From: tom@womack.net (Tom Womack) Subject: Re: Growth of the conductor of elliptic curves Date: 21 Mar 00 17:35:22 GMT Newsgroups: sci.math.numberthy Summary: [missing] > Let N(r) denote the smallest conductor of elliptic curves with rank r. > >From the tables of Cremona one finds: > > r N(r) > ------- > 0 11 > 1 37 > 2 389 > 3 5077 > > Now I would like to know: > (1) Has this been extended in general? I don't think so, since doing an exhaustive search over conductors would take an awfully long time even if you were only going to N=234446 (which is the conductor of the curve $y^2 + xy = x^3 - x^2 - 79x + 289$ of rank 4 and lowest known conductor). The computations for distinct prime conductor are independent, so a distributed approach isn't completely out of the question, but I fear that, for $N=234446$, even the linear-algebra-on-Manin-symbols step on its own would be a fairly challenging problem. > (2) Are there known bounds on N(r)? I'll boast that N(6) <= 7208218544 (y^2 = x^3 - 9907x + 306370), which I found this morning after running MWRANK and Cremona's conductor-finding code over a large set of curves with lots of integer points which I'd found by running a sieve method overnight. I've put up a garishly polychrome Web page at http://tom.womack.net/conductors.html to summarise what I've been able to find out so far; if anyone can contribute higher ranks or lower conductors, tell the list and I'll summarise. Tom ============================================================================== From: tom@womack.net (Tom Womack) Subject: Mordell curves of reasonably high rank and small K Date: 1 May 00 17:56:00 GMT Newsgroups: sci.math.numberthy Summary: [missing] http://tom.womack.net/maths/mordellc.htm has the results of my search for Mordell curves of rank 7 and 8. The minimum |K| I've found for rank 7 is K=-78051887, for rank 8 is K=5703943325. A rank-7 curve of smaller |K| would have to have fewer than 23 integer points with X co-ordinate less than 2^19. Tom ============================================================================== From: victor@idaccr.org (Victor S. Miller) Subject: An Elliptic Curve over Q with Rank at least 24 Date: 2 May 00 17:30:32 GMT Newsgroups: sci.math.numberthy Summary: [missing] I'm passing on this announcement from Roland Martin and William McMillen. I've verified (using gp) that the regulator of this set of points is very far from 0 [ed.] An Elliptic Curve over Q with Rank at least 24 By Roland Martin and William McMillen January 2000 Using a construction similar to those in [3] and [5], we have constructed an elliptic curve over Q with rank at least 24. Also see [1], [2], [4], and [6]. Theorem. The elliptic curve y2 + xy + y = x3 - 120039822036992245303534619191166796374x + 504224992484910670010801799168082726759443756222911415116 over Q has rank at least 24. The following points P1,...,P24 are independent points on the curve. P1 = (2005024558054813068, -16480371588343085108234888252) P2 = (-4690836759490453344, -31049883525785801514744524804) P3 = (4700156326649806635, -6622116250158424945781859743) P4 = (6785546256295273860, -1456180928830978521107520473) P5 = (6823803569166584943, -1685950735477175947351774817) P6 = (7788809602110240789, -6462981622972389783453855713) P7 = (27385442304350994620556, 4531892554281655472841805111276996) P8 = (54284682060285253719/4, -296608788157989016192182090427/8) P9 = (-94200235260395075139/25, -3756324603619419619213452459781/125) P10 = (-3463661055331841724647/576, -439033541391867690041114047287793/13824) P11 = (-6684065934033506970637/676, -473072253066190669804172657192457/17576) P12 = (-956077386192640344198/2209, -2448326762443096987265907469107661/103823) P13 = (-27067471797013364392578/2809, -4120976168445115434193886851218259/148877) P14 = (-25538866857137199063309/3721, -7194962289937471269967128729589169/226981) P15 = (-1026325011760259051894331/108241, -1000895294067489857736110963003267773/35611289) P16 = (9351361230729481250627334/1366561, -2869749605748635777475372339306204832/1597509809) P17 = (10100878635879432897339615/1423249, -5304965776276966451066900941489387801/1697936057) P18 = (11499655868211022625340735/17522596, -1513435763341541188265230241426826478043/73349586856) P19 = (110352253665081002517811734/21353641, -461706833308406671405570254542647784288/98675175061) P20 = (414280096426033094143668538257/285204544, 266642138924791310663963499787603019833872421/4816534339072) P21 = (36101712290699828042930087436/4098432361, -2995258855766764520463389153587111670142292/262377541318859) P22 = (45442463408503524215460183165/5424617104, -3716041581470144108721590695554670156388869/399533898943808) P23 = (983886013344700707678587482584/141566320009, -126615818387717930449161625960397605741940953/53264752602346277) P24 = (1124614335716851053281176544216033/152487126016, -37714203831317877163580088877209977295481388540127/59545612760743936) References [1] Stéfane Fermigier, An elliptic curve over Q of rank 22, Internet posting. [2] Martin and McMillen, An Elliptic Curve over Q with Rank at least 23, Internet posting. [3] J.-F. Mestre, Courbes elliptiques de rang 11 sur Q(T). C. R. Acad. Sci. Paris, 313, ser. 1, 1991, p. 139-142. [4] J.-F. Mestre, Courbes elliptiques de rang 12 sur Q(T). ibid., 313, ser. 1, 1991, p. 171-174. [5] 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. [6] 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. -- Victor S. Miller | " ... Meanwhile, those of us who can compute can hardly victor@idaccr.org | be expected to keep writing papers saying 'I can do the CCR, Princeton, NJ | following useless calculation in 2 seconds', and indeed 08540 USA | what editor would publish them?" -- Oliver Atkin ============================================================================== From: john.cremona@maths.nottingham.ac.uk (John Cremona) Subject: Re: An Elliptic Curve over Q with Rank at least 24 Date: 3 May 00 11:03:00 GMT Newsgroups: sci.math.numberthy Victor, To correctly verify with gp that those 24 points are independent, you should work with the minimal model since the height pairing algorithm used in gp requires this. In fact the model you posted is minimal, since one can easily check that gcd(c4,delta)=1. The discriminant has the rather large composite factor 121186187324634274722191791046317677651078393842196124697808418768214982402918637227743 which I have not yet factored -- if anyone out there can do so please let me know the factors! I also checked independently (!) that the points were independent, by showing that their images under the map E(Q) -> E(Q)/2E(Q) -> \sum_{i=1}^{21}E(Q_p_i)/2E(Q_p_i) =~= (F_2)^25 are independent over F_2, where p_1,...p_21 are the primes 7,19,23,37,43,47,53,59,61,67,73,79,83,89,97,101,109,137,163,173,179. (For p=67,79,89,97 the local quotient has dimension 2; omitting p=179 does not work.) This method has the advantage of requiring only the evaluation of Legendre symbols and linear algebra over F_2, rather than computing the 24x24 height pairing matrix to sufficient precision to be sure that it has nonzero determinant. See http://www.maths.nottingham.ac.uk/personal/jec/papers/filter.ps (or filter.ps.gz) for a description of this method, which is also described by Silverman in @Article{Silverman-xedni, author = "J. H. Silverman", title = {The {xedni} Calculus and the elliptic curve discrete logarithm problem}, journal = {Design, Codes, and Cryptography}, year = {2000}, volume = {10}, pages = {5--40}, } John Cremona ============================================================================== From: "Troy Kessler" Subject: Re: Quadratic Fields With 3-rank 5 Date: Sat, 2 Sep 2000 10:40:27 -0700 To: "Dave Rusin" Summary: [missing] If the d<0 is even then y^2=x^3+d should have high rank. If d<0 is odd then y^2=x^3+16d should have high rank. -----Original Message----- From: Dave Rusin Newsgroups: sci.math.numberthy To: kesslert@surfree.com Date: Friday, September 01, 2000 8:52 PM Subject: Re: Quadratic Fields With 3-rank 5 > >Could you remind me how we get elliptic curves of high rank from these >quadratic fields? Thanks >dave >