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.