298020667812024521$$

of $E$, which is one of the 511 $2$-coverings returned by running the command TwoDescent(E) magma. Running the command FourDescent(C2) returns 256 $4$-coverings of $E$. One of them is the curve $C_4$ defined by the intersection of the quadrics

$$46500x^2 + 74693xy + 54170xz + 647076xw - 121026y^2 - 196538yz + 862965yw - 212375z^2 - 238791zw + 333744w^2$$

and

$$722768x^2 - 2936122xy + 3336517xz - 2782182xw - 2731148y^2 - 13360024yz - 950117yw - 4385375z^2 + 2688700zw - 199207w^2$$

in $\mathbb{P}^3(\mathbb{Q})$. Running PointsQI(C4,2^11) returns a single point $Q = (2834:53:2444:376)$. We can map $Q$ to a rational point on $E$ using the map given as the second return value of the command AssociatedEllipticCurve(C4:E:=E), and the point we get is the final point above.