The cardinality of the curve E: Y^2 = X^3+4589*X+91128 
modulo (p=10^1499+2001) is p+1-t with

t=-33869807502019604962429491732215909675312109702927
   38256660675992888798047358665277537955891121925282
   04312584535186270889492848925526604477183857980923
   59477665742381275305428214011551504715014013677567
   26419740446725166854919715964302583573538365053023
   79848948669059223574594199242770519859918901305463
   56315123760066014938693077158287600028457826986887
   95318782320991052056671119858045473657308508388388
   25841475058655269609931826424414231889685224914589
   24896470910210262180251945525555489587430454422785
   21589418839490929443491392338326471404212463987422
   20667685409798264040019116263291700594455504306667
   01089059814033633752435141603224931835752822032445
   78882193392994255212234498834012240698002236270116
   43294029774897132053258354006762051244824374486430

Why perform such a computation? It was tempting to measure the impact
of technology on the feasability of such a computation using the
classical SEA algorithm. After all, the preceding record of 500
decimal digits was done 10 years ago [1]. 

Rewriting the program in NTL, we get the following equivalent
cumulated CPU time on an AMD 64 Processor 3400+ (2.4GHz):

 500dd	   10 hours
1000dd	  180 hours
1500dd   6400 hours

These timings do not take into account the time needed to compute the
relevant modular equations. However, during the computations for the
1000dd, it appeared that computing these equations was more and more
costly, the reason being that the classical O(ell^4) method using
series expansions and chinese primes was just too slow. Fortunately,
one of us (AE) came in with a new faster method. Briefly, one looks
for the modular equation of some function built from Hecke operators
T_r, as described for instance in Mueller's thesis. The complexity of
AE's method is then O(r ell^3), which is much faster than O(ell^4), if
r is small, which happens very frequently. More details are given in
AE's forthcoming paper. We content ourselves with just one numerical
example. For the prime ell = 2713, it took 41 hours to compute the
modular equation. This should be compared to the 3 hours required to
compute X^p mod Phi_2713 and to the 3 more for performing the
remaining computations.

As can be seen from the data file [2], some effort was put in finding
t mod ell for Atkin primes that are sometimes rather large, using the
techniques of Dewaghe.

	AEnge, PGaudry and FMorain

[1] http://listserv.nodak.edu/cgi-bin/wa.exe?A2=ind9501&L=nmbrthry&P=R503

[2] http://www.lix.polytechnique.fr/Labo/Francois.Morain/SEA/d1500x.t