From:  James Propp[SMTP:propp@math.wisc.edu]
Subject:  the MW group of the taxicab cubic
Date:  Monday, November 08, 1999 7:50 PM
To:  NMBRTHRY@LISTSERV.NODAK.EDU

I think most of us know the story about Ramanujan and Hardy and the taxi
and the equation x^3+y^3=1729, and the fact (12,1) and (10,9) are both on
the curve.  But what doesn't seem to have entered the peripheral folklore
of this problem is the answer to the question: do these points generate
the Mordell-Weil group of this particular elliptic curve?

Noam Elkies tells me, courtesy of the program mwrank, that the rank is
indeed 2.  I gather that with more work, someone a bit more conversant
than I am with elliptic curves could get mwrank to actually determine
whether or not these two points generate the full MW group.

Thanks,

Jim Propp
==============================================================================

From: rlward1@orion.ncsc.mil (Robert L. Ward)
Subject: Re: the MW group of the taxicab cubic
Date: 10 Nov 99 16:16:22 GMT
Newsgroups: sci.math.numberthy

I used MAGMA to answer this question.

The minimal model of this curve is:

   Y^2 + Y = X^3 - 20178727,

its discriminant is -3^9*7^4*13^4*19^4, and its conductor is
80714907 = 3^3*7^2*13^2*19^2.

The transformation to get this equation from x^3 + y^3 = 1729 is

   X = 5187/(y+x),
   Y = (7780*y-7781*x)/(x+y),
   x = (7780-Y)/(3*X),
   y = (7781+Y)/(3*X).

I issued the following commands:

   E := EllipticCurve([0,1,0,0,-20178727]);
   MW,h := MordellWeilGroup(E);
   P := h(MW.1);
   Q := h(MW.2);

Then P turned out to be (273,409) and Q to be (399,6583), which correspond
to the Ramanujan solutions of the original equations.  Thus MAGMA asserts that
generators of the Mordell-Weil group of the curve are indeed the Ramanujan
points.

Robert L. Ward

----------
[previous article quoted --djr]
==============================================================================
From: twomack@chiark.greenend.org.uk (Tom Womack)
Subject: Ranks of Fermat curves
Date: 16 Nov 99 19:46:10 GMT
Newsgroups: sci.math.numberthy
Keywords: curves defined by  x^3+y^3=N

For an MMath dissertation, I computed the two smallest values of N such
that x^3+y^3=N had precisely 5 solutions with x,y both positive - these
are N_1=48988659276962496 and N_2=490593422681271000.

I think I'm the third independent discoverer of N_1 (D J Bernstein has it
buried somewhere on his home page, and
http://www.research.att.com/~njas/sequences/JIS/wilson10.html
is an article by D W Wilson describing how he found both it and N_2); I'm
sure it ought to be more widely known. During the dissertation, I also
found the upper bound for N(6 solutions) given in Dr Wilson's paper.

I've just started a PhD at Nottingham under John Cremona, who gave me
access to some clever software for computing EC ranks; his findinf program
found seven independent points on the minimal Weierstrass model for N_1,
and at least five for N_2. Apecs gives an upper bound of 7 for the rank of
the curve for N_1 and of 8 for N_2, so we have rank(N_1) = 7 and know
rather little about N_2.

What's the largest known rank for this class of curve? Elkies says that
rank 12 is known for sextic twists, but that this is a cubic twist and
that he doesn't know the record in that case.

Tom
==============================================================================

From: jbuddenh@texas.net (James Buddenhagen)
Subject: Re: Ranks of Fermat curves
Date: 18 Nov 99 14:18:12 GMT
Newsgroups: sci.math.numberthy

I don't know the answer to your questions, but one of my favorite
elliptic curves is the Ramanujan curve (if I may coin a name):  U^3 +
(U+21)^3 = 8^3 + V^3.  It is certainly related to your family, so
possibly might be of interest.  It is noteworthy because in the form
given above it is the 'simplest' rank 7 curve that I know of.  I found
it playing with APECS in 1993.
--Jim Buddenhagen

Tom Womack wrote:
[previous article quoted --djr]
==============================================================================
[The curve  x^3 + y^3 = N  is transformed to the normal form 
 Y^2 = X^3 - (432 N^2)  under the transformation
 X=12*N/(x+y), Y=36*N*(x-y)/(x+y)  --djr]