From fermigie@mathp7.jussieu.frMon Jul  8 09:02:31 1996
Date: Sun, 7 Jul 1996 21:00:48 -0400
From: Stefane fermigier <fermigie@mathp7.jussieu.fr>
To: Multiple recipients of list NMBRTHRY <NMBRTHRY@listserv.nodak.edu>
Subject: An elliptic curve of rank >= 22 over Q.

                   An elliptic curve with rank >= 22 over Q.

                         S. Fermigier, 19 May 1996.


Let

E : y^2 + xy + y = x^3 - 940299517776391362903023121165864 x
 + 10707363070719743033425295515449274534651125011362

(or in GP notation,
E = [1, 0, 1, -940299517776391362903023121165864,
10707363070719743033425295515449274534651125011362]
).

Let P_1,...,P_22 the point in this list :
pts = [
[32741153161482344264/3025, -223089674587110979578532169697/166375],
[215521674613198983365/24649, -6872949155061353554235704378947/3869893],
[637312541911044643/81, -1420356190129296832193564087/729],
[-11906250919327880080/361, -16580788535875788634285886853/6859],
[-136152345735493381/4, -14482270545045735913281693/8],
[-27830298157016213012252/7134241, 72099692861364392796183359497454267/190555577
11],
[4127671322151440, 2626107692045613116291646],
[6175679781777296, 2266254335997033124678449],
[12047255022287093, 1061993236525943920980477],
[416685837455186583191/32761, 5321268222786709669160311587369/5929741],
[149915813139075767108024/10220809, 8704326838108646949177663157917117/326759263
73],
[58759417448623559/4, 2030968553150713398654657/8],
[237195157887349854919517/16024009, -11477798111611307979707215505421441/6414410
8027],
[9568474434078537574436/687241, 319520556343135681977874272805086/569722789],
[1725892668710258675291/177241, 117378050663464845770966453025039/74618461],
[-35277008506980340471/1024, 48766027143946934186731674507/32768],
[-2752742763529705669/121, 6000532252185982381233585699/1331],
[-18552633109178014, -4665466215824339436717966],
[-113251707338691187737649969/3304065361, 31015252789483147082000987237322934173
9/189920981015641],
[-7572001778163591251/729, -86590661426506799357663502953/19683],
[-380526048554032285152211/11242609, 73081235744931307684790623068490233/3769646
7977],
[-1503889497722021588110681/42784681, -160705885170116750151534640924719585/2798
54598421]
]

Then, according to GP/PARI, P_1,...,P_22 are independent in E(Q), so E
is an exemple of an elliptic curve defined over Q with rank >= 22.

This example was found using methods similar to those described in

Koh-ichi Nagao, An example of elliptic curve over $\Q$ with rank $\geq 20$,
Proc. Japan Acad., Ser. A, 69, No.~8, 1993, p.~291--293;

Koh-ichi Nagao et Tomonori Kouya, An example of elliptic curve over $\Q$
with rank $> 21$, Proc. Japan Acad. Ser. A, 70, No.~4, 1994, p.~104--105;

or

Stefane Fermigier, Un exemple de courbe elliptique definie sur $\Q$ de
rang $\geq 19$, C. R. Acad. Sci. Paris, t.~315, s'erie~I, 1992, p.~719--722;

but starting from an infinite family of curves of with >= 13 over Q(t)
discovered by Mestre. For the computations, I used a few tens of
(Sparc) workstations for about 1 week.

Since Mestre has recently found an infinite family of curves of rank >= 14
over Q(t), this could even lead to better ranks.


Best regards to all,

        S. Fermigier

PS : See

http://www.eleves.ens.fr:8080/home/fermigie/dvi/rang22.dvi

for a DVI version of this announce. See also

http://www.eleves.ens.fr:8080/home/fermigie/elliptic.html

for a collection of links to articles about elliptic curves (and related
topics). Feel free to E-mail me new references or suggestions.