[Just a brief announcement, before we leave 2005. Actually an addendum
to our March post to NMBRTHRY.]

The cardinality of the curve E: Y^2 = X^3+4589*X+91128 
modulo (p=10^2004+4863) is p+1-t with

t=-145 74736621811672498083077358121717122155122186271605
       53980454232680076248744978623581194348521775049050
       37580978920502744864569999307190135779172711605580
       08909039210574477242791187309790267838630940351206
       81949467168481146028118277485482639925461925715085
       12016725683021884503690112509899054251851329358179
       72004038466598937197623489340897109986151370825034
       82598338885929150898255092196008854686648504068502
       44734518954016827799071558956366047031592844309861
       51330882520924380524943393549994615049124552490740
       13097757787322381090209036127869762638189268507040
       13248163666196322527673693652953065601561120962652
       28857922912417022547576236670596959627810743700965
       97469144590037734199525927467178676517860191437906
       60864816823332766224332418777339131080592388580826
       61364334513712208232938769646296040131535716890875
       83454492581869100270919621948234421776518765028157
       13612281407057923131526798495202623953640032994519
       46592856265426556409782965032638341741834404083835
       52835836543869391164614140827319838307446969929268

As can be seen from the data file [2], some effort was put in computing large
degree isogenies, in order to push the modular equations program
further, and to test faster algorithms for computing the
eigenfactor. Several papers are in preparation describing the
improvements.

Here are the timings on an AMD 64 Processor 3400+ (2.4GHz), with our NTL
implementation, for some large numbers (excluding the time for
computing modular equations):

what  | 500dd | 1000dd | 1500dd | 1600dd | 1700dd | 2005dd
-----------------------------------------------------------
X^p   |  6h   | 134h   |  35d   |   46d  |  60d   | 133d
Total | 10h   | 180h   |  77d   |   63d  |  80d   | 195d

Happy new year 2006,

	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/d2005x.t

[3] http://listserv.nodak.edu/cgi-bin/wa.exe?A2=ind0503&L=nmbrthry&T=0&F=&S=&P=324