From: duje@math.hr (Andrej Dujella)
Subject: torsion = Z2, rank = 15
Date: 26 Apr 01 16:00:07 GMT
Newsgroups: sci.math.numberthy
Summary: Sample rational elliptic curve of high rank and torsion Z2

I found an elliptic curve over Q with torsion group Z2
and rank = 15. This improves the previous record rank =14
(Fermigier (1996), Kulesz - Stahlke (2000)). The curve
is found by Fermigier's method and the rank is computed
using APECS and RATPOINTS.

The curve is:

y^2 + xy + y = x^3 + 34318214642441646362435632562579908747*x
               + 3184376895814127197244886284686214848599453811643486936756

        Torsion points:

O, [-55741267008740887705/4, 55741267008740887701/8]

        Independent points of infinite order:

P1 = [-5955399047526089895, -52619192486073556789679851928]
P2 = [4996845231479851005, 58996807068911558580932032822]
P3 = [-13155206566829859045, -21360729170232157127198638028]
P4 = [5982316535030750730, 60031451151089353115173694947]
P5 = [3520990345094746477605, 208928587539794577855401843236822]
P6 = [-149780582516304339030/289, -276460561394858085500066301116014/4913]
P7 = [5537764707520796477505/256, 485495189997228202630782626087287/4096]
P8 = [10379720384859947873670/529, 1299762167535132813050160826170149/12167]
P9 = [3607902715536254330407755/36481,
     6876286874413935169019164791798028712/6967871]
P10 = [575451914344737045120/18769,
      145126149873708232941281933935318916/2571353]
P11 = [-635343181823720560310/81, -35955391603910411255538526302302/729]
P12 = [385275433846822770303250/9, 239142619084196570639847351097336094/27]
P13 = [11630065797764473356485380/349281,
      41921913401808378926412528792277717237/206425071]
P14 = [2130862087205394565011555/206116,
      6377017660319811010371576798520557433/93576664]
P15 = [361886218060793196368152192851/82159049956,
      1377397812507014979022748799194701395570639671/23549577125088104]


I also found two (new) curves with torsion group Z2 and rank =14:


y^2 + xy = x^3 - 7372911492530406268416156245*x
           + 243594391906613268628507257344604677608161


y^2 + xy = x^3 - 9468594009199910821775026235*x
           + 391555244342706113038825183927385153742225


        Andrej Dujella
        Zagreb, CROATIA

        http://www.math.hr/~duje/
==============================================================================

From: John.Cremona@nottingham.ac.uk (John Cremona)
Subject: Re: torsion = Z2, rank = 15
Date: 1 May 01 15:56:22 GMT
Newsgroups: sci.math.numberthy

A couple of remarks concerning Andrej Dujella's posting of an elliptic
curve over Q with 2-torsion and rank 15:

It was not quite clear from the posting that the curve had rank
exactly 15, or just at least 15.  However, a run of mwrank confirms
that the rank is exactly 15.  [This only takes a few seconds to
verify, but it is essential to use the version of mwrank which uses
multiprecision real arithmetic, otherwise the result is embarrassingly
wrong: the 2-torsion remains undetected and there is a catastrophic
overflow yielding a decidedly wrong result.]

Secondly, I noticed that the regulator of Dujella's 15 points differed
from  the regulator of the 15 points output by mwrank, in the ratio
4/9.  In fact there is a relation

3*P1 + 2*P5 -P8 -P9 + P10 -P11 -P13 + 2*P14 -P15 + 2*Q1 = T

between Dujella's points P1,...,P15, where

 Q1=[7007445993440694361372245/34969,
     18561410601719170080618893537906456866/6539203]

is the first point output by mwrank and T is the point of order 2.
Replacing P15 by Q1 enlarges the group generated by index 2.  I have
checked that the index of the group these points generate is not
divisible by 2, 3, 5, 7 or 11 and I expect that P1,...,P14,Q1,T do
generate the full Mordell-Weil group.

As far as I know this curve is the first known to have 2-torsion and
exact rank 15.  Fermigier produced several curves which were
candidates for this status, and which certainly have 2-Selmer rank 15,
but I have not been able to verify this, since mwrank gets bogged down
in the second descent; the advantage of Dujella's curve is that all 15
generators are found without the need for a second descent.

John Cremona
--
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