Karim BELABAS on Thu, 4 May 2000 13:25:47 +0200 (MET DST) |
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Re: large rank and torsion group |
[Igor:] > On Wed, May 03, 2000 at 03:07:30PM -0400, Andrej Dujella wrote: > > I am searching for Diophantine triples (i.e. rationals a,b,c > > such that ab+1, ac+1 and bc+1 are all perfect squares) > > with the property that the corresponding elliptic curve > > > > y^2=(ax+1)(bx+1)(cx+1) > > > > has large rank and/or large torsion group. > > > > I found the following (interesting) examples: > > > > {a,b,c}={217/69, 355/69, 368851912/328509} > > torsion group = Z/2Z * Z/2Z, rank = 8 > > > > {a,b,c}={119/60, 3398759/864000, -864000/3398759} > > torsion group = Z/2Z * Z/4Z, rank = 5 > > > > {a,b,c}={145/408, -408/145, -145439/59160} > > torsion group = Z/2Z * Z/8Z, rank = 3 > > > > I would like to know what are current records for the ranks of > > elliptic curves over Q with torsion groups Z/2Z * Z/mZ for m=2,4,8. > > > > > > Andrej Dujella > > duje@math.hr > > I couldn't resist checking the above with gp, and guess what, > gp shows torsion group = Z/2Z * Z/2Z, for the 2nd curve, which is not > what Andrej claims. Remembering that there was a problem with the old > implementation of elltors(), I tend to suspect a precision problem here. > > Any comments? Which GP version, architecture ? And finally, on which precise input ? Given the functions E(a,b,c) = [0, (b + c)*a + c*b, 0, c*b*a^2 + (c*b^2 + c^2*b)*a, c^2*b^2*a^2] [rationaly equivalent to the original one] Tors(a,b,c) = elltors(ellinit(E(a,b,c)), 1) test() = Tors(119/60, 3398759/864000, -864000/3398759) In current CVS + 32bit UltraSparc, I get the following results : (13:08) gp > \p28 realprecision = 28 significant digits (13:08) gp > test *** precision too low in torselldoud. after increasing the precision, I get the correct result: (13:08) gp > \p100 realprecision = 105 significant digits (100 digits displayed) (13:08) gp > test / / / 3758759 11354445338831 \ / 14400 \ \ \ %1 = | 8 (4 2) | | ------- -------------- | | ----- 0 | | | \ \ \ 1216800 1480602240000 / \ 28561 / / / (with prettyprinter off, this reads: [8, [4, 2], [[3758759/1216800, 11354445338831/1480602240000], [14400/28561, 0]]] ) On my machine, all three examples react in the same way, yielding the expected result at \p100. Cheers, Karim. P.S: The old Nagell-Lutz implementation is unable to give the result in decent time (stopped after one hour, needed 160MB of stack). __ Karim Belabas email: Karim.Belabas@math.u-psud.fr Dep. de Mathematiques, Bat. 425 Universite Paris-Sud Tel: (00 33) 1 69 15 57 48 F-91405 Orsay (France) Fax: (00 33) 1 69 15 60 19 -- PARI/GP Home Page: http://www.parigp-home.de/