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

I am trying to find the next rank 8 curve with the torsion subgroup Z/6 using Kihara's family as described in https://arxiv.org/pdf/1503.03667.pdf. Meanwhile, I came across a curve generated by $$t=629/3287$$ (or $$t=6202/8089$$, $$t=-8089/1772$$, $$t=-23009/1258$$).
Magma Calculator (http://magma.maths.usyd.edu.au/calc/) and mwrank return 6 generators for this curve.

SetClassGroupBounds("GRH");
E := EllipticCurve([1, 0, 1, -134523401167995213138670219183146040563810987418811883, 66402369909929526433604564866758135700820111823876373971833120805994125518227306]);
MordellWeilShaInformation(E);


Sagemath 8.4 returns 7 for the upper bound of analytic rank.

E = EllipticCurve([1,0,1,-134523401167995213138670219183146040563810987418811883,66402369909929526433604564866758135700820111823876373971833120805994125518227306])
E.analytic_rank_upper_bound(max_Delta=2.8,root_number="compute")


Is there a way to find one more generator?
A working piece of any code would be greatly appreciated.
Max

Yes. A 7th generator has $$x$$-coordinate $$181265389257356655988118224516379188326810855287159053664052560/3919647209484520988422390115383428889.$$ Knowing the $$6$$ generators Magma finds (let's say they are P1, P2, P3, P4, P5, P6), the Magma command

twocovers := TwoDescent(E : RemoveTorsion := true, RemoveGens := {P1,P2,P3,P4,P5,P6});


finds the unique $$2$$-cover which should have a rational point, but on which we haven't yet found one. Then, one can obtain $$4$$-covers for this via

fourcovers := FourDescent(twocovers : RemoveTorsion := true, RemoveGensEC := {P1,P2,P3,P4,P5,P6});


This command takes about 20 minutes to run on my machine, and it returns an intersection of two quadrics on which one needs to find a point. Point searching to a height of $$10^{8}$$ turns up a point, and this leads to a point P7 on $$E$$ with $$\hat{h}(P7) \approx 171.3$$. Subtracting off a linear combination of P1, P2, $$\ldots$$, P6 one obtains the point with $$x$$-coordinate above and canonical height $$\approx 94.34$$.

If this hadn't worked, a potentially feasible (but quite time-consuming) strategy is the following. Since $$E$$ has a rational point of order $$6$$, the image of the mod $$3$$ Galois representation is quite small and doing a $$3$$-descent is feasible. Magma has an implementation of Tom Fisher's algorithm (see his 2008 paper from the Journal of Algebra) of combining $$3$$-covers and $$4$$-covers to obtain $$12$$-covers, and point searching on those can yield rational points on $$E$$ with canonical height in excess of $$1000$$.

• Thank you! I played with the provided generator and found that the height could be dropped to ≈ 80.48 for x7 = 67791786397761343898028760401120208532053460906140337645553935/539679746694387917^2. Also, the shortest x7 = 178847722743710765459745080724093213046094425206654805/7597559985243^2 happens at height ≈ 94.14. Aug 18 '19 at 19:03
• My guess is that it is better (in general) not to use RemoveGens, as the height in a given coset might be much greater than in the minimal one. Jan 31 '20 at 4:41