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.

E := EllipticCurve([1, 0, 1, -134523401167995213138670219183146040563810987418811883, 66402369909929526433604564866758135700820111823876373971833120805994125518227306]);

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

E = EllipticCurve([1,0,1,-134523401167995213138670219183146040563810987418811883,66402369909929526433604564866758135700820111823876373971833120805994125518227306])  

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


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[1] : 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 $1000$.

  • $\begingroup$ 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. $\endgroup$ Aug 18 '19 at 19:03
  • 1
    $\begingroup$ 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. $\endgroup$ Jan 31 '20 at 4:41

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