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I need to calculate rank of the some elliptic curves,(espicially getting generators or finding a rational point on the elliptic curves) but I cannot do this by my computer.

I am interested in calculating the rank of the following elliptic curves.

The elliptic curve is $$ Y^2=X^3 - (3h^2)X^2 + 3h(h^3-h)X -(h^3-h)^2 $$

for $h=967, 1198, 1787, 1987$,

$h=2459, 2572, 2711, 2797, 2971, 4999$

please if possible find a generator for some of these elliptic curves.

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    $\begingroup$ try cloud.sagemath.com (examples) $\endgroup$
    – reuns
    Commented Jan 2, 2017 at 22:01
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    $\begingroup$ The standard Sage implementation relies on Cremona's mwrank, and mwrank "searches" for appropriate two-covers, rather than directly computing them. I think Magma is better for things like this, and the online calculator may handle most of the OP's cases. $\endgroup$
    – Jeremy Rouse
    Commented Jan 2, 2017 at 23:10

2 Answers 2

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The following results were found using a mixture of my own codes and Denis Simon's excellent ellrank code.

The curves have trivial torsion so Mwrank might take a long time - I haven't tried. The two heights for the rank 1 curves are for the two possible height normalisations.

h=967 Rank=1 generator=[238501273696/245025, 900632541139856/121287375]

h=1198 Rank=0

h=1787 Rank=0 or 2

h=1987 Rank=0 or 2

h=2459 Rank=1 Height=37.4/74.8

h=2572 Rank=2 generators= [60035809/9, 302757191/27] and [3435573760731933430513/381659437643236, 27488556048550361767336062809879/7456139229698648679016]

h=2711 Rank =0

h=2797 Rank=1 Height=28.1/56.2

h=2971 Rank=0

h=4999 Rank=1 Height=29.7/59.4

If I get peace from my grandchildren I might try the three rank 1 curves!!

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  • $\begingroup$ What that means rank 0 OR 2 ? $\endgroup$
    – Ofra
    Commented Jan 3, 2017 at 19:56
  • $\begingroup$ Means that the rank is even and 0<=rank<=2, based on output from ellrank. $\endgroup$
    – Allan MacLeod
    Commented Jan 3, 2017 at 20:08
  • $\begingroup$ . . . and assuming the parity conjecture. Presumably conductor much too large for Heegner-point constructions in the rank-1 cases. $\endgroup$
    – Noam D. Elkies
    Commented Jan 3, 2017 at 20:26
  • $\begingroup$ ِDear Prof Allan Mac Leod; $\endgroup$
    – mehdi baghalaghdam
    Commented Jan 3, 2017 at 23:05
  • $\begingroup$ ِDear Prof Allan Mac Leod; Thank you very so much for your times and your reply and your support..Thank you..I am very glad for these results; Please, if possible, writte the generetors for each case that rank is positive. Please ,if possible, find the generators of the other elliptic curves for me..I need generators of these elliptic curves when rank is positive.. Thank you so much for your times ..sincerely yours.mehdi baghalaghdam $\endgroup$
    – mehdi baghalaghdam
    Commented Jan 3, 2017 at 23:23
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The following finish off the results:

(a) h = 2459

x = 2455940168334175449299068876662469864/403764781843031846693075441721

Conductor = 329061105621720

(b) h = 2797

x = 18256234369/2304

Conductor = 550823321110248

(c) h = 4999

x = 38932053386017900293094583125/1502165941669975655844

Conductor = 401464366065

By the way, if you set $x=z+h^2$ the curve reduces to the very simple Weierstrass form \begin{equation*} y^2=z^3-3h^2z-h^2(h^2+1) \end{equation*}

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  • $\begingroup$ Dear Prof Allan Mac Leod: $\endgroup$
    – mehdi baghalaghdam
    Commented Jan 4, 2017 at 14:03
  • $\begingroup$ I wolud like to thank Professor Allan Mac Leod, Known Mathematician, for so much help me. Thank you so much for your times..These results were gereat......sincerley yours.... mehdi baghalaghdam. $\endgroup$
    – mehdi baghalaghdam
    Commented Jan 4, 2017 at 14:13
  • $\begingroup$ I would like to thank Prof Allan Mac Leod for useful helps in my paper....thank you ..sincerely.yours $\endgroup$
    – mehdi baghalaghdam
    Commented Jan 4, 2017 at 14:24
  • $\begingroup$ Dear Prof Allan Mac leod.The solutions given for the x-coorditane s are the generators x-coorditanes for elliptic curves, Is it right? $\endgroup$
    – mehdi baghalaghdam
    Commented Jan 4, 2017 at 16:42
  • $\begingroup$ Dear Prof Allan Mac Leod. I wish to know how to get Denis Simon' ellrank code for caculating rank of elliptic curves with the coefficients large? Thank you for your giude. $\endgroup$
    – mehdi baghalaghdam
    Commented Jan 4, 2017 at 22:16

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