From: victor@ccr-p.ida.org (Victor S. Miller)
Newsgroups: sci.math.numberthy
Subject: An Elliptic Curve/Q of rank 23
Date: 16 Mar 98 16:14:56 GMT

I'd like to pass on the following announcement from Roland Martin and
William McMillen of NSA, about an elliptic curve of rank 23 over Q.
I've verified, using pari, that the determinant of the the gram matrix
formed from the height-pairing is, indeed, not 0 (in fact very far).

                Victor Miller -- moderator Number Theory Net

An Elliptic Curve over Q with Rank at least 23
By Roland Martin and William McMillen
June 1997

Using a construction similar to those in [2] and [4], we have
constructed an elliptic curve over Q with rank at least 23. Aso see
[1], [3], and [5].

Theorem. The elliptic curve

y2 + xy + y = x3 - 19252966408674012828065964616418441723 x
                   + 32685500727716376257923347071452044295907443056345614006

over Q has rank at least 23. The following points P1,...,P23 are
independent points on the curve.

P1 = (16902136044621724275584661392595/119224493521,
     -69455519784971993679807552308609739430858248812/41166906143372569)
P2 = (6647882272466103821634772046571/30891226081,
     -17137023844710987140049387309945953892946213544/5429411004770479)
P3 = (1277229332035649706664846727592/2961427561,
      1443380843339272397458721030742392016696304046/161157926442059)
P4 = (1754834771916476982132090651/369369961,
      49412130720987886904443301152758710388796/7098921280459)
P5 = (902743031953703698667092998/307406089,
      6538434104009303265024749952830709029353/5389750958437)
P6 = (103579510135061476534950819/45091225,
      230697883363551870088729854504374414548/302787575875)
P7 = (31762044569407766003397375255/14054813809,
      1411381089291349753164768808558921002947204/1666240341498377)
P8 = (29436984213667648723395/17956,
      5657335012046240705357319452802233/2406104)
P9 = (1127027270330215920, 3523978127407100674110377602)
P10 = (686464244502821899711515/139129,
      -394563651945882403580468873435105816/51895117)
P11 = (11962675953816366561795/1369,
      -1167962768316319592876571517317044/50653)
P12 = (30520680805402695175757355/3345241,
      -151915114589061403100759698106532333112/6118445789)
P13 = (11449775538050756019357635316/967521025,
      -1150775031908416918955115365651634494501651/30094741482625)
P14 = (4969418243982621661795591770/1285294201,
       184569435055535326363669745422918052707327/46079082400051)
P15 = (480465113537612829840777315/160801,
      -10531550647702714814852169224678207441368/64481201)
P16 = (97907154284679777917982542166035/57601436172721,
       964874722537391293613786748114488474882993683572/437169613520472565481)
P17 = (249989354826313432718977195/4397409,
       3941156276776007263792745630379334937996/9221366673)
P18 = (54840074123086507808388135/3996001,
       387496978790653709721061294119215460988/7988005999)
P19 = (9690141319063801580189469420/87590881,
      -953144078079942906360903670036536669593542/819763055279)
P20 = (882142442406602738753880/76729,
      -775394556680837651292166377698874734/21253933)
P21 = (2812175950395226936581984/24025,
       4712624271973109965160039085789391367/3723875)
P22 = (7126269737101017406079752337071371/2947180538019481,
       83015454575998684006900205726968222686505350799684/
       159996363164349841378621)
P23 = (2143448685801212487450/841, 10099849221189668277354753748208/24389)

References

[1] Stefane Fermigier, An elliptic curve over Q of rank 22, to appear.

[2] J.-F. Mestre, Courbes elliptiques de rang 11 sur
Q(T). C. R. Acad. Sci. Paris, 313, ser. 1, 1991, p. 139-142.

[3] J.-F. Mestre,  Courbes elliptiques de rang 12 sur Q(T). ibid.,
313, ser. 1, 1991, p. 171-174.

[4] Koh-ichi Nagao, An example of elliptic curve over Q with rank  20,
Proc. Japan Acad., Ser. A, 69, No. 8, 1993, p. 291-293.

[5] Koh-ichi Nagao and Tomonori Kouya, An example of elliptic curve
over Q with rank  21, Proc, Japan Acad. Ser. A, 70, No. 4, 1994,
p. 104-105.